Classical Groups
for Physicists


Brian G. Wybourne


Department of Physics
University of Canterbury
Christchurch, New Zealand.


A WILEY-INTERSCIENCE PUBLICATION


JOHN WILEY & SONS, New York · London . Sydney · Toronto




Copyright @ 1974 by John Wiley & Sons, Inc.
All rights reserved. Published simultaneously in Canada.
No part of this book may be reproduced by any means, nor
transmitted, nor translated into a machine language with-
out the written permission of the publisher.
Library of Congress Cataloging in Publication Data:
Wyboume, Brian G
Classical groups for physicists.
"A Wiley-Interscience publication."
Includes bibliographical references.
1. Lie groups. 2. Lie algebras. I. Title.
QCI74.5.W9 1974 512'.55
ISBN 0-471-96505-7
73-17363
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1

To J.
R.
L.
M.
A. I.
B.

Preface
The usefulness of a knowledge of Lie groups and algebras to physicists can
scarcely be disputed: their application to physical problems has pervaded
most branches of modern physics. In this text I have aimed at giving an
introduction to some of the principal ideas in the theory and application of
continuous groups, assuming that the reader has attended the normal
undergraduate courses in quantum mechanics and mathematics. A nod-
ding acquaintance with some of the elementary notions of finite groups is
also assumed.
The approach adopted is along the traditional path of Cartan and Weyl,
with some attention to the modifications and extensions introduced by
Dynkin. The emphasis is on exposition accompanied by illustrative ex-
amples and exercises. A minimum of mathematical tools is required, and
proofs of a number of theorems are either briefly sketched or omitted-
adequate references being given to enable the interested reader to pursue
the subject further.
Having taught a wide range of students as well as groups of interested
physicists and chemists, I have found that many are bewildered by the
diversity and immensity of the subject, and need to be guided through a
preliminary course prior to embarking on a detailed exploration that
makes full use of the whole range of mathematical tools. I have decided to
select certain topics for discussion and am all too well aware of the fact
that many important topics have been either omitted entirely or only
briefly discussed. In the final three chapters I have endeavored to present
in some detail three case studies of the application of group theory to
problems in physics, with the aim of bringing together the various tools
developed earlier.
This course has developed over a period of years commencing in 1964 as
a course to a group of chemists at Argonne National Laboratory who had
a keen, if somewhat peripheral, interest in the subject. I am grateful to the
Vll

Vll1 PREFACE
many students who have offered helpful criticisms.
This book could not have been completed without frequent periods of
leave granted by the Council of the University of Canterbury and the
forbearance of my family, to whom it is dedicated. As usual I am indebted
to my secretary, Mrs. M. A. Sewell, who has patiently, and with unusual
skill, gone through more than one draft of the manuscript.
BRIAN G. WYBOURNE
University of the Pacific
Stockton, California
July 1973

Contents
1 Introduction
1
2 Symmetry and Quantum Numbers
2.1 Symmetry and Atomic Quantum Numbers, 13
2.2 Hierarchies of Symmetry, 
3
3 Groups Composed of Regular Matrices
7
3.1 The Group Postulates, 7
3.2 Regular Matrix Groups, 8
3.3 Matrix Properties, 10
3.4 Continuous Matrix Groups, 10
3.5 Matrix Exponential Functions, 14
4 Local Properties of Lie Groups
4.1 Parameterization of the Group Elements, 17
4.2 Connectivity, 18
4.3 The Beginning of Lie Groups, 19
4.4 Infinitesimal Group Generators, 20
4.5 The Two-Dimensional Rotation Group SO(2), 122
4.6 Infinitesimal Rotations, 23
4.7 General Infinitesimal Transformations, 23
4.8 Infinitesimal Operators of a Lie Group, 25
4.9 Examples of Infinitesimal Operators, 28
4.10 Structure Constants of Lie Groups, 31
4.11 Generation of Finite Group Elements, 33
4.12 Finite Transformations, 37
17
IX

X CONTENTS
5 Lie Groups and Lie Algebras
40
5.1 Lie Algebras, 40
5.2 Transformation of Basis, 41
5.3 Homomorphisms and Isomorphisms, 42
5.4 Automorphisms and Endomorphisms, 43
5.5 Lie Algebras and Subalgebras, 43
5.6 Ideals and Proper Ideals, 44
5.7 Adjoint Representations of Lie Algebras, 44
5.8 Complex Extensions of Real Lie Algebras, 45
5.9 Simple and Semisimple Lie Algebras, .46
5.10 The Killing Form and Cartan's Criterion for: Semisimple
Lie' Algebras, 46
5.11 Example of SO( 4), 48
5.12 Example of E 2 , 49
5.13 Derivations of Lie Algebras, 50
5.14 Solvable Lie Algebras, 50
5.15 Nilpotent Lie Algebras, 51
5.16 Direct and Semidirect Sums, 52
5.17 Antisymmetric Tensors, 53
5.18 The Casimir Operators, 53
5.19 Generalizations of the Casimir Operators, 55
5.20 Compact and Noncompact Lie Algebras, 55
5.21 Lie Groups and Lie Algebras, 55
6 Root Vectors and the Classical Lie Algebras
6.1 Introduction, 57
6.2 Standard Form of the Semisimple Lie Groups, 57
6.3 Properties of Roots, 59
6.4 Symmetry of the Roots, 60
6.5 The Standard Form Obtained, 60
6.6 Further Theorems Concerning Roots, 62
6.7 Cartan-Weyl Normalization, 65
6.8 Graphical Representation of Root Vectors, 65
6.9 Lie Algebras of Rank 2, 67
6.10 Lie Algebras of Rank 1>2, 70
6.11 The Exceptional Lie Algebras, 71
57
7 Simple Roots and Dynkin Diagrams
7.1 Simple Roots, 73
73

CONTENTS Xl
7.2 Examples of B 2 and B 3 , 75
7.3 Dynkin Diagrams, 76
7.4 The Cartan Matrix, 177
7.5 Examples of Cart an Matrices, 79
7.6 The Cartan Matrix and the Enumeration of Roots, 83
7.7 Application to G 2 , 83
7.8 Construction of Some Simple Lie Algebras, 84
8 The Chevalley Basis
8.1 Co-Weights and the Chevalley Basis, 87
8.2 Phases in the Chevalley Basis, 89
8.3 The Algebra ,su (3) in the Chevalley Basis, 90
87
9 Representations of Lie Groups
and Lie Algebras
9.1 Group Representations, 92
9.2 Real and Complex Representations, 93
9.3 Contragredient Representations, 94
9.4 Adjoint Representations, 94
9.5 Unitary and Nonunitary Representations, 95
92
10 Weights and the Labeling
of Irreducible Representations
97
10.1 Weights and Weight Spaces, 97
10.2 Theorems Concerning Weights, 99
10.3 The Weyl Reflection Group, too
10.4 Weights and the Classification of Irreducible
Representations, 101
10.5 Computation of the Complete Set of Weights, 102
10.6 Examples of Computations of Weights, 105
11 Kronecker Products
109
11.1 Definition, 109
11.2 Kronecker Product of Representations, 110
11.3 The Weight Space for Kronecker Products, 110
11.4 Decomposition of the Kronecker Product, I,ll

XU CONTENTS
12 Representations, Weights, and Labeling
12.1 Basic Representations, 113
12.2 Kronecker Powers, 114
12.3 Elementary Representations, 116
12.4 Weights of Elementary Representations, 1118
12.5 Spin or Representations and the Groups Bn and Dn' 121
12.6 Labeling of Irreducible Representations, 123
12.7 A Matter of Notation, 126
113
13 The Exceptional Groups 127
13.1 Basic Representations of the Exceptional Groups, 127
13.2 Labeling of Representations for the
Exceptional Groups, 130
14 Dimensions of Irreducible Representations 133
14.1 Scalar Products of Basic Weights, 133
14.2 Dimensions of Irreducible Representations, 135
15 The Casimir Invariants
139
15.1 Eigenvalues of the Quadratic Casimir Operators, 139
15.2 Generalized Casimir Invariants, 140
15.3 Invariants for Nonsemisimple Lie Groups, 142
15.4 Casimir Operators for SO(3) and SO(2, 1), 143
16 Some Global Properties of Lie Groups 150
16.1 Topological Neighborhoods, 150
16.2 Topological Spaces, 151
16.3 Examples of Topological Spaces, 152
16.4 Homeomorphisms, 153
16.5 Products of Topological Spaces, 153
16.6 Hausdorff Spaces, 154
16.7 Metric Spaces, 154
16.8 Connected Spaces, l55
16.9 Compact Spaces, 155
16.10 Homotopic Paths, 156
16.11 Simply Connected and Multiply Connected Spaces, 158
16.12 The Fundamental Group, 159

CONTENTS XIll
16.13 Universal Covering Spaces, 11160
16.14 Topological Groups, 161
16.15 Products of Topological Groups, 163
16.16 Isomorphism of Topological Groups, 163
16.17 Topological Subgroups, 163
16.18 Invariant Topological Subgroups, 164
16.19 Coset Spaces and Factor Groups, 164
16.20 Homogeneous Spaces, J66
16.21 Manifolds and Lie Groups, 167
16.22 Real Simple Lie Groups and Lie Algebras, 167
16.23 Isomorphisms of Lie Groups and Lie Algebras, 173
16.24 Universal Covering Group, 174
17 Representations of Some Three-Parameter Lie Groups 176
17.1 The Three Parameter Lie Groups, 176
17.2 The Standard Form, 1\77
17.3 The Casimir Invariants, 178
17.4 The Elementary Representations, 179
17.5 Basis for the Spinor Representation, 180
17.6 Realization in Terms of Boson Operators, 181
17.7 Construction of Other Representations, 182
17.8 The Unitary Representations, t86
17.9 Matrix Elements of L 12 and L, 188
17.10 Finite Transformations, 190
17.11 Diagonalization of a Noncompact Generator, 195
17.12 Coupling Coefficients, 195
17;13 Specialization to SO(3), 198
17.14 Coupling Coefficients for SO(2, 1), 201
17.15 Coupling Coefficients and Analytic Continuation, 203
18 Some sufi, I)-Type Spectrum-Generating Algebras 207
18.1 Introduction, 207
18.2 A Realization of su(l, 1), 208
18.3 Discrete Eigenvalue Spectrum, 209
18.4 Continuous Eigenvalue Spectrum, 211
18.5 Three-Dimensional Isotropic Harmonic Oscillator, 2111
18.6 The Generalized Kepler Problem, 212
18.7 The Two-Dimensional Kepler Problem, 214
18.8 The Morse Potential, 215
18.9 Limitations of su(l, 1), 216

XIV CONTENTS
19 The Wigner-Eckart Theorem and Tensor Operators 218
19 .1 Introduction, 218
19.2 Some Notation, 219
19.3 Tensor Operators, 220
19.4 Tensor Operators in SO(3), 221
19.5 Tensor Operators for Semisimple Lie Groups, 222
19.6 Coupling Coefficients, 222
19.7 Coupling to the Identity Representation, 223
19.8 The Wigner-Eckart Theorem, 225
19.9 Selection Rules, 227
19.10 Application to SO(3), 228
19.11 Generalized Recoupling Coefficients, 230
19.12 Recoupling Coefficients for SO(3), 232
19.13 Coupling Coefficients for SO(4), 236
19.14 Racah's Factorization Lemma, 240
19.15 Isoscalar Factors, 242
19.16 Adjoint Tensor Operators, 243
19.17 Symmetry Properties of Coupling Coefficients, 245
19.18 Reciprocity and Isoscalar Factors, 248
19.19 Phase Conventions, 249
19.20 Simple Isoscalar Factors, 250
19.21 The Building-Up Principle, 251
19.22 Alternative Calculation of Isoscalar Factors, 261
19.23 Coupled Tensor Operators, 263
19.24 Coupled Tensor Operators for SO(3), 265
20 Case Study I:
The Isotropic Harmonic Oscillator 268
20.1 Introduction, 268
20.2 Second Quantization and the Harmonic Oscillator, 269
20.3 The Groups U(3) and SU(3), 270
20.4 Rotational Symmetry, 271
20.5 Some SU(3) Tensor Operators, 272
20.6 Reduced Matrix Elements, 275
20.7 The Quadratic Casimir Operator, 277
20.8 Ladder Operators in SU(3), 278
20.9 Some Further SU(3) Tensor Operators, 279
20.10 Commutation Relations, 279
20.11 A Larger Group for the Oscillator, 282
20.12 Subgroups of Sp(6,R), 283

CONTENTS XV
20.13 A Further Group for the Oscillator, 286
20.14 A Dynamical Group for the Oscillator, 286
20.15 Group Contractions and the Dynamical Group, 288
20.16 The N-Dimensional Isotropic Harmonic Oscillator, 290
20.17 Tensor Operators for the SO(2, 1) X SO(3) Subgroup, 290
20.18 Matrix Elements of Multiple Operators, 292
21 Case Study II:
The Hydrogen Atom 297
21.1 Introduction, 297
212 SO(4) and Hydrogen Energy Levels, 300
21.3 Spherical Tensors and SO(4), 302
21.4 Reduced Matrix Elements of A, 302
21.5 Ladder Operators in SO(4), 304
21.6 Boson Operators and SO(4), 306
21.7 Dynamical Group of the Hydrogen Atom, 307
21.8 The Casimir Operators, 311
21.9 The SO(4, 1) Subgroup, 312
21.10 Further Subgroups of SO (4, 2) 313
21.11 SO(4,2) Bases and Hydrogenic Atoms, 314
21.12 A Coordinate Realization of SO(4,2) 319
21.13 A Physical Realization of SO(4,2), 320
21.14 Tilted States of the Hydrogen Atom, 321
21.15 A Dilatation-Operator Realization of
SO 1(2, 1) X S02(2, 1), 323
21.16 The Electric Dipole Operator, 325
21.17 Galilean Boosts, 329
21.18 Lorentzian Boosts, ,331
21.19 Infinite-Component Wave Equations, 332
21.20 Example of Hydrogen, 337
21.21 A Finite-Dimensional Realization of SO(4,2), 340
21.22 Reformulation of the Dirac Theory of the Electron, 343
21.23 The Hydrogen Atom with Spin, 344
21.24 The Conformal Group and SO(4,2), 345
21.25 Concluding Remarks, 348
22 Case Study III:
Fermions and Shell Structure 349
22.1 Introduction, 34'9

XVI CONTENTS
22.2 States of a Fermion Shell, 350
22.3 The Supergroup, 352
22.4 Two Important Subgroups, 353
22.5 A Unitary Subgroup, 354
22.6 Tensor Operators and Annihilation and
Creation Operators, 355
22.7 A Coupled Tensor Operator, 356
22.8 Further Subgroups, 357
22.9 Classification for the j = 7 /2 Shell, 367
22.10 Seniority, 359
22.11 The Quasi-Spin Formalism, 360
22.12 Quasi-Spin Classification of States, 361
22.13 Quasi-Spin for Annihilation and Creation Operators, 362
22.14 Symmetry Classification of Operators, 363
22.15 Interaction of Particles in a Central Field, 366
A.I
A.2
A.3
A.4
A.5
A.6
A.7
A.8
A.9
A.IO
A.II
A.12
Appendix Schur Functions and Young Tableaux 372
Introduction, 372
S-Functions, 373
Outer S-Function Multiplication, 375
S-Function Division, 377
Inner Multiplication of S-Functions, 378
Characters of Groups as S-Functions, 378
Reduction of the Number of Parts of an S-Function,
Branching Rules, 380
Kronecker Products for Continuous Groups, 381
Outer Plethysm of S-Functions, 382
Inner Plethysm of S-Functions, 385
Machine Calculation of S-Function Properties, 386
379
References
387
Author Index
407
Subject Index
413

1
Introduction
Over the past two decades developments in theoretical physics have
tended to draw heavily on the properties of Lie algebras and Lie groups. 1,2
These developments have essentially been an outgrowth of the early
recognition by Yamanouchi,3 Weyl,4 Wigner,5,6 van der Waerden,7 and
Racah, 8 among many others, of the significance of symmetry transforma-
tions in describing physical phenomena. The early applications were
primarily concerned with the shell structure of atoms, 8,9 and later of
nuclei, 1 0-13 while more recently extensive applications have been made in
the field of elementary-particle physics. 14,15 However, applications have by
no means been limited to fundamental physics; for example, much of the
classical theory of special functions is now being treated using the methods
of Lie groupsl6-19 Hoffman 20 ,21 has discussed the application of Lie
groups to problems of visual perception, while Non0 22 ,23 has considered
their use in describing the stored-energy function of a hyperelastic
ma terial.
The usefulness of a knowledge of Lie groups and algebras to working
physicists is now beyond dispute. However, the student first approaching
the subject is bewildered by its immensity and by the many diverse
approaches possible. The expositor of the subject must choose whether to
expound with great mathematical rigor or to attempt an interpretation of
the subject, leaving questions of rigor to the professional mathematician.
Here we choose the latter approach and try to present some of the
principal ideas and illustrate them by examples and exercises.
1

2 INTRODUCTION
I assume that the reader has attended the normal undergraduate courses
on quantum mechanics and has a nodding acquaintance with the ele-
mentary properties of finite groups. 5,24
The development of the theory of Lie groups is characterized by a
number of distinctive approaches, each possessing advantages and disad-
vantages to the student. There is the approach formulated by Schur 25 using
the- properties of invariant matrices. This method underwent extensive
development by Littlewood,26 Mumaghan,27 and Robinson 28 using the
earlier results of Frobenius 29 and Young. 30 Its applications in atomic
spectroscopy have been expounded elsewhere, 3 1,32 and here are confined
to a brief appendix. Generally, physicists have tended to follow the
tradition of Elie Cartan, 3 3 making extensive use of the roots associated
with the relevant Lie algebras, this being the path followed by Weyl,34 van
der Waerden,7 and Racah. 8 The Cartan-Weyl approach has undergone
considerable development by the Soviet school led by Dynkin,35-37 with
his strong emphasis on the properties of the simple roots.
In this book we develop, in good company, the Cartan-Weyl approach
as modified by Dynkin 3 5-37 and Chevalley.38 We develop the theory
without attempting to supply rigorous proofs, to which adequate references
are given. Our aim is to give a guided tour through the subject, illustrating
the various points with examples. Clearly it is hopeless, in the space of one
volume, to attempt an exhaustive treatment, taking account of the full
diversity of the subject. Our approach naturally reflects personal pre-
ferences, if not prejudices. In this way I hope the reader can see the wood
without getting hopelessly lost in the trees.

2
Symmetry and Quantum Numbers
2.1 SYMMETRY AND ATOMIC QUANTUM NUMBERS
The quantum mechanics of complex atoms usually starts by considering
the central-field equation 39
N
 [ :: Vt+ U(r;) ]1/1=£1/1
( 2.1 )
as a starting approximation. The eigenvalues E are highly degenerate, there
being just one eigenvalue for each electron configuration. We are thus
faced with the problem of finding a suitable set of quantum numbers to
characterize the eigenfunctions associated with each degenerate eigenvalue.
These quantum numbers will be "good" quantum numbers if their opera-
tor representatives commute with the central-field Hamiltonian 3C CF ' Two
such operators are 8 2 and L 2 . From these two operators we obtain the
familiar spin and orbital quantum numbers Sand L. Furthermore, 3C CF
also commutes with J2 = L 2 + 8 2 and lz = Lz + Sz. Thus we could attempt to
use the quantum numbers SLIM to designate the central-field basis states.
Once we have defined a suitable set of basis states, we can use these to
calculate the matrix elements of perturbing interactions. It may, of course,
then occur that matrix elements of the perturbing terms will couple basis
states associated with different sets of quantum numbers, and they will
cease to be good quantum numbers.
3

4 SYMMETRY AND QUANTUM NUMBERS
The quantum numbers SLIM constitute a complete set for labeling the
central-field eigenfunctions in the p-shell, but are inadequate in dis-
tinguishing all the eigenfunctions for configurations having three or more
equivalent electrons (or holes) if 1>2. We could simply construct an
orthonormal set of states making an arbitrary separation of the duplicated
terms. Such an approach, while perfectly feasible, leads to no simplifica-
tions when we come to calculate the perturbation matrix elements. Al-
ternatively, we could try to enlarge the set of operators that commute with
the zero-order Hamiltonian until we obtain a set of "quantum numbers"
sufficient to label all, or nearly all, of the degenerate eigenfunctions. As
becomes apparent later, the eigenvalues of the commuting operators label
not only the eigenfunctions, but also particular irreducible representations
of particular symmetry groups. Thus it becomes possible to speak of a set
of degenerate eigenfunctions as transforming under the symmetry opera-
tions of a group according to a specific representation of the group. At first
this might seem to be just an academic maneuver; the practical advantage
comes when we attempt to evaluate the matrix elements of the perturbing
interactions. The perturbation terms may also be resolved into
symmetrized parts having well-defined transformation properties under the
symmetry operations of the same groups as were used to label the eigen-
functions. When this is the case, it is then possible to use the powerful
Wigner-Eckart theorem (see Section 19.8) to predict which matrix elements
are necessarily zero (i.e., to obtain selection rules) and to obtain re-
lationships between different matrix elements.
EXERCISES
2.1 Use the method of determinantal states to show that the d 3 electron con-
figuration contains two 2 D terms. 3 9
2.2 Show that for M s =!, M L =2,
IDMs=tML=2>
= - t[ { 2 2 - 2 } + { 2 i-I} - { 2 1 - i } - { 2 0 0 } ]
IDMs=tML=2>
= -1 [ s{ 2 2 - 2 }-3{ 2 i -1 }-{ 21- i}
v'84
+4{ 2 i - i }+3{ 2 0 0 }+2V 6 {i 1 o}]

HIERARCHIES OF SYMMETRY S
represents a possible separation of the two 2D terms of d 3 .
2.3 Show that in the g3 electron configuration there are two 4F terms. 40
2.4 Investigate 41 the possibility of finding a classifying operator that is diagonal
in the two 4F terms of g3.
2.2 IDERARCHIES OF SYMMETRY
As a simple example of the methods of finding additional quantum
numbers, consider an atomic state with J = 6 placed in a crystal at a site
having C 3v symmetry. For the free atom or ion having J = 6, the thirteen
degenerate eigenfunctions will transform with respect to rotations in a
three-dimensional space as the (6) representation of R3' Placing the atom
or ion in a site of C 3v symmetry lifts the degeneracy in accordance with the
R3C3v branching rule
(6)31r + 2 1 r +42r
123
(2.2 )
where we follow Bethe's r notation 42 and indicate the residual degeneracy
as a left superscript. Equation 2.2 shows that the 13-fold degenerate atomic
level splits into nine sublevels. However, we have only three representa-
tions of C 3v (Ir l' lr 2' 2r 3) to label the nine sublevels.
Now consider the case for octahedral symmetry f) h' We obtain under
R3e h
(6)lr +lr +2r +3r +2 3 r
JJ 1 2 4 4 5
(2.3)
We could, if desired, establish a hierarchy of symmetries and make the
reduction R3f) hC3v' When we do this we find, for f) hC3v'
lrllrl
lr2lr2
2r32r3
3r 42r 3 + lr 2
3r52r3+1rl
Thus we could use the irreducible representations of (9 h as additional
labels for the basis states in C 3v ' designating the three If 1 states of C 3v as
(Ir l ) If l' (r5) lr l , and (r5) lr l

6 SYMMETRY AND QUANTUM NUMBERS
the two 1 r 2 states as
(Ir 2 ) lr 2 and (3r 4 ) lr 2
and the four 2r 3 states as
(2r 3) 2r 3' (3r 4) 2r 3' (r 5) 2r 3' and (r 5) 2r 3
where the 0 h representations are enclosed in brackets and the two 3r 5
states of 0 h have been arbitrarily separated into r 5 and tr 5' Thus by
seeing how the eigenfunctions behave under the reduction R3f)hC3v'
we have obtained additional quantum numbers that allow us to distinguish
the different eigenfunctions of C 3v ' Of course, it should be noted that
whereas the quantum numbers of C 3v are rigorously good quantum num-
bers, the additional quantum numbers we have introduced are not, since
the C 3v crystal-field potential will mix the states based on e h' However, it
is frequently possible to simplify the calculation by computing first the
basis functions for e h symmetry and then using these to compute the
additional perturbation produced by the distortion from f) h to C 3v sym-
metry.
The method we have indicated for crystal point groups exploits the
properties of the eigenfunctions under finite transformations such as
discrete rotations and reflections. To produce additional labels, we estab-
lished a hierarchy of nested symmetry groups and examined the reduction
of the irreducible representations as the symmetry was restricted.
The same basic ideas may be applied to the classification of atomic or
nuclear states, except that we must then consider continuous groups
involving matrix transformations in a multidimensional space rather than
finite groups involving discrete transformations.
EXERCISE
2.5 The symmetry of the crystal field about the rare-earth ion in the rare-earth
double nitrates is known to be C 3' The major groups of the crystal-field levels
can be understood by assuming that the crystal field has approximately
icosahedral symmetry. Show 43 that the structure within the groups can be
explained by assuming an approximate symmetry Th with a small distortion to
C 3 .

3
Groups Composed of Regular Matrices
3.1 THE GROUP POSTULATES
A collection of elements A, B, C,... is said to form a group G if the
elements can be combined together in such a way as to satisfy the four
group postulates:
1. Identity. Among the collection of elements there is an element,
known as the identity element (or the unit element), such that for any
element A of the group,
A =A=A
(3.1 )
2. Closure. The product of any two elements of the group itself corres-
ponds to a unique element of the group.
3. Inverses. For any element A of the group there exists an inverse
element A-I such that
AA - 1 = A - lA = 
(3.2)
4. Associativity. If three or more elements are combined under group
multiplication, then the order of multiplication is immaterial, that is,
A (BC) = (AB)C=ABC
(3.3 )
In forming the elements of a group, it is essential that all four postulates
be satisfied. The collection of elements may be finite, in which case we
7

8 GROUPS COMPOSED OF REGULAR MATRICES
have a finite group. If the elements are denumerable infinite, then the group
is said to be an infinite discrete group. If, however, the elements form a
continuum, in a topological sense,44 then the group is said to be a
continuous group.
EXERCISES
3.1 Show that the four elements + 1, + i form a finite group under multiplication.
3.2 Show that the set of permutations
(  2 : ), (  2 : ),c 2  ),
2 1 3
(  2  ), (  2  ), (  2  )
2 I 3
forms a finite group.
3.3 Show that the set of matrices
(  ), (  o ) ( - V3/2 ),
- I ' V3 /2
( _1 -V3 /2 ),( -t V3/2 ),( -! - V3/ 2 )
2
- \"3 /2 t - V3 /2 - t V3 /2
forms a finite group.
3.4 Show that the infinite set of matrices
( cosO sin 0 )
- sin (J cos(J
produced by the continuous variation of (J between 0 and 2", forms a
continuous group.
3.5 Show that the infinite set of triangular matrices
( :)
where a is real and unbounded, forms a continuous group.
3.2 REGULAR MATRIX GROUPS
Under certain conditions we may show that square n X n matrices A
satisfy the aforementioned group postulates:

REGULAR MATRIX GROUPS 9
1. The unit element is the n X n identity matrix,
I 0
1
= (3.4)
0 I
2. The existence of the inverse element A-I is assured by restricting our
attention to nonsingular matrices, that is,
det IA I =FO
(3.5 )
3. The laws of matrix multiplication are such that the associative law of
multiplication is satisfied.
4. The set of matrices is such that closure is assured.
The groups involving regular matrices may be finite or infinite, be
discrete or continuous, and have real (R) or complex (C) elements. The
variables in the real space R n are designated x = (x l' . . . ,x n ), and in the
complex space C n as z = (z 1"" 'Zn)' A regular matrix of degree n acting in
R n or C n will produce a transformation xx' or zz'. In problems in
physics we are frequently interested in classes of transformations that leave
invariant some functional form of x or z. For example, in an isotropic
three-dimensional Euclidean space we may wish to consider transforma-
tions that hold xi + x + x as an invariant, or in a four-dimensional
Lorentzian space the form xi + x + x - x.
EXEROSES
3.6 Show that the transformations produced by the matrices
( cosf}
- sin f}
sin f} )
cosf}
(0< f} <2'17")
acting in R 2 leave invariant the form XT + x.
3.7 Show that in three dimensions transformations of the type
r'=Ar+a
where A is a nonsingular matrix and a is a vector, form a 12-parameter group
(the so-called affine group).

10 GROUPS COMPOSED OF REGULAR MATRICES
3.3 MATRIX PROPERTIES
We briefly summarize a number of properties of matrices for later use.
The inverse, transpose, complex conjugate, and Hermitian conjugate of a
matrix A are denoted by A -1, tA, A *, and A t, respectively. The properties
of a number of special matrices are now tabulated.
Matrix relation
Name of matrices
A=tA
A +tA =0
tAA = 
A=A*
A=-A*
A=At
A+At=O
AtA = 
Symmetric
Skew symmetric
Orthogonal
Real
Imaginary
Hermitian
Skew Hermitian
Unitary
3.4 CONTINUOUS MATRIX GROUPS
Consider a group whose elements comprise all regular nonsingular real
matrices of degree 2,
( all
a 21
a I2 )
a 22
(3.6)
Apart from the non singularity restraint
all a 22 =F a 1 2 a 2 1
(3.7)
the range of the elements of the matrix is unrestricted. Let us rewrite the
matrix elements ai} of Eq. 3.6 as
a.. = .. + a..
l} l} l}
(3.8 )
If all ai} = 0 we simply obtain the identity element
=( )
(3.9)
Clearly we could treat the ai} as real independent parameters and generate
all the elements of the group by a continuous variation of the ai}' The

CONTINUOUS MATRIX GROUPS 11
range of the parameters is unbounded and limited only to the extent
demanded by the nonsingularity condition of Eq. 3.7. Any element of the
group could then be designated by giving its associated values of the
parameters aij'
We may form a variety of groups from regular matrices A of degree n
involving real or complex elements. In what follows we list some of the
important continuous matrix groups. Extensive descriptions have been
given by Chevall ey 45 and Helgason. 46
(a) The general linear group. The most comprehensive linear matrix group
is the complex general linear group GL(n, C) of regular invertible complex
matrices of degree n. A particular matrix is characterized by its n 2
elements. Each element may contain a real and an imaginary part. The
continuous variation of the 2n 2 parts (i.e., the n 2 real and the n 2 imaginary
parts) will generate the entire matrix group, and hence the group is of
dimension 2n 2 and may be characterized by 2n 2 real parameters.
If we restrict the elements of GL(n, C) to real values only, we obtain the
n 2 -parameter subgroup GL(n,R). Clearly,
GL(n, C):) GL(n,R)
(3.10)
The group GL(n, C) will obviously contain many other subgroups.
(b) The special linear group. Making the restriction that the complex
matrices of GL(n,C) be of determinant + 1, we obtain the complex special
linear group SL(n, C), which is characterized by 2(n 2 - 1) parameters. The
real special linear group SL(n,R) formed by real matrices of determinant +
1 has n 2 - 1 parameters. Clearly,
GL(n, C)::> SL(n, C)::> SL(n,R)
(3.11 )
and
GL(n,R):) SL(n,R)
(3.12 )
The special linear group is sometimes referred to as the special unimodular
group.
(c) The unitary groups. The unitary matrices A of degree n form the
elements of the n 2 -parameter unitary group U(n) that leaves the Hermitian
form
n
 z.z:cc
 I I
i= 1
(3.13)
invariant. Since the unitarity of the matrices A requires that

12 GROUPS COMPOSED OF REGULAR MATRICES
A tA = 
(3.14 )
the range of the matrix elements aij is restricted by the requirement that
 aitalj = 8ij
t
(3.15 )
and hence laijl2 1. Thus in this case the domain of the n 2 parameters is
bounded. As we see later, we have here an example of a compact group.
The group of matrices in GL(p + q, C) which leaves invariant the Her-
mitian form
-ZlZ-'" -zpZ;+Zp+lZ;+l +... +zp+qZ;+q
(3.16 )
is designated as the group U(p,q), where U(n,O) - U(O,n) = U(n). Clearly,
GL(p+q,C):J U(p,q)
(3.17 )
and
GL(n,C):J U(n).
(3.18 )
(d) Special unitary groups. If we limit our attention to unitary matrices of
determinant + 1, we obtain the (n 2 - I)-parameter special unitary group or
unitary unimodular group SU(n), where
SU(n) = U(n) n SL(n,C)
(3.19 )
Similarly,
SU(p,q) = U(p,q) n SL(p +q, C)
(3.20)
The matrices in SL(2n, C) which commute with the transformation cp of
C 2n given by
cp
(zl,.."Zn,Zn+l"",z2n)  (z:+1"",Z1n' -zr,...,-z:)
(3.21 )
form a group usually designated as SU*(2n).
(e) The orthogonal groups. The group of complex orthogonal matrices of
degree n form a n(n - 1 )-parameter group designated as O(n, C). Since
tAA = , we have IA 1= + 1, and thus the group decomposes into two
disconnected pieces and we cannot go continuously from one to the other.
The orthogonal matrices of determinant + 1 form a subgroup of O(n, C),
namely, the n(n -I)-parameter special complex orthogonal group, SO(n, C).
The matrices of SO(n, C) have the important property of leaving invariant
the complex quadratic form

CONTINUOUS MATRIX GROUPS 13
n
 z; (3.22)
;= 1
Clearly,
SO(n, C) = SL(n, C) n O(n, C)
(3.23 )
(j) The special orthogonal groups. The set of real orthogonal matrices of
degree n forms the n(n-l)j2-parameter real orthogonal group O(n,R),
while the set of real orthogonal matrices of determinant + I forms the real
special orthogonal group SO(n,R). Again, O(n,R) consists of two discon-
nected pieces, with SO(n,R) occurring as a subgroup. The real special
orthogonal matrices leave invariant the real quadratic form
n
 X;2
;= 1
(3.24 )
The matrices in SL(p + q, R) that leave invariant the quadratic form
p q
-  x; +  xl
;=1 j=p+1
( 3 .25 )
form the elements of the group SO(p,q).
Finally, the group of matrices in SO(2n, C) which leave the skew
Hermitian form
-ZlZ+l +zn+lzi-'" -znzn+z2nz
(3.26 )
invariant form the elements of the group SO*(2n).
(g) The Symplectic Groups. The symplectic group Sp(2n, C) is the 2n(2n +
I)-parameter group of regular complex matrices which leave invariant the
nondegenerate skew-symmetric bilinear form
n
 (x;y;-x;y;)
;= 1
( 3.27 )
of two vectors x - (xl,...,xn'x,...,x) and y (Yl""'Yn'Y""'Y)'
Clearly, GL(n, C) Sp(2n, C), and the matrices need not be unitary.
Restriction to real matrices gives the n(2n+ I)-parameter group Sp(2n,R).
The symplectic group Sp (2n) = U (2n) n Sp (2n, C) is known as the unitary
symplectic group. This group, like Sp(2n, R), is a n(2n + I)-parameter group.
Note that symplectic groups arise only for even-dimensional vector spaces.

14 GROUPS COMPOSED OF REGULAR MATRICES
EXERCISES
3.8 Show that for a matrix A to leave the bilinear form given in Eq. 3.27
invariant, we must have
AtJA =J
( 3.28 )
where
J = ( 0 n )
-n 0
(3.29 )
3.9 Investigate the properties of matrices that hold invariant the bilinear form
XIY2 - X2YI' where (XI,X2) and (YI,Y2) are the coordinates of a pair of points
in a plane.
3.10 Show that the matrices
( cosh(J sinh(J ) (3.30)
sinh (J cosh (J
leave invariant the real quadratic form x  - x and constitute the group
SO(I, I).
3.5 MATRIX EXPONENTIAL FUNCfIONS
The notion of the exponential of a regular matrix A of complex numbers
plays an important role in the subsequent development of the theory of
continuous groups. Formally, we define the exponential function of a
matrix A by the series
A 2 A 3
e A =  + A + - + - + . . .
n 2! 3!
00
= L  (3.31)
p=o
where AO=  n' the identity matrix of degree n.
The following theorems relate to the exponentiation of matrices. De-
tailed proofs abound in the literature. 45 - 48
Theorem 3.1
The exponential series in Eq. 3.31 is convergent only if the absolute values laijl
of the matrix elements of A have an upper bound.
Proof: Let JL be an upper bound such that laijl < JL for all ai' and let at be

MATRIX EXPONENTIAL FUNCTIONS 15
the matrix elements of A P (0< p < 00). We assert that
I at I < ( nIL )p
This is certainly true for p = 1. If it holds for p, then it must hold for p + 1:
n
laG+ l l=  afka <nlL(nlL)P=(nlL)P+1
k=l
Thus the series is convergent. (N.B. It is important to note that the series is
bounded only if the aij are bounded. As is seen later, this is indeed the case
for any compact group.)
The following theorems are consequential on Theorem 3.1.
Theorem 3.2
If A and B are two commuting matrices, then
e A +B = eAeB
(3.32)
Theorem 3.3
If B is a regular matrix of degree n, then
Be A B- 1 = e BAB - 1
(3.33 )
Theorem 3.4
If AI"" ,An are characteristic roots of A the characteristic roots of e A are
AI \.
e , . . . ,e
( 3.34 )
Theorem 3.5
The exponential series of Eq. 3.34 satisfies the usual exponential function
properties
eA.=(e A )*, et...=t(e A ),
eAt=(eA)t, e- A =(e A )-1
(3.35 )
Theorem 3.6
The determinant of e A is e trA .

16 GROUPS COMPOSED OF REGULAR MATRICES
Theorem 3.7
If A is skew symmetric, e A is orthogonal, while if A is skew Hermitian, e A is
unitary.
EXERCISE
3.11 Show that if A and B are any two matrices of order n, then
e-ABe A = B+ II! [B,A] + i! [[B,A ],A] +. ..
(3.36)
(This corresponds to the so-called Campbell-Hausdorff formula.)

4
Local Properties of Lie Groups
4.1 PARAMETERIZATION OF THE GROUP ELEMENTS
We have already seen that the elements of groups composed of regular
nonsingular matrices of degree n may be represented, under certain cir-
cumstances still to be discussed, in terms of r parameters a j such that
A = A ( a}, . . . , a r )
( 4.1 )
The identity matrix  n is normally characterized by the null set of
parameter values
 n = A (0, . . .,0)
( 4.2 )
The. continuous variation of the r parameters may then generate the entire
group manifold.
In the case of the group GL(n, C) we may regard the n 2 elements of the
matrices A as labeling points in an n 2 -dimensional complex Euclidean
space. Alternatively we may consider the n 2 real and n 2 imaginary com-
ponents of the matrices as labeling points in a real 2n 2 -dimensional
Euclidean space. Variations in the 2n 2 parameters then take us from one
point to another point in the Euclidean space. The various r-parameter
subgroups of GL(n, C) may be represented by r-dimensional subspaces of
the 2n 2 -dimensional Euclidean space.
17

18 LOCAL PROPERTIES OF LIE GROUPS
4.2 CONNECfIVITY
A group is said to be connected if we can take an arbitrary element A
and reach the identity element  by a continuous variation of the r
parameters. This amounts to being able to connect any pair of points in
the group space by an arc generated by the continuous variation of the
group parameters.
The rotation group SO(n) is clearly connected, whereas the full ortho-
gonal group O(n) is not, since it is not possible to pass continuously from
the orthogonal matrices of determinant + 1 to those of determinant - 1. In
this case the group O(n) is said to consist of two disjoint pieces. The piece
that is connected to the identity element forms a group by itself, in our
example the group SO(n).
We can construct each disjoint piece by taking the piece connected to
the identity element and taking its product with one of the elements of the
disjoint piece. Thus in the case of 0(2) we can take the elements
( cos ()
- sin ()
sin () )
cos()
of SO(2) and multiply them by the element
( -)
to create the complete set of elements with determinant - 1.
The group O(n) is an example of a mixed continuous group. In these
cases the elements must be labeled by a set of r continuous parameters
together with a set of discrete labels equal to the number of disjoint pieces.
Thus in O(n,R) we may designate the group elements by n(n-l)j2 real
continuous parameters together with the sign -of the determinant of the
element.
EXERCISE
4.1 Show that physically the elements of the connected piece of O(n, R) generates
rotations, whereas those of the disjoint piece generate improper rotations (i.e.,
rotations combined with a reflection).

rHE BEGINNING OF LIE GROUPS 19
4.3 THE BEGINNING OF LIE GROUPS
Let us suppose that the elements A of a group G can be expressed in
terms of r continuous parameters and write
A ( Q ) = A ( aI' . . . , a r )
( 4.3 )
where the identity element A (0) of the group is identified with the set of
null parameters.
The closure of the group elements requires that the product of any two
group elements, say A (n) and A ( ), be itself an element A ('Y) of the
group, where
A ( Y ) = A ( n )A (  ) = A ( Y ( n ,  ) )
( 4.4 )
and hence
'Y = f( n ,  )
( 4.5)
Continuity of the group parameters 'Y will be assured if they are con-
tinuously differentiable functions of all the parameters n and . It follows
from Eq. 4.5 that for the identity element A (0) we must have
'Y = f( Y ,0) = f( 0, Y )
( 4.6)
The existence of the inverse element A ( Q ) - 1 = A ( Q') requires that the
parameters n' be continuously differentiable functions of the parameters
Q.
Finally the associative postulate
A (Q) (B(  ) C( Y)) - (A (n ) B(  )) C( Y)
( 4.7)
requires that
f [ n ;f(  , Y ) ] = f [ f( n ,  ), Y ]
( 4.8)
Continuous groups satisfying the above requirements are referred to as Lie
groups.
Note: Although we occasionally make use of the concept of continuous
differentiability of functions of the group parameters, it is sufficient to
assume that the group elements are continuous functions of the group
parameters. Hilbert suggested in 1900, at the Paris Congress of Mathema-
tics, that Lie's concept of continuous transformation groups should be
capable of development without the assumption of the differentiability of
the functions defining the group. The history of the solution of Hilbert's
so-called "fifth problem" has been outlined by Maurin 49 and by Mont-
gomery and Zippin. 50

20 LOCAL PROPERTIES OF LIE GROUPS
EXERCISES
4.2 Show that the matrices associated with rotations in two dimensions form a
one-parameter Lie group.
4.3 Show that the transformations associated with translation along a line form a
one-parameter Lie group.
4.4 INFINITESIMAL GROUP GENERATORS
We have assigned the null parameters to the identity element A (0). Let
us now investigate the properties of the group elements in the neigh-
borhood of the identity element. For sufficiently small values of the
parameters we may represent an element A (Q) lying close to the identity
by a Taylor expansion
r ( aA )
A(a)=A(O)+ klak aa k a.-O
1 r r ( aA ) ( aA ) 3
+ 2 kl '1 aka, aak a.-O aa, a,=O + O( a )
(4.9 )
r 1 r r 3
=A(O)+  akXk+ 2   aka,XkX,+O(a )
k=l k=l/=l
( 4.10)
where we write
x - ( M )
k- aa k a=O
( 4.11 )
The X k are referred to as the infinitesimal group generators of the group
elements about the identity. If the inverse element A (a)-l is also in the
neighborhood of the identity, then writing
1 r 1 r r
A(a)- =A(O)-  akXk+ 2   ak lX ,X k X,+O(a 3 ) (4.12)
k=l k=l/=l
we have
A (a) -1 A (a ) =A (0) + O( a 2 )
( 4.13)

INFINITESIMAL GROUP GENERATORS 21
Let us define the commutator of two oup elements A() and A(Y)
lying near the identity A (0) as A (  ) - IA (y) - IA (  )A (y). The commutator
must itself define a group element A (, y) lying close to A (0). Using Eqs.
4.10 and 4.12 for sufficiently small values of the group parameters, we find,
to second order in  and y ,
-I -I
A() A(r) A( )A(y)=A(O)+fik'Y/[Xk'X/]
(4.14 )
where
[Xk,X/] =XkX/-X/X k
( 4.15)
is the commutator of the group generators X k' Xl' But from Eq. 4.10 we
must have
A (, y) =A (Q) =A (0) + amXm +...
( 4.16)
Equations 4.14 and 4.16 must be identical, and comparing terms we have
[Xk,X/] = CXm
( 4.17)
where
am = C{3k 'Y/
( 4.18 )
The quantities C are termed the structure constants of the infinitesimal
Lie group.
The structure constants of an infinitesimal Lie group have a number of
important properties.
1. They are antisymmetric in their lower indices, that is
C m - C m
k/ - - /k
( 4.19 )
2. Since the infinitesimal generators satisfy the Jacobi identity
[[Xk,X/ ],X m ] + [[X/,X m ],X k ] + [[Xm,X k ],X/] =0
( 4.20)
the structure constants must satisfy the requirement
CC::'n + C/:" Ckn + C;:'k Cf" = 0
(4.21 )
Equations 4.17 to 4.21 later form the basis for the development of the
theory of Lie algebras;I,S2 However, before going further it is as well for
us to investigate the properties of the group generators in greater detail.
This we do by first considering some simple examples.

22 LOCAL PROPERTIES OF LIE GROUPS
4.5 THE 1WO-DIMENSIONAL ROTATION GROUP 80(2)
Consider a vector r whose terminus is given by the coordinates (x,y) and
which lies in the xy plane. Rotate the vector r through an angle ()
(anticlockwise) to give a new vector r' whose terminus is given by (x',y').
y
(x, y)
x
Then
= ix' (x,y; ()
( 4.22 )
x' = x cos () - y sin ()
y' = x sin() + y cos()
= 1;" (x,y; () )
Thus the transformation is induced by the two-dimensional orthogonal
matrix of determinant + 1,
A ( () ) = ( cos ()
sin ()
- sin() )
cos()
(4.23 )
The length r 2 = x 2 + y2 is left invariant, and the transformation matrices
A «()) are functions of the single parameter () (0  () < 2'1T). These matrices
clearly form a one-parameter compact Lie group. The identity element 2
is obtained by putting () = O. We also note that
( cos () - sin () ) ( cos () , - sin ()' ) _ ( cos ( () + (J')
sin (J cos (J sin (J , cos (J , sin ( (J + (J')
- sin ( (J + (J') )
cos ((J + (J')
-that is, the product of two elements is itself an element of the group.
Furthermore, the group is commutative or Abelian. However, this is not the
case for the higher-dimensional rotation groups.

GENERAL INFINITESIMAL TRANSFORMATIONS 23
EXERCISE
4.4 Give examples of the noncommutativity of sequences of rotations In a
three-dimensional real Euclidean space.
4.6 INFINITESIMAL ROTATIONS
Let us now consider the effect of an infinitesimal transformation. An
infinitesimal rotation is an orthogonal transformation of the coordinate axes
in which the components of a vector are almost the same in both sets, that
is, they differ by infinitesimal amount. Thus the transform of the ith
component xl of an n-dimensional vector r is practically the same as Xi:
n
xl = Xi + €nXl +... + €inXn = Xi + L €ijX}
j=l
( 4.24 )
where the €ij's are infinitesimals. For example, in SO(2) we would have
x' = X - Y 8(}
y'=x8(}+y
(since cos(}l as (}O, and sin()8(} as (}O). Equation 4.24 may be
rewri tten as
n
X:=  ( 8..+€.. ) x.
I  lj lj J
j=l
(4.25 )
But 8ij is simply the unit matrix tJ n , and hence in matrix notation we may
wri te
x' = ( u= + I: ) x
where I: is a matrix of infinitesimals and  the identity matrix.
( 4.26)
4.7 GENERAL INFINITESIMAL TRANSFORMATIONS
We now consider more general transformations Ta that carry a point
X = (Xl"" ,x n ) into another point x' = (x 1 ',... ,x:) lying in an n-dimensional
space. The transformation Ta will be specified by r parameters
a = (a l' . . · ,a r )

24 LOCAL PROPERTIES OF LIE GROUPS
Thus
x' =xT a = j(x; a)
( 4.27)
Another transformation is
x" =x'T b = j(x'; b)
( 4.28 )
We assume that the transformations are unique, that is, there is one, and
only one, transformation that will take x into x'.
Clearly, if we write
x' = x T
a
then
x" =x'Tb=xTaTb=xTc
( 4.29)
where
c p = <p P ( a, b )
(p = I,. .. ,r)
( 4.30 )
or for brevity, c = <p(a, b). For example, in SO(2),
x' = xT(J = j( x,y; (J)
x" =x'T(J'= j(x',y'; (J')
= xT(J T(J' = xT(J+(J' = j( x,y; (J + 0')
It follows from Eqs. 4.29 and 4.30 that
x TaTb = XTcp(a,b)
and
j [j( x; a) ; b] = j [x; <p (a, b) ]
The <p(a, b) are assumed to be continuously differentiable functions to all
orders.
If the transformations Ta are to form a group, then the functions <p must
satisfy the following conditions:
1. There must be a unit element a o such that
a = <p ( a, a o ) = <p ( a o , a)
If we take the origin as a o we have a o = 0, so that
a = <p ( a, 0) = <p ( 0, a)
( 4.31 )

INFINITESIMAL OPERATORS OF A LIE GROUP 25
2. There exists an inverse transformation Ta- l = Ta' such that
Ta-1Ta = TaTa- 1 = 
cp " ( a' , a) = cp " ( a, a') = 0
( 4.32)
3. The associative law of multiplication is obeyed, that is,
Ta(TbTc) = (TaTb)Tc
which implies that
cp [ a; cp (b, c ) ] = cp [ cp ( a, b ) , c ]
( 4.33 )
Thus we have three equations (Eqs. 4.31, 4.32, and 4.33) that must be
satisfied by the functions cp(a, b) of Eq. 4.30.
4.8 INFINITESIMAL OPERATORS OF A LIE GROUP
Let us first consider the simple case of a one-parameter group in one
variable x. The initial point is given as
x
x+dx
xo
xo=f(xo;O)
We can get to x by making the transformation
x=f(xo;a)
( 4.34 )
We could go to x + dx by making the transformation
x + dx = f(x o ; a + da)
( 4.35)
However, we could also get there by first going to x and then to x + dx by
an infinitesimal parameter change a, that is,
x + dx = f(x; 8a)

26 LOCAL PROPERTIES OF LIE GROUPS
Expanding the above result gives
( aj( x; a) )
dx= aa a=oa=U(x)a
( 4.36)
The connection between da and a can be established by first noting that
a + da = cp( a, 8a)
and hence
( acp( a, b) )
da= ab b-O a= U(a) a
( 4.37)
N ow consider the general case of r parameters and n variables. The
analog of Eq. 4.36 is
. L ( aji(x;a) )
dx'= ao
aa o
(1 a=O
( (J = 1, . . . ,r; i = I, . . . ,n )
or
dx i = U(x) 8a o
( 4.38)
where
U(x) = ( at;:a) t-o
( 4.39)
The analog of Eq. 4.37 is simply
,J 0 = ( acp 0 ( a, b) ) 8 p
ua ab p a
b=O
or
dao = V;(a)8a P
( 4.40 )
where
( acp 0 ( a, b) )
V;(a) = ab P b-O
(4.41 )
The inverse of V O will be A P where A PV O = 8 0 . The inverse of the trans-
P 'r 'r P 'r
formation produced by Eq. 4.40 is thus
8a P = A:( a) da'r
( 4.42)

INFINITESIMAL OPERATORS OF A LIE GROUP 27
Substituting Eq. 4.42 into Eq. 4.38, we find
dxi=U(x)A:(a)daP
or
:; = U(x)A:(a)
( 4.43 )
The infinitesimal transformation xx + dx induces in F(x) the trans-
formation F(x)F(x) + dF(x). Now
aF .
dF(x) = ----: dx'
ax'
= aF. u i a(J
ax' (J
(from Eq. 4.38)
=8a(J U i aF.
(J ax'
dF(x) =8a(J X(JF
( 4.44 )
The operators
X =Ui
(J (J ax i
( 4.45)
are called the infinitesimal operators of the group. The operator Sa that
effects the infinitesimal transformation F(x)F(x) + dF(x) is
Sa = 1 + 8a (J X (J
( 4.46)
EXERCISE
4.5 Repeat the above analysis, but instead of considering infinitesimal transfor-
mations of the variable x, consider infinitesimal changes in the group elements.
Thus show that the elements of the group lying in the neighborhood of the
identity are generated by the infinitesimal group generators
a ( a<p a ( a, b) ) a
XfJ = V;(a) aaa = ab fJ aaa
b-O
( a = 1, . . . ,r)

28 LOCAL PROPERTIES OF LIE GROUPS
4.9 EXAMPLES OF INFINITESIMAL OPERATORS
A. The Rotation Group SO(2)
The group SO(2) is a one-parameter (8) group, and we now obtain the
form of the single infinitesimal operator. The infinitesimal transformation
IS
x'=x-y88
y'=x88+y
8x = - y 88
8y = x 88
Thus from Eq. 4.38,
8 8y
U(x) = 8 = -y and U(y) = 80 =x
Using these results in Eq. 4.45 gives the infinitesimal operator X of SO(2)
as
a a a a
x= U(x) ax + U(y) ay = -y ax +x ay
a a
X=x- -y-
ay ax
( 4.47)
In the quantum theory of angular momentum we put
J = -i ( X _y1- )
z ay ax
( 4.48 )
and hence have
X=-iJ
z
( 4.49)
with
So = I - i 88 Jz
( 4.50 )
B. Infinitesimal Operators of SO(3)
The elements of the group SO(3) are formed by orthogonal matrices A
of determinant + I and of degree 3. The orthogonality requirement implies
that
tAA=3

EXAMPLES OF INFINITESIMAL OPERATORS 29
The infinitesimal rotations have a transformation matrix of the form
A =  3+B
where  3 is the degree-3 unit matrix and B is a matrix that has all its
elements in the neighborhood of zero.
For the transformation to preserve the orthogonality we must have
 3=tAA = (3+tB) (3 + B) 3 +tB + B
that is,
tB + B = 0
Thus B must be a skew-symmetric matrix with three independent com-
ponents, say
o
- )
B= ( -:
a
-c
But x' = ( 3 +. £ )x, that is,
( ;:: ) = [(    ) + ( -
z+dz 0 0 I b
o
c
](:)
a
-b
-c
o
dx=ay-bz
dy a x+cz
dz=bx-cy
Thus the infinitesimal operators of 80(3) are
a ( cz ) a a ( - cy ) a a a
x = -+ -=z--y-
1 ac ay ac az ay az
a ( - bz ) a a ( bx ) a a a
x 2 = ab ax + ab az = x az - z ax
a(ay) a a( -ax) a a a
x 3 = aa ax + aa ay = y ax - x ay

30 LOCAL PROPERTIES OF LIE GROUPS
We note that the infinitesimal operators of 80(3) are closed under com-
mutation:
[X 1 ,X 2 ] =X 3 [X 2 ,X 3 ] =X 1 [X 3 ,X 1 ] =X 2
( 4.51 )
The normal operators associated with the quantum theory of angular
momentum are J k = - iX k , and the corresponding commutation relations
become
[J 1 ,J 2 ]=iJ 3
[ J 2' J 3 ] = iJ 1
[ J 3' J 1 ] = iJ 2
( 4.52)
EXERCISES
4.6 Construct the infinitesimal operators for the group of transformations
X' = ax
y' = by
and for
x' = ax
I 1
y =-y
a
4.7 Obtain a set of infinitesimal operators for SO(4) and write out their commu-
tation relations. Show that the six infinitesimal operators can be divided into
two subsets of three operators each which are closed under commutation. 5 3
4.8 Consider the group of real linear transformations
x' = ax + b
Show that
a a
x =x- and X b = ax
a ax
with
[Xa,X b .] = -X b
4.9 Show that the rotation group in n dimensions, SO(n), is characterized by
n(n -1)/2 infinitesimal operators which may be written in the general form
J pr = - i(Xp a: r - X r a: p ) (p,r= 1,.. .,n and r >p).
4.10 Establish 54 the commutation relations for the infinitesimal operators of
SO(5).
4.11 Show that for the group SO(2, 1) there are three infinitesimal operators
X l' X 2' X 3 which satisfy the commutation relations
[X 1 ,X 2 ] =X 3
[X 2 ,X 3 ]=-X 1
[X 3 ,X 1 ] =X 2 .

STRUCTURE CONSTANTS OF LIE GROUPS 31
4.10 STRUCfURE CONSTANTS OF LIE GROUPS
We saw earlier, in Eq. 4.17, that the commutators of the infinitesimal
group generators are expressible as a linear combination of the infinite-
simal generators-no new quantities appear: that is, the set of infinitesimal
generators is closed under commutation. We now establish this result for
the infinitesimal operators in terms of the properties of infinitesimal
transformations.
We recall Eq. 4.43:
;; = U(x)A:(a)
(i= l,...,n;o,p= 1,...,r)
( 4.43 )
This expression describes the change in the point x produced by an
infinitesimal displacement from its initial position x(O) where a = O.
In order to obtain a finite displacement we require that Eq. 4.43 be
integrable. The condition for integrability is 55
a 2X i a 2X i
-
aa 'Ta a P aa pa a 'T
(4.53 )
Substituting Eq. 4.43 into 4.53 gives
aAO ( a ) a Ui ( x ) a j a\O ( a ) a Ui ( x ) a j
Ui ( x ) P + a , 2- A O(a) = Ui(x) ''7 + o. 2-A;(a)
a aa'T ax} aa'T P a aa P ax} aa P
Grouping terms together gives
( a u i a u i ) ( aA a aA a )
U! - U! \PAo+ U i  - -2- =0
P ax} a ax} ''7 P a aa'T aa P
where for brevity we suppress x and a and note Eq. 4.43.
Now multiply by U[U; and sum over T and p, noting that
U[A; = 8[, etc,
to obtain
au i au i ( a\O A )
j 11 j  _ ''7 aA 'T p i_a i
U ---:- - U ---:- - _ a P - _ a 'T U U lI U o - ClI ( a) U o
ax} ax} a a
( 4.54 )
U(x) is independent of a, and hence if we differentiate the left-hand side
with respect to a P , we get zero, and
Ui ( CO ( a )) =0
a aa P 

32 LOCAL PROPERTIES OF LIE GROUPS
and thus the c(a) are independent of the parameters a.
Taking account of Eq. 4.45, we have
[Xa,X p ] = XaXp - XpXa
= u-LUj-L - Uj-Lu
ax' Pax} p ax} ax'
( ,aut . a u ) a
= u'--u'- -
a ax i p ax i axJ
Comparison with Eq. 4.54 gives
[X",X p ] =c;p u;-L
ax}
[X a , X p ] = c;pX"
( 4.55)
as was indeed found in Eq. 4.17.
We note that Eq. 4.55 is valid for both infinitesimal operators and
generators. The antisymmetry requirement of the structure constants c;p
ensures that the infinitesimal operators or generators are self-commuting,
since necessarily c;a = O.
EXERCISES
4.13 Show that the infinitesimal operator X associated with the linear translation
xx+a
is of the form
a
x=-
ax
Show that the infinitesimal operators a lax, a lay, a laz associated with
translations in a three-dimensional Euclidean space, taken together with the
infini tesimal opera tors
a a
x- -y-
ay ax '
a a
y az - z ay ,
a a
z- -y-
ax az
associated with rotations in the same space, are closed under commutation
and thus define a Lie group (the Euclidean group E3 in three dimensions).

GENERATION OF FINITE GROUP ELEMENTS 33
4.11 GENERATION OF FINITE GROUP ELEMENTS
The elements A ( Q) lying in the neighborhood of the identity element
A (0) of an r-parameter Lie group are defined, in Section 4.4, in terms of
the infinitesimal group generators
_ ( aA ( n ) )
X k -
aak
Q==O
( k = 1, .. . ,r )
( 4.11 )
where
r
A(n)=A(O)+  arXk+O(n2)
k=l
( 4.56)
In the particular case of the one-parameter group 80(2), we have, for
the element A (8fJ) infinitesimally removed from the identity,
A(80)( )+80( -)=;j2+89X9 (4.57)
Let us write the infinitesimal angle as
fJ
8fJ = -
N
where N is an arbitrarily large number, so that Eq. 4.57 becomes
A(80);j 2+  X9
( 4.58)
We may generate the group element A (fJ) associated with a finite angle fJ
by applying A (8fJ) N times, that is,
A(O)(;j 2+  X9 f
In the limit of Noo we obtain the exact result
A ( fJ ) = e fJX,
( 4.59)
If we formally expand the exponential, as in Eq. 3.31, and note the
common series associated with sines and cosines, we obtain the well-
known result for a finite angle:

34 LOCAL PROPERTIES OF LIE GROUPS
A ( (} ) = ( cos (}
sin (}
- sin(} )
cos(}
( 4.60 )
Thus starting with the infinitesimal group generator, we may generate
the entire continuum of group elements connected to the identity element.
For a Lie group characterized by r infinitesimal group generators X k , the
analog of Eq. 4.59 becomes
A (Q ) = e akXk
(k = 1, . ..,r)
( 4.61 )
a result that may be equivalently obtained by integrating Eq. 4.11. We note
that Eq. 4.61 holds only for the group elements connected to the identity
element. To obtain the elements of disjoint pieces it is necessary to
introduce the discrete group elements and to apply them in turn to the
elements of the connected piece. In general the infinitesimal group genera-
tors X k will be noncommutative, and hence in any particular application
the sequence of the generators must be preassigned.
The above remark may be illustrated by consideration of the group of
rotations in a three-dimensional Euclidean space. This essentially corres-
ponds to the three-parameter group 80(3).
At first we might be inclined to parameterize the group elements in
terms of the familiar Euler angles \fJ, (}, cp to give
A (\fJ,(J,cp)
cos cp cos () cos \fI- sin cp sin \fI
sin cp cos () cos \fI + cos cp sin \fI
- sin () cos \fI
- cos cp cos () sin \fI- sin cp cos \fI
- sin cp cos () sin \fI + cos cp cos \fI
sin () sin \fI
cos cp sin ()
sin cp sin ()
cos(}
( 4.62 )
with the parameters falling in the intervals - 71' < \fJ < 71', 0 < (} < 71', - 71' < cp
< 71'. However, this parameterization suffers from a number of significant
shortcomings. If (} = 0, only cp + \fJ is determined, while if (} = 71', only cp - \fJ is
determined, and thus at these singular points in the parametric space, cp
and \fJ no longer define a rotation matrix uniquely. Singular points arise in
any parameterization scheme for the rotation matrices. The Euler parame-
terization is made particularly inappropriate by the occurrence of the
singularity about the identity element of the group. The disastrous con-
sequences of this choice are seen by differentiating Eq. 4.62 to give the
"generators" as

GENERATION OF FINITE GROUP ELEMENTS 35
0 -1 0 0 0 1
X =x = 1 0 0 and x- 0 0 0
t/I <p e-
O 0 0 -1 0 0
These "generators" commute among themselves and do not yield the
familiar angular-momentum commutation rules found in Eq. 4.51.
An alternative and more appropriate parameterization can be obtained
by making the first rotation through an angle a 1 about the x axis followed
by a rotation a 2 about the y axis and finally a rotation a 3 about the z axis.
In this case we have
A (aI' a2, (3)
cosa2cos a 3
-cosa2slna3
- slna2
- sinal sina2 cosa3 + cosal sina3
cosal cosa3 + sinal sina2sina3
- sInal cosa2
cosal sina2 cosa3 + sinal sina3
- cosal sina2sina3 + sinal cosa2
cosalcosa2
( 4.63 )
'IT 'IT
where -'IT<a 1 <'IT, -'IT<a 2 <'lT, - 2 <a3< 2' The singular points in the
parametric space no longer occur about the identity element, but rather at
a3 = + 'IT /2. The infinitesimal group generators are now
o 0
X al = 0 0
o I
o
-1
o
o
X = 0
a2
-I
o
o
o
I
o
o
o
X = 1
a)
o
-1
o
o
o
o
o
(4.64 )

36 LOCAL PROPERTIES OF LIE GROUPS
and satisfy the well-known commutation relationship
[X,X] = f.ijkXak
( 4.65)
It follows from Eq. 4.61 that Eq. 4.63 may be equivalently written as
A (aI' a 2 , a 3 ) = exp (alX al + a2Xa2 + a 3 X (3 )
( 4.66)
In the case of the two-dimensional unimodular unitary group SU(2), the
infinitesimal group generators must be defined in terms of two-
dimensional matrices. Let us write
A ( a  a ) = eaIXl+a2X2+a3X3
I' "'2' 3
( 4.67)
where the X k are just i times the Pauli spin matrices, that is,
XI = (  ),
X 2 = ( 
- ),
X 3 = ( 
_ ) ( 4.68)
It is not difficult to show that Eq. 4.67 is equivalent to the parameteriza-
tion
A (a 1 ,a 2 ,a 3 )
=[
( cos a 1 cos a 2 + i sin a I sin a 2 ) e ia 3
cosal sina 2 + isina l cosa 2
- cos a I sin a 2 + i sin a I cos a 2 ]
(cosa l cosa2 - isina l sina2)e-ia3
( 4.69)
where - '1T  a I < '1T, - '1T  a 2 < '1T, 0  a 3  '1T. The infinitesimal genera tors
now satisfy the commutation rule
[Xi'  ] = 2f.ijk X k
( 4.70)
Had we defined the group generators as i /2 times the Pauli spin matrices,
we would have obtained a commutation rule identical to that given by Eq.
4.65 for SO(3). In this case the angles appearing in Eq. 4.69 would have
been halved.

EXERCISES
FINITE TRANSFORMATIONS 37
4.15 Show that the infinitesimal group generators of SU(n) are characterized by
traceless skew Hermitian matrices of order n.
4.16 Show that the infinitesimal generators of SO(n) are characterized by real
skew-symmetric matrices of order n.
4.17 Show that the matrices forming the elements of SU(3) can be expressed in
terms of eight independent parameters a i taking the group generators Xi as
Xl=(!
X4=(
X7=(
so that
1
o
o
o
o
o
o
o
}
}
-}
X - ( 
2- I
o
X 5 =(;
o
o
o
Xg= --..L ( 
V3
o
A ( 0: ) = e a iX,
4.12 FINITE TRANSFORMATIONS
;
-;
} X3=(
-:} X6=(
-! }
 !}
We saw in Section 4.8 that the infinitesimal transformation
o
o
! -)
F(x)F(x) +dF(x)
is effected by the infinitesimal operator
Sa = 1 + a (J X (J
In the particular case of SO(2) we may write
S(J = 1 + 9 X(J
Proceeding as before, we find for a finite rotation
S(J( 9) = e(Jx(J
( 4.46)
( 4.71 )
(4.72 )

38 LOCAL PROPERTIES OF LIE GROUPS
where now X(J is the infinitesimal operator found in Section 4.9. The
operator e(Jx(J will generate any finite transformation connected to the
identity transformation by a continuous path.
For a Lie group characterized by r infinitesimal operators Xo, the analog
of Eq. 4.72 becomes
Sa (a) = e aaxa
(a = 1, .. .,r)
(4.73 )
In the case of the modified parameterization adopted In the preVIOUS
section for SO(3) we have
S ( a flI a ) = e alX1 +a2X2+a3X3
1,U.2' 3
(4.74 )
where the infinitesimal operators Xi are as derived in Section 4.9.
In general, the generation of finite transformations proceeds by first
defining a suitable parameterization of the continuous group elements. The
choice of parameterization is not unique, but ideally should be made to
avoid singularities in the parameter space about the identity element. A
suitable choice can usually be made by first obtaining a matrix representa-
tion of the Lie group's associated Lie algebra; these matrices are then
taken as the infinitesimal group generators. The finite group elements
connected to the identity can then be produced in parameterized form by
exponentiation of the group generators as in Eq. 4.67. If the matrix form of
the infinitesimal group generators is known, then the corresponding in-
finitesimal operators may be readily found. The finite transformation
operators then follow by exponentiation of the infinitesimal operators as in
Eq. 4.73, using the same group parameters as determined in the representa-
tion of the group elements. As usual, transformations associated with the
appropriate disjoint pieces of the Lie group will be obtained from those of
the connected piece by application of the discrete operations of the group
A general study of the parameterization of the unitary and rotation
groups has been made by Murnaghan. 56 The representation of finite
transformations for U(3) has been made by Chacon and Moshinsky,57
while Holland 58 has investigated the case of SU(3) and the extension 59 to
SU(n).
Of course, once we consider finite group elements and transformations,
we are becoming concerned with the global structure of groups and must
consider the group topology, as we do later.

EXERCISE 39
EXEROSES
4.19 A general element of SU(I, I) corresponds to a unimodular unitary matrix
( pa. :.)
with a and {3 complex numbers. Show 6o ,
expressed as the product of three matrices,
(;. :.)=(ej2 e-/2)(:=;
61 that these matrices can be
sinhr/2 ) ( e;p.'/2 0 )
cosh r / 2 0 e - ;p.' /2
where -2'1T<1L, 1L'<2'1T, O<r<oo, and that for r*O every element of
SU(I, I) is obtained twice. Show that there is a singularity at r=O such that
only IL + IL' is determined. Find an alternative parameterization that avoids
the singularity about the identity element. 62
4.19 The Euclidean group in the plane, £2' relates a point (x,y) to a point (x',y')
in the plane by the transformation 18
x' = X cos () - y sin () + a
y' = x sin () + Y cos () + b
where () is an angle of rotation in the plane about the origin and a and bare
the x and y components of a translation in the plane.
1. Show that each point (x,y) in the plane may be associated with a vector
(x,y, I), which is transformed into (x',y', I) by the matrix
( cos ()
sin ()
o
- sin ()
cos()
o
)
2. Show that the infinitesimal group generators are
X9=( -I :), Xa=( : 0 } Xb=( : 0 )
0 0 0
0 0 0
and satisfy the commutation rules
[X(J,X a ] = X b , [X(J,X b ] = - Xa, [Xa,X b ] =0 ( 4.75)

5
Lie Groups and Lie Algebras
5.1 LIE ALGEBRAS
We have found that there are associated with any r-parameter Lie group
r infinitesimal operators which are characterized by their commutation
properties. We could say that the r infinitesimal operators X T span a real
r-dimensional vector space characterized by quantities T aTX T , where the
aT are real numbers. The algebra of our r- dimensional vector space is
defined by the requirement that the infinitesimal operators X T satisfy the
conditions
[X p ' Xo ] = c:OX T
(5.1 )
with c T = - c T
po op
[Xp,X p ] =0
(5.2 )
and the Jacobi identity
[Xp,[Xo,X p ]] + [Xo,[Xp,X p ]] + [Xp,[Xp,X o ]] =0 (5.3)
Under the above conditions the r infinitesimal operators X T are said to
form the Lie algebra of the corresponding Lie group. For every Lie group
there is a Lie algebra, and for every subgroup there is a subalgebra. The
study of Lie algebras and their sub algebras is of fundamental importance
in the study of Lie groups. Indeed, in many respects physicists have been
more interested in algebraic than group structures.
40

TRANSFORMATION OF BASIS 41
Formally, we may define a Lie algebra as follows: Let A be an
r-dimensional vector space over a field K in which the law of composition
for vectors is such that to each pair of vectors X and Y there corresponds a
vector Z = [X, Y] in such a way that
[aX + ,8Y,Z] =a[X,Z] +,8 [Y,Z]
(5.4 )
[X, Y] + [Y,X] =0
(5.5 )
[X,[ Y,Z]] + [Y,[Z,X]] + [Z,[X, Y]] =0
(5.6 )
for all a,,8,..., E K and all X, Y, Z,..., EA. A vector space A satisfying the
above commutator relationships will be said to constitute a Lie algebra. A
given Lie algebra is said to be real if K is the field of real numbers and
complex if K is the field of complex numbers. The Lie algebra associated
with a Lie group is always rea1. 44 In general we designate the Lie algebra
associated with a given Lie group by the same letter as for the group, but
in lowercase.
5.2 TRANSFORMATION OF BASIS
Equations 5.4 to 5.6 do not uniquely determine the infinitesimal opera-
tors of a given group. Weare still free to replace the basis Xo by another,
X , - pv
0- aoAp
(5.7)
where at is a nonsingular matrix. We now have
[ X' X' ] = C'T X'
p' 0 po '1'
= [a;Xp,a;X>J
=a;a;[Xp,XJ
_ P A " X
- apaoc pA "
''1' x , P A "X
Cpo '1' = apa(J CPA "
But X'T = aT"X" and hence
C'T a" = aPaAc"
po '1' P 0 PA
'\ -1
C'T = a P al\.c" ( a" )
po p 0 PA '1'
(5.8 )

42 LIE GROUPS AND LIE ALGEBRAS
EXERCISE
5.1 Show that the two Lie algebras
[ Xl' X 2 ] = X 3' [ X 2' X 3 ] = Xl' [ X 3' Xl] = X 2
and
[ Xl' X 2 ] = X 3' [ X 2, X 3 ] = - Xl' [ X 3' Xl] = - X 2
are related by a simple transformation of basis.
5.3 HOMOMORPIDSMS AND ISOMORPIDSMS
Let A and A I be two Lie algebras defined over a common field K. A
mappingp of A into A' is said to be a homomorphism of A into A' if p is a
linear transformation and preserves the operation of the commutator
product as follows:
p(aX + {3Y) =apX + {3pY for any (X, Y EA,a,{3 EK) (5.9 )
and
p[X, Y] = [p(X),p( Y)] for any (X, YEA) (5.10)
If the mapping is one to one, then p is said to be an isomorphism of pA
into A'. We note here that different Lie groups may have the same
structure constants and as a result the same Lie algebra, but be radically
different groups in the large. Groups possessing a common Lie algebra are
said to be locally isomorphic, that is, isomorphic in the neighborhood of the
identity. Thus the Lie algebras su(2) and so(3) are isomorphic, while the
Lie groups SU(2) and SO(3) are only locally isomorphic and certainly not
isomorphic in the large. For example, the matrix ( - 1 0 ) is a
o -1
member of the group SU(2) and is certainly not to be found near the
identity (  ). A more dramatic example of local isomorphism is to
note that the rotation group in two dimensions, SO(2), and the translation
group T(l) along a straight line both involve a single self-commuting
infinitesimal operator and thus have trivially isomorphic Lie algebras, and
yet the groups are obviously not isomorphic in the large.

LIE ALGEBRAS AND SUBALGEBRAS 43
EXERCISES
5.2 Verify that the Lie algebras associated with SU(2) and SO(3) are isomorphic.
5.3 Show that the Lie algebras associated with SO( 4) and SO(3) X SO(3) are
isomorphic.
5.4 AUTOMORPHISMS AND ENDOMORPHISMS
An isomorphism of a group G with itself is called an automorphism of G.
All automorphisms of a group G form a group known as the group of
automorphisms of G, and commonly denoted as Aut(G). The isomorphic
mapping of the group G into itself by a fixed element a of the group such
that
baba - 1
(all bEG)
(5.11 )
is known as an inner automorphism. All other automorphisms are referred
to as outer automorphisms. 63 The inner automorphisms form a subgroup of
Aut( G). An automorphism cp such that cp2 =  is termed an invo/utive
automorphism. 44 If the group G can be mapped homomorphically onto one
of its subgroups H, then the mapping is said to form an endomorphism.
EXERQSES
5.4 Prove that the set of all inner automorphisms of a group G is a subgroup of
Aut(G).
5.5 Show that the group of permutations on three objects, the symmetric group
S 3' has six inner automorphisms and no outer automorphisms.
5.6 Show that the mapping AA t, where A is an arbitrary element of SO(n),
corresponds to an involutive automorphism.
5.5 LIE ALGEBRAS AND SUBALGEBRAS
A subset Z of a Lie algebra A is called a subalgebra of A if Z is a linear
subspace of A and
[X, Y] EZ for any (X, Y EZ) (5.12)
A subalgebra Z of A is said to be Abelian if
[X, Y] =0 for any (X, Y EZ)
(5.13 )

44 LIE GROUPS AND LIE ALGEBRAS
Equation 5.12 amounts to specifying a subset of the infinitesimal operators
X'T that is closed under commutation; hence the close analogy between a
subgroup and a subalgebra. For an Abelian group the structure constants
must all vanish, that is, c:O = O. The group SO(2) is a trivial example of an
Abelian group with an associated Abelian algebra so (2). Every group
contains a trivial Abelian subgroup, since [Xp,X p ] = O.
5.6 IDEALS AND PROPER IDEALS
A subset Z of A is said to form an ideal or invariant subalgebra of A if Z
is a linear subspace of A and
[X, Y] EZ
for any (X EZ, YEA)
(5.14 )
that is,
[Xp,X o ] = c;(JX'T ( p, T E Z, 0 EA) (5.15)
If the algebra contains members that are not in the ideal, then the ideal
is said to be a proper ideal. In this case it is important to note that the
identity element is always a member of the algebra. By restricting our
attention to proper ideals, we eliminate the improper ideals formed by the
whole algebra and by the subset {O} containing the identity element.
The set of all elements X(J of an algebra A that satisfies the condition
[Xp,X(J] =0
(p, EZ, oEA)
( 5.16)
is said to form the maximal ideal or center of the algebra. It follows that
the elements of the center of a Lie algebra form an Abelian subalgebra,
and that the center commutes with all the elements of the group G.
EXERCISES
5.7 Establish that the Lie algebra sl(n, C) is an ideal of the Lie algebra gl(n, C).
5.8 Show that o(n, C) is a subalgebra of gl(n, C) but not an ideal.
5.9 Prove that if E is a sub algebra and I an ideal of a Lie algebra A, then E + 1 is
a subalgebra of A, and En I is an ideal of E.
5.7 ADJOINT REPRESENTATIONS OF LIE ALGEBRAS
Any fixed element of X of a Lie algebra A defines a linear transfonna-
tion

COMPLEX EXTENSIONS OF REAL LIE ALGEBRAS 45
ad (X): Z[X,Z]
for any (Z EA)
( 5.17)
of the Lie algebra onto itself. Consider any K EA; then
[ad( Y),ad(Z)]K = ad( Y)ad(Z)K -ad(Z )ad( Y)K
= ad( Y) [Z,K] - ad(Z) [Y,K]
= [Y,[Z,K]] - [Z,[ Y,K]]
= [ [ Y, Z ], K]
= ad ( [ Y, Z ] ) K ( 5.18 )
where we have made use of the Jacobi identity. The mapping "ad" gives a
representation of the Lie algebra known as the adjoint representation.
5.8 COMPLEX EXTENSIONS OF REAL LIE ALGEBRAS
A Lie algebra may be defined over the field of real or complex numbers.
The complex extension [R] of a real Lie algebra R is the set of all elements
of the form Z = X + iY, where X, Y E Rand i denotes the complex unit
i= v=T . Addition is defined in [R] by
Zl + Z2 = (Xl + iY I ) + (X 2 + iY 2 )
=(X t +X 2 )+i(Y I +Y 2 )
(5.19)
and multiplication by a complex number y = a + i{3 as
yZ= (aX - f3Y) + i(aY + {3X)
(5.20 )
The Lie commutator for [R] becomes
[Zl' Z2] = [Xl + iY I ,X 2 + iY 2 ]
= [X I ,X 2 ] - [ Y I ,X 2 ] + i[ Xl' Y 2 ] + i[ Y I ,X 2 ]
( 5.21 )
where the structure constants are the same as for the real Lie algebra.
We later take up the problem of the classification of all complex
semisimple Lie algebras. The corresponding classification of all real semi-
simple Lie algebras is complicated by the fact that several real Lie algebras
may have a given complex Lie algebra as their complex extension. For
example, the real Lie algebras so(3) and so(2, 1) have the same complex
extension. 6 4

46 LIE GROUPS AND LIE ALGEBRAS
EXERCISES
5.10 Verify that so(3) and so(2, 1) have the same complex extension.
5.11 Show that the complex extension of g/(n, R) is g/(n, C).
5.9 SIMPLE AND SEMI SIMPLE LIE ALGEBRAS
A Lie algebra is said to be simple if it contains no proper ideals, while
the algebra is said to be semisimple if it contains no Abelian ideals except
{O}. A simple algebra is necessarily semisimple, though the converse need
not hold. We note that simple and semisimple groups are necessarily of
more than one dimension. The concept of a semisimple Lie algebra attains
significance through the following theorem. 5 2
Theorem 5.1
A Lie algebra A is semisimple if and only if A may be written as a direct sum
A=A EB...EBA
1 n
where Ai is an ideal of A, with each ideal forming a simple Lie algebra.
Given an arbitrary Lie algebra A, it is useful to establish a criterion for
deciding if A is semisimple.
5.10 THE KILLING FORM AND CARTAN'S CRITERION FOR SEMI-
SIMPLE LIE ALGEBRAS
It is convenient to define a symmetrical tensor
gOA = gAo = C;pCtT
(5.22 )
which is known as the metric tensor or Killing form after the early work of
Killing. 65 Associated with any Lie group or its associated Lie algebra there
is a metric tensor defined in terms of its structure constants.
Cartan 33 has given a simple test for deciding if a Lie algebra is
semisimple.
1beorem 5.2
A Lie algebra A is semisimple if, and only if,
det I gOA I * 0
( 5.23 )

THE KILLING FORM AND CARTAN'S CRITERION FOR SEMISIMPLE LIE ALGEBRAS 47
The proof amounts to showing that for a nontrivial Abelian subalgebra to
exist we necessarily require
det I gOA I = 0
Suppose a Lie algebra possesses an Abelian ideal whose indices we
distinguish by attaching primes.
Then
_ T p
gOA' - COpCA'T
= c;P,Ct,'T (since ct- T = 0 if p is not contained in the sub algebra )
- C T C P'
- - p'o A'T
- C T ' C P'
-- I '\."
po AT
= 0 (since C{,'T' = 0 for an Abelian subalgebra)
Hence the row A' of the determinant of gOA' is zero, and
det I gOAl = o.
The Cartan condition given in Eq. 5.23 is just the condition that the
inverse gOA of gOA exists, and hence for semisimple algebras we may write
g OA g = 8 A
oA a
(5.24 )
Before proceeding further let us evaluate gOA for two simple cases.
Consider first the Lie algebra so(3) whose elements satisfy the commuta-
tion relations
[X I ,X 2 ] =X 3 ,
[X 2 ,X 3 ] =X I ,
[X 3 ,X I ] =X 2
We have from Eq. 5.22
g II = C [ pC f T = ci 2 c i 3 + ci 3 c i 2 = ( 1 ) ( - 1 ) + ( - 1 ) ( 1 ) = - 2
Continuing, we find
gOA = - 28 0A
( 5.25 )
and hence so(3) is semisimple and the Killing form is negative definite.
Now consider the Lie algebra so(2, I) whose elements satisfy
[Xl'X 2 ] =X 3 ,
[X 2 ,X 3 ]=-X I ,
[X 3 ,X I ] =X 2

48 LIE GROUPS AND LIE ALGEBRAS
We now find that
-2
gOA = 0
o
o 0
2 0
o 2
( 5.26)
that is,
detl gOAl = -8
and hence so (2, 1) is also semisimple.
EXERCISE
5.12 Show that the Lie algebras so(3) and so(2, 1) are both simple Lie algebras.
S.ll EXAMPLE OF 80(4)
The properties of the four-dimensional rotation group provide a good
example of some of the preceding points. The infinitesimal operators of
SO(4) may be written in terms of the variables (x,y,z,t) as 53
a a a a a a
M =z--y-' M =X--Z-' M =y--x-
1 ay az ' 2 az ax ' 3 ax ay
a a
N =x- -t-.
1 at ax '
a a
N =y--t-.
2 at ay ,
a a
N 3 =z at -t az (5.27)
from which we deduce the commutators
[M;, M.i ] = f.ijkMk' [M;, N; ] = 0, [M;, Nj ] = f.ijkNk' [N;, Nj ] = f.ijkMk (5.28)
These commutators may be thrown into a more lucid form by making the
linear transformation to a basis consisting of the terms
M.+N.
I I
J;= 2 '
M,-N,
K,= I I
I 2
( i = 1, 2, 3 )
(5.29 )
giving the simpler commutators
[J;, ] = f.ijkJk'
[K;,  ] = f.ijkKk'
[K;,]=O
(5.30 )
The operators J;,K; form the elements of the Lie algebra so (4). The
operators (J 1 ,J 2 ,J 3 ) and (K 1 ,K 2 ,K 3 ) are separately closed under commuta-
tion, each describing a sub algebra of so(4), namely that of so(3). Thus the

EXAMPLE OF E 2 49
Lie algebra so(4) is the direct sum of two so(3) Lie algebras. This splitting
of the so(4) Lie algebra into two so(3) subalgebras is directly associated
with the local isomorphism of the Lie group SO(4) with the direct product
group SO(3) X SO(3).
The triads (J t ,J 2 ,J 3 ) and (K t ,K 2 ,K 3 ) each form proper ideals in so(4),
and thus the Lie algebra so(4) is not simple. The commutation relations of
Eq. 5.30 show that these two ideals are non-Abelian and hence the so(4)
algebra is semisimple. The algebra has been divided into two parts, each of
which forms an ideal and of course a subalgebra. The two ideals them-
selves form two simple algebras whose properties can be treated separately.
5.12 EXAMPLE OF E 2
The Euclidean group in the plane is associated with the Lie algebra
defined by the commutation relations (cf. Eq. 4.75)
[X t ,X 2 ] =X 3 ,
[X t ,X 3 ] = -X 2 ,
[X 2 ,X 3 ] =0
(5.31 )
The metric tensor is readily found to be
-2 O  O  )
gOA = 0
o
(5.32 )
and is obviously singular. Thus we can immediately conclude that E 2 does
not have a semisimple Lie algebra and contains a non-trivial Abelian
subalgebra comprising the two elements X 2' X 3' This means that the Lie
algebra cannot be reduced to a direct sum of simple Lie algebras; rather,
we write the Lie algebra as a semidirect sum
E 2 = T2 fB sR2
(5.33 )
where T 2 is the Abelian ideal and R 2 the subalgebra formed by Xt. We
take up the case of semi direct sums shortly.
EXERCISE
5.13 Show that the Euclidean group in three dimensions, E 3 , does not have a
semisimple Lie algebra, and that it may be written as a semidirect sum of the
Abelian Lie algebra associated with the group of translations T3 and of the
Lie algebra so(3). Extend this result to the Euclidean group in n dimensions.

so LIE GROUPS AND LIE ALGEBRAS
5.13 DERIVATIONS OF LIE ALGEBRAS
We may construct the derived algebra A(l) of a Lie algebra A by forming
the set of all linear combinations of elements that can be expressed as
commutators of the elements of A. Formally we write
A(l)=[A,A]
( 5.34 )
Clearly A(l) forms an ideal of A.
In the case of the Lie algebra associated with the Euclidean group E 2
(cf. Eq. 5.31), the derived algebra consists of just the two elements X 2 ,X 3 .
Starting with a Lie algebra A, we may form a whole series of derived
algebras. If we write for the kth derived algebra
A(k)= [A k-l,A k-l]
(5.35).
then the series
A A (l) A (k) .. ·
, ,..., ,
( 5.36 )
is called the derived series of the Lie algebra A.
EXEROSE
5.14 Verify that 66
A ( 1) = [ g, ( n ), g, ( n ) ] = s I ( n )
5.14 SOLVABLE LIE ALGEBRAS
If for some positive integer k we have
A (k) = 0
( 5.37)
the Lie algebra A is said to be a solvable Lie algebra. For the case of £2
above we have
E(2) = [ X X ] =0
2 2' 3
(5.38 )

NILPOTENT LIE ALGEBRAS 51
and hence the Lie algebra appropriate to the Euclidean group E 2 IS
solvable.
Solvable Lie algebras have the following important property: 51,52
1beorem 5.3
If A is a solvable Lie algebra, so is every Lie subalgebra and every
homomorphic image of A.
It also follows that a solvable Lie algebra cannot contain any simple
subalgebras.
EXERCISES
5.15 Show that the Euclidean group E 3 , unlike E 2 , is not associated with a
solvable Lie algebra.
5.16 Show that the Lie algebras formed by all upper triangular matrices are
solvable.
5.15 NILPOTENT LIE ALGEBRAS
If A is a Lie algebra we may define
A 2 =A(l)= [A,A],
An=[A,An-l]
(n>l)
(5.39)
where the A n are ideals of A. The series
A >A 2 >A 3 > · · ·
( 5.40)
is called the descending central series or descending sequence of ideals. If the
series terminates for some positive integer n, then the Lie algebra A is said
to be nilpotent.
A nilpotent Lie algebra is necessarily solvable, but a solvable Lie algebra
need not be nilpotent. Thus the algebra associated with E 2 is solvable but
certainly not nilpotent. Again we may ShOW 51 ,52:
1beorem 5.4
If A is a nilpotent Lie algebra, so is every Lie subalgebra and every
homomorphic image of A.

52 LIE GROUPS AND LIE ALGEBRAS
EXERCISE
5.17 Show that the Lie algebras formed by all upper triangular matrices with
equal diagonal elements are nilpotent and solvable.
The nilpotent and solvable Lie algebras form a class of algebras apart from
the semisimple Lie algebras, and in a sense the classification of these two
classes of Lie algebras may proceed separately.
5.16 DIRECT AND SEMIDIRECT SUMS
A Lie algebra A will be said to be splittable into a direct sum of Lie
subalgebras
A =AtEBA2EB'" EBAn
if for every pair of subalgebras A;,Aj we have A; n Aj = O.
If a Lie algebra A has two subalgebras At,A2 such that
[At, At] CAt, [A 2 ,A 2 ] cA 2 , [At,A2] CAt
(5.41 )
(5.42 )
then the Lie algebra A is said to be the semidirect sum of A t and A 2'
Clearly A t is an ideal of the semidirect sum. Normally we write a
semidirect sum by first giving the ideal and then the residual subalgebra.
Thus
A = A t EB sA 2
(5.43 )
We have already noted that the Lie algebras associated with the Euclidean
group in n dimensions provide examples of semidirect sums. With the
introduction of the semidirect sum, the distinction between solvable and
semisimple Lie algebras assumes added importance via the following
theorem:
Theorem 5.5
Any Lie algebra A may be written as a semidirect sum
A=PEBsS
(5.44 )
of a solvable Lie algebra P and a semisimple Lie algebra S.
Again, the Euclidean group forms a trivial example of the above
theorem.

THE CASIMIR OPERATORS 53
5.17 ANTISYMMETRIC TENSORS
Let us define a new tensor
_ A
C ollp - gOA C llp
(5.45)
Using Eq. 5.22 we have
_ T P A_ TAp
C OIlP - COPCATC,...P - CopCIlPCAT
( 5.46 )
We now make use of the Jacobi identity for the structure constants (Eq.
4.21) to obtain
_ TAP TAp
COmP - - Cop C"",CpA - Cop C TIl CAP
_ TAP + TAP
C OIlP - CopC"".CpA CpoCTIlCAP
( 5.47)
The right-hand side is invariant under any cyclic permutation of the
indices. But since gOA is a symmetric tensor and c;" is antisymmetric in J1.
and P, it follows from Eq. 5.47 that c01JJ' is totally anti symmetric under any
interchange of its indices.
5.18 THE CASIMIR OPERATORS
Let us define a quantity
C= g poX X
p 0
( 5.48 )
where the X T are the elements of a Lie algebra A. Then
[C,X T ] = gpo[XpXO'X T ]
= gPOX p [XO,X T ] + gpoc;,XAX O
= gpoc;'XpX A + gpoc;,XAX O
- po A X X + op A X X
- g COT P A g COT A p
= gpoc;(XpX A + XAX p )
(5.49)
where we have made a change in the variables (J and p. It follows from Eq.
5.45 that for a semisimple Lie algebra
C A - g AP c
OT POT

54 LIE GROUPS AND LIE ALGEBRAS
and hence Eq. 5.49 becomes
[C,X,,] = gPOgcp(J,,(XpXA + XAX p )
(5.50 )
But cp(JT is antisymmetric, while the quantity in brackets is symmetric in p
and A, and hence the right-hand side of Eq. 5.50 must vanish to give
[C,X,,] =0
for all (X"EA)
(5.51 )
Thus the operator C has the important property of commuting with all the
elements of a semisimple Lie algebra; it is known as the Casimir operator 67
of the Lie algebra. The Casimir operator plays an important role in what
follows, since we know from Schur's lemma 5 that any operator that
commutes with all the elements, of a group must be a multiple of the
identity.
It should be noted that the Casimir operator is defined only for semi-
simple Lie algebras. This does not, however, preclude the construction of
operators that commute with all the elements of a nonsemisimple Lie
algebra. For example, in the Euclidean group E3 we have, in terms of the
angular momentum J and the linear momentum P, the nonsemisimple Lie
algebra
[J;, ] = f.ijkJk'
[P;, 1J] = 0, [P;, ] = f.ijkPk'
( i,j, k = 1, 2, 3 )
[ Pi' J; ] = 0
( 5.52)
with p2 and p.J commuting with all the components of P and J.
EXERCISES
5.18 Show that the Casimir operator for so(3) may be written as
c= - t(Xf+xi+xj)
(5.53 )
5.19 Show that the Casimir operator for so(2, 1) may be written as
c= -t(Xf-Xi-Xi)
(5.54 )
5.20 Establish that for so( 4) we may construct two Casimir operators
F= !(M 2 +N 2 ) and G=M.N
(5.55 )
( 5.56)
or
F=J 2 +K 2 and G=K2_J2
[see Section 5.11 for the relevant details on so(4)].

LIE GROUPS AND LIE ALGEBRAS 55
5.19 GENERALIZATIONS OF 1HE CASIMIR OPERATORS
Racah 8 has suggested the generalization of Casimir's commuting opera-
tor by considering the operators
I = c /3 2 C /3 3 ... c /3 I X:X IX a 2. · · X a"
n al/31 a2/32 a,,{J,.
( 5.57)
which also have the property of commuting with all the elements of the
semisimple Lie algebra. We refer to the generalized Casimir operators
when we take up the problem of state labeling.
5.20 COMPACT AND NONCOMPACT LIE ALGEBRAS
A Lie group is commonly said to be compact if its parameterization
consists of a finite number of bounded parameter domains; otherwise the
group is said to be noncompact. The Lie algebra associated with a compact
(or noncompact) Lie group is likewise said to be compact (or noncompact).
Generally, a Lie algebra A over the field of real numbers will be called
compact if its Killing form is negative definite. Obviously, compact Lie
algebras will necessarily be semisimple. For complex Lie algebras the
Killing form is indefinite, and hence all complex Lie algebras are noncom-
pact.
EXERCISES
5.21 Show that so(3) is a compact Lie algebra.
5.22 Show that so(2, 1) is a noncompact Lie algebra.
5.23 Show that the Euclidean group has a noncompact Lie algebra.
5.21 LIE GROUPS AND LIE ALGEBRAS
Much of our previous discussion of Lie algebras can be recast in terms
of the analysis of the structures of Lie groups. We now define a few
properties of Lie groups:
1. A Lie group G is said to be Abelian if all its elements commute. The
infinitesimal operators associated with the group will form an Abelian Lie
algebra.
2. A subgroup H of G is defined as a subset of transformations contained
in G which by themselves satisfy the group postulates.

56 LIE GROUPS AND LIE ALGEBRAS
3. A subgroup H is said to be an invariant subgroup if it contains all the
conjugates of its elements, that is, if whenever Sa E H is an element of the
subgroup, so is SbSaSb IS; I. In this case the associated Lie algebra will
contain an ideal or invariant subalgebra.
4. A group is said to be simple if it contains no invariant subgroup
besides the identity element. A group is said to be semisimple if it contains
no invariant Abelian subgroup besides the identity element. It follows that
simple Lie groups are necessarily semisimple, but not conversely.
5. If the infinitesimal operators X,. of a Lie group G can be decomposed
into the sum of two sets, each closed under commutation and with the
members of different sets commuting, then the elements of the groups H
and K associated with these two sub algebras commute, and the group G is
called the direct product of Hand K and is denoted by H X K.
6. If the Lie algebra associated with a given Lie group can be split only
as a semidirect sum, then the Lie group is said to be the semidirect product
of a solvable Lie group H and a semisimple Lie group K and is denoted by
HK.

6
Root Vectors and the Classical Lie
Algebras
6.1 INTRODUcnON
We have seen in the previous chapter that an arbitrary Lie algebra may
always be resolved into the semidirect sum of a solvable and a semisimple
Lie algebra. Here we take up the problem of classifying all complex simple
(and consequentially all semisimple) Lie algebras.. Our approach is largely
along the traditional Cartan-Weyl route. The classification of the complex
simple Lie algebras later provides a means of classifying all simple real Lie
algebras.
6.2 STANDARD FORM OF THE SEMISIMPLE LIE ALGEBRAS
We noted in Section 5.2 that the basis of a Lie algebra could be linearly
transformed into another basis. We now seek a standard form for the
commutators of the elements X T of a semisimple Lie algebra.
Let A be an arbitrary linear combination of the X T such that
A =aP-X
P-
( 6.1 )
57

58 ROOT VECTORS AND THE CLASSICAL LIE ALGEBRAS
Suppose X is some other linear combination such that
X=bPX
P
( 6.2 )
and
[A,X] =pX (6.3)
This equation has the form of an eigenvalue equation where p is the
corresponding eigenvalue and X the corresponding eigenvector. Writing
Eq. 6.3 in full, we have
a P-b Pc'" X = P b ". X
P-P ". ".
Since the X". are linearly independent, we have
(aP-c;, - p: )b P = 0
( 6.4 )
from which we obtain the secular equation
det la P- c ;' - p: I = 0
(6.5)
For an r-element Lie algebra, Eq. 6.5 does have not more than r roots.
However, degeneracies of the roots may exist. Cartan 33 has shown that if A
is chosen so that the secular equation has the maximum number of
different roots, then for semisimple algebras only p = 0 is degenerate. If
p = 0 is I-fold degenerate, then I is called the rank of the semisimple
algebra.
The root corresponding to p = 0 will have associated with it I linearly
independent eigenvectors H; which span an I-dimensional subspace of the
r-dimensional vector space.
Thus
[A,H;] =0
( i = 1,2, . . . ,1)
(6.6)
The eigenvectors Ea associated with the remaining r -I distinct roots will
span a (r - I)-dimensional subspace of the r-dimensional vector space, and
we have
[A,Ea] = aEa
( 6.7)
Since A commutes with H;, we may write
A ='A;H.
,
( 6.8)

PROPERTIES OF ROOTS 59
6.3 PROPERTIES OF ROOTS
We now examine some of the properties of the roots with the eventual
aim of cataloging all the possible roots for each complex semisimple Lie
algebra
Consider the commutator
[A, [H;,Ea]] = [A,H;Ea] - [A,EaH;]
= [A,H; ]Ea + H; [A,Ea] - [A,Ea]H; - Ea[A,H;]
= ex[H;,Ea]
( 6.9)
where we take into account the results of Eqs. 6.6 and 6.7. Thus if Ea is an
eigenvector associated with the eigenvalue ex, there are I eigenvectors
[H;,Ea] belonging to the same eigenvalue. But the ex are nondegenerate,
and hence the eigenvectors [H;,Ea] must each be proportional to Ea' that
IS,
[H;,Ea] =ex;Ea
c,'" = a,'"
la I a
(6.10)
( 6.11 )
Comparing Eqs. 6.7, 6.8, and 6.10, we have
ex=A;ex,
I
( i = 1, . . . ,1)
(6.12 )
The ex; may thus be regarded as the covariant components of a vector Q in
an I-dimensional space.
To continue our development of the properties of roots we use the
Jacobi identity,
[A, [Ea,Ep]] + [Ea' [Ep,A]] + [Ep, [A,Ea]] =0
( 6.13 )
Using Eqs. 6.7 and 6.13 we find that
[A, [Ea,Ep]] = (ex + 13) [Ea,Ep]
(6.14 )
which demonstrates that the eigenvector [Ea,Ep] is associated with the root
ex +13 if a+f3 is nonvanishing. If 13= -ex, then [Ea,Ep] is a linear combina-
tion of the H;:
[Ea,E-a] =c,_aH;
( 6.15 )

60 ROOT VECTORS AND THE CLASSICAL LIE ALGEBRAS
with c;f3=O for T=I=ex+f3. If ex+f3 is a nonvanishing root, then
[ E a' E f3] = N af3 Ea + f3 or c: f3 + f3 = N af3
( 6.16)
6.4 SYMMETRY OF TIlE ROOTS
Consider the metric tensor
gaT = c: TJ CT
( 6.17)
The summation is over J.L and 11 subject to the restrictions of Eqs. 6.11 and
6.15 :
_ a TJ + ,.,. -a +  a+f3 f3
gaT - C a11 C Ta C a - aCT,.,.  C a f3 C a + f3
/3=#= - a
( 6.18 )
But each term can only exist if T= -ex (see Eqs. 6.11 and 6.15)
gaT = 0
if T =1= - ex
( 6.19 )
Thus if - ex is not a root, then det I gaT I = 0 and Cartan's criterion for a
semisimple Lie algebra is not satisfied.
Theorem 6.1
For every nonvanishing root ex of a semisimple Lie algebra there is a root
-ex.
6.5 THE STANDARD FORM OBTAINED
We are free to normalize the Ea of Eq. 6.7 so that
ga -a = 1
Let us now order our basis so that
1
gik
- - 1- - - - - - - - - - -
'01
I 1 0
1
I
101
'10
( 6.20)
gciA =

THE STANDARD FORM OBTAINED 61
Then since det I gC11 =1=0, we have
det I gikl =1=0
Using Eq. 6.10 we obtain
gik =  Ci:C ka =  aia k
a a
(6.21 )
and thus gik may be used as the metric tensor of the I-dimensional space
spanned by the vectors a. We note that
C i -a= g ikc
a a -ak
= g ikc
ka-a
= gikcta
(by Eq. 6.20)
= gikak = a i
(by Eq. 6.20)
( 6.22 )
( 6.23 )
[Ea,E-a] =aiHi
where the a i are the contravariant components of the vector a. We are
now in a position to write the standard form (frequently referred to as the
Cartan-Weyl basis 33 ,34) of the commutation relations for a semisimple Lie
algebra as
[Hi,Hk] =0
(i,k= 1,...,/)
[ E a' E 13] = N al3 Ea + 13
( if a + ,8 =1= 0 )
( 6.24a )
( 6.24b )
( 6.24c )
(6.24d)
[HiEa] =aiEa
[Ea,E-a] =aiHi
The Cartan-Weyl basis is most frequently used by physicists, though as we
see later there are other bases that sometimes have advantages over that of
Cart an and Weyl.
We note that Eq. 6.24a amounts to constructing from the elements of the
semisimple Lie algebra A a commutative sub algebra, which is frequently
referred to as the Cartan subalgebra, and is the maximal Abelian sub al-
gebra in A .

61 ROOT VECTORS AND THE CLASSICAL LIE ALGEBRAS
6.6 FURTHER THEOREMS CONCERNING ROOTS
We now return to our study of the properties of roots and give the
results of a number of important and well-known theorems concerning
roots. 8,51
Theorem 6.2
If a and {3 are roots then 2(a,{3)/(a,a) is an integer and {3-2a(a,{3)/(a,a)
is also a root.
Note: The notation (a,{3) is used to indicate the scalar product of the
root vectors a and {3, that is, (a,{3) = a i {3i'
Proof: We follow Racah's proof quite closely.8 Let us suppose {3 is a root
and y is such that a + y is not a root. Then from Eq. 6.24c,
[E_a,Ey] =N -ayEy-a=E-a
[E-a,E-a] =E-2a
[E -a' E-ja] = E_ (j+ l)a
( 6.25 )
where we use a prime to indicate that we are disregarding the normaliza-
tion of the E/3. There can only be a finite number of E/3' and hence Eq
6.25 must terminate after g steps.
[E-a,E-ga] =E_(g+l)a=O
( 6.26 )
Using Eq. 6.16, we also have
[Ea,E-(j+l)a] =ILj+1E-ja
( 6.27)
To evaluate ILj+ 1 we eliminate E-u+ l)a from Eq. (6.25) and Eq. 6.27 and
use the Jacobi identity
ILj+ lE-ja = [Ea' [E -a,E-ja]]
(from Eqs. 6.25 and 6.27)
= - [E-ja' [Ea,E-a]] - [E-a' [E-ja,Ea]]
= - [E_ja,a iH i ] + ILj[ E -a,E-(j-l)a]
(using Eqs. 6.24c and 6.27)
= ai[Hi,E_ja] + JJyE-ja (using Eq. 6.25)
(6.28)

FURTHER THEOREMS CONCERNING ROOTS 63
We now obtain a recursion formula using Eqs. 6.28 and 6.24 to give
JLj+ 1 = (ex, y) - j( ex, ex) + JLj
( 6.29)
This holds for j I, since ILo is not defined by Eq. 6.25. However, if we put
ILo = 0
then Eq. 6.29 is also valid for j = O.
Using Eqs. 6.29 and 6.30, we obtain
. j(j-I)
ILj = J ( ex, y ) - 2 ( ex, ex )
It follows from Eq. 6.26 that
ILg + 1 = 0
( ) _ g(ex,ex)
ex, y - 2
j ( g - j + I) ( ex, ex)
JLj= 2
( 6.31 )
If f3 is any root, there exists some integer j;;> 0 such that y = f3 + jex is a root,
but y + ex is not a root, then owing to Eq. 6.31 we have
(a,{3)= (g- 2J l(a,a)
-that is,
2(ex,f3) .
= g - 2J
(ex, ex)
(6.32 )
This quantity is clearly integral, which proves the first part of the theorem.
Now if (ex, ex) were zero for some root ex, this root would be orthogonal to
every root, by Eq. 6.31. But as the roots span the entire I-dimensional
space, this would violate Cartan's criterion for a semisimple algebra. Thus
we can write
2(ex,y)
g=
( ex, ex )
( 6.33 )

64 ROOT VECTORS AND THE CLASSICAL LIE ALGEBRAS
and hence for every pair a and y for which a + y is not a root, there exists
a string of roots
y, y - a,..., y - ga
(6.34 )
which is invariant under reflection with respect to the hyperplane through
the origin perpendicular to the vector a. Since every f3 belongs to one of
these strings, we have that f3-2a(a,f3)/(a,a) is also a root, and thus the
proof of the theorem is complete.
Theorem 6.3
If a is a root, then the only integral multiples ka of a are ex, 0, and - ex.
This result follows directly from Eq. 6.24c, since
[Ea,Ea] =0
and thus 2a cannot be a root. Any value of Ikl> I would give rise to a
string of roots containing 2a, which cannot be a root, and hence the
statement of Theorem 6.3 follows.
Theorem 6.4
The a-string containing {3 (a, f3 =t= 0) contains at most four roots, and hence
2(a,f3)
( ) = 0, + 1, + 2, + 3
a,a
Proof: We may assume f3 =t= + a, since the a-string containing a consists
of three roots a,O, - a. Assume we have at least five roots. Relabeling
these, we may suppose that {3 - 2a, {3 - a, {3, {3 + a, f3 + 2a are roots. Then
2a=(f3+2a)-f3 and 2(f3+a)=(f3+2a)+f3 are not roots. Hence the f3
string containing f3 + 2a has just one term f3 + 2a. Therefore (f3 + 2a, {3)
= O. Similarly f3 - 2ex - {3 and f3 - 2a + f3 are not roots, so that (f3 - 2a, {3)
= O. Adding, we obtain (f3, f3) = 0, which is possibly only if f3 = O. Thus we
can have at most four roots. If the string of roots is as in Eq. 6.34, we have
(using Eq. 6.32)
2( a,f3) .
=k-J
( a, a )
with k+j+ 1<4

GRAPHICAL REPRESENTATION OF ROOT VECTORS 65
Therefore k,j  3, and thus
2(a,f3)
( ) =0, + 1, + 2, + 3
a, a
( 6.35)
6.7 CARTAN-WEYL NORMALIZATION
The determination of the N a ,f3 of Eq. 6.24c remains to be considered. We
have from Eqs. 6.25 and 6.27 for /3 = y - ja that
JLj E a + 13 = [E a' [ E - a' E a + 13 ] ]
= N _ a,a+ 13 [Ea' Ef3] (by Eq. 6.24c)
= N -a,a+f3Na,f3Ea+f3
j(g-j+I)(a,a)
N a ,/3N -a,a+/3 = J.Lj = 2
where we have made use of Eq. 6.31. This result may be written in a
slightly more convenient form by putting g = k + j to give
N N _ j(k+ l)(a,a)
a, /3 - a, a + 13 - 2
( 6.36)
where we have the string of roots
13 + ja,f3 + (j -I )a,..., 13,...,13 - ka
( 6.37)
Equation 6.36 demands that a choice of phase be made for the various
N a ,/3. A consistent choice arises from recognizing the anti symmetry of the
N a ,f3 and that the following equalities hold:
N =-N =-N =N
a, 13 13, a - a, - 13 - 13, - a
( 6.38)
6.8 GRAPIDCAL REPRESENTATION OF ROOT VECTORS
We recall Eq. 6.12:
a=Aia.
,
(;=1,...,/)
(6.12 )

66 ROOT VECTORS AND THE CLASSICAL LIE ALGEBRAS
where a is known as a root vector and contains I covariant components a;
lying in an I-dimensional weight space. We may plot out the root vectors to
form a I-dimensional root-vector diagram. Van der Waerden 7 has shown
that to each diagram there corresponds one, and only one, root-vector
system, and he was able to give a complete classification of the simple Lie
algebras.
Van der Waerden's method makes use of the results derived previously.
We recall that:
1. If a is a root vector, so is - a.
2. If a and {3 are root vectors, then 2(a,{3)/(a,a) is an integer.
3. If a and {3 are root vectors, then so is
( 6.39)
( 6.40)
f3 - 2a (a,f3)
( a, a)
( 6.41 )
The angle cp between two roots a and {3 is given by
( a, {3 )
cos cp =
v ( a,a) ( {3,{3)
( 6.42 )
or
2
2 ( a, {3 ) 0 1 1 3 1
cos cp = - or
(a,a)( {3,{3) - '4,1:'4,
using Theorems 6.2 and 6.4.
Because of Eq. 6.39 we need only consider positive angles, and thus we
are restricted to the angles
(6.43 )
cp = 0° , 30° ,45 ° , 60° , 90°
( 6.44 )
which in turn restricts the ratios of the scalar products as follows:
1. cp = 0°. This case only arises for a = {3 and is thus trivial.
2. cp=30°. Then (a,{3)/(a,a) = ! or t, and hence (a,{3)/({3,{3)= t or t,
respectively, and therefore ({3,{3)/(a,a)= t or 3.
3. cp=45°. Then (a,{3)/(a,a) = -1 or 1, and hence (a,{3)/({3,{3) = 1 or t,
respectively, and therefore ({3,{3)/(a,a)= t or 2.
4. cp=60°. Then (a,{3)/(a,a)=!, and hence (a,{3)/({3,{3) = t, and
therefore (a, a) = ( {3, {3).
5. cp=90°. Then (a,{3)=O, and hence (a,a)/({3,{3) is indeterminate.
The ratio k a /3 of the lengths of the root vectors a and {3 are given by

LIE ALGEBRAS OF RANK 2 67
Yfii a,a)
k -
af3 - ( f3, f3 )
( 6.45 )
Thus we have
cp = 30° , k = 3
cp=45°, k=2
cp=60°, k= 1
cp = 90° , k undetermined
We now have sufficient information to construct root-vector diagrams for
all the simple Lie algebras.
For 1= 1 we have from Eqs. 6.24 and 6.39 that there are just two nonzero
roots + 'a, and hence the only diagram (cp = 0°) is
-ex
o
o
o
ex
o
There is only one Lie algebra of rank 1 associated with this diagram,
namely su(2), which is isomorphic with so(3). This Lie algebra is normally
designated as AI'
6.9 LIE ALGEBRAS OF RANK 2
We now consider Lie algebras of rank 1=2. The root-vector diagrams
span a two-dimensional weight space.
A. cp=30°
Suppose a is a root vector, and coordinates of its terminus being (1,0).
Then there will be another root vector f3 of length Y3 at an angle of 30°
to a, with its terminus at (1, Y3 /2). It follows from Eq. 6.39 that - a and
-f3 will also be root vectors. Taking (a,f3)/(f3,f3)=t, we have from Eq.
6.41 that f3 - a is also a root, and of course so is a - f3. These two roots
have terminii at (t, Y3 /2) and (- t, - Y3 /2), respectively. Continuing in
this way we finally obtain the highly symmetrical "Star of David" root-
vector diagram containing 12 nonzero roots-and of course two null roots

68 ROOT VECTORS AND THE CLASSICAL LIE ALGEBRAS
(0,0), since 1=2. The Lie algebra associated with these root vectors was
designated by Cartan as G 2 .
/ \
/ \
/ \
/ \
/ \
\
\
\
\
\
/
/
/
/
/
/
G 2
B. cp=45°
Proceeding as before, we readily arrive at the figure
B 2
corresponding to Cartan's Lie algebra B 2 . There are 10 root vectors
(including the two null root vectors), which may be associated with the
root-vector scheme of the so(5) Lie algebra.

LIE ALGEBRAS OF RANK 2 69
c. cp=60°
The resulting root-vector diagram is the hexagon
\
\
\
\
/
/
/
/
A2
corresponding to Cartan's A 2 Lie algebra. There are eight root vectors
(including the two null root vectors), which may be associated with the
roots of the su (3) Lie algebra.
D. cp=45°
Two distinct diagrams arise:
D2
The first diagram is identified with Cartan's C 2 algebra, which is
isomorphic to B 2 and differs only by a rotation of the root figure through
45 0 . The algebra associated with C 2 is that of the infinitesimal operators of
the symplectic group Sp(4) in four dimensions and is written as sp(4).
The vector diagram of D 2 corresponds to a Lie algebra having six
operators. It may be developed into two sets of mutually orthogonal roots,
and thus represents the algebra of so(4), which is isomorphic to the direct
sum of two so(3) algebras.

70 ROOT VECTORS AND THE CLASSICAL LIE ALGEBRAS
6.10 LIE ALGEBRAS OF RANK I > 2
Van der Waerden 7 has shown how to extend the root-vector diagrams
for 1 = 2 to all possible complex simple Lie groups of higher rank.
(a) Bf' Let us introduce two unit vectors
e} = ( I, 0) and e 2 = (0, I )
We may now construct the diagram for B 2 from the vectors + e}, + e 2 , and
+ e} + e 2 , where the signs are arbitrary. We obtain the coordinates
( + 1,0) ; (0, + 1) ; ( 1, + 1); ( - 1, + 1 )
giving eight root vectors, which together with the two null vectors yield the
vector diagram for B 2 .
Consider the diagram for B3' We introduce three mutually orthogonal
unit vectors
e} = (1,0,0), e 2 = (0, 1,0), e 3 = (0,0, 1)
and form the root vectors + e} and + e; + e j to obtain 18 root vectors. If we
add to these the three null vectors, we have the 21 roots of B3 or (as we see
later) so(7).
Thus, in general, for Bf we take 1 mutually orthogonal unit vectors and
plot out the root vectors + e; and + e; + e j (i,j= 1,2,...,/) in an 1-
dimensional space. This gives 2/ 2 roots, which together with the 1 null
vectors represent an algebra of order 1(2/+ 1) which is in fact that of
so (21 + 1), where 1 is integral.
(b) C f . For C f we may use the same unit vectors as for B 1 , but we take the
root vectors for the diagram as
+ 2e; and + e; + e j
C f has the same order as Bf and corresponds to the Lie algebra sp(2/).
(c) Df' For 1 > 2 we take the root vectors to be
+ e + e ,
- ;- J
( i,j = 1, 2, .. . ,1 )
There are 2/(1-1) vectors, and the algebra is of order 1(2/-1). Df gives a
diagrammatical representation of the roots of the algebra so(2/).

THE EXCEPTIONAL LIE ALGEBRAS 71
(d) AI' The root-vector diagram for Al may be constructed by taking 1+ 1
mutually orthogonal unit vectors and forming all possible root vectors of
the form
e.-e.
I J
( i,j = 1, 2, . . . , I + 1 )
in an (I + 1 )-dimensional space and then projecting them onto a suitable
I-dimensional subspace. There are 1(1 + 1) vectors, which together with the I
null vectors correspond to an algebra of order 1(1 + 2), namely, that of
su(1 + 1).
6.11 THE EXCEPTIONAL LIE ALGEBRAS
The four root-vector schemes AI' B I , C I , and DI correspond to the four
classical Lie algebras su (I + 1), so (21 + 1), sp (2/), and so (2/), respectively.
Van der Waerden has shown that apart from these four root-vector
schemes there are only five other possible schemes. These correspond to
Cartan's exceptional Lie algebras G 2 , F 4 , E 6 , E 7 , and E8'
The root-vector diagrams of F 4 , E 6 , E 7 , and E8 may be constructed as
follows:
F 4 - Add to the roots of B4 the 16 root vectors
!( + e l + e 2 + e 3 + e 4 )
There are 48 root vectors in addition to the 4 null root vectors, making a
total of 52 roots, and hence the Lie algebra of F4 is of order 52. Clearly F4
contains B 4 as a subalgebra.
E6' Add to those of As the root vectors + v'2 e 7 and all the root vectors
1 e 7
!( + e l + e2 + e3 + e4 + eS + e6) + -
v'2
where in the first term we take three signs positive and three negative.
There are 72 nonnull root vectors, and the algebra is of order 78. Clearly
the Lie algebra E7 contains as a sub algebra so(6)EBsu(2).
E7' Add to the root vectors of A7 all the roots
! ( + e l + e 2 + e 3 + e 4 + e s + e 6 + e 7 + e 8 )
with four signs positive and four negative. There are 126 nonnull root

72 ROOT VECTORS AND THE CLASSICAL LIE ALGEBRAS
vectors, and hence the algebra E7 is of order 133. Clearly E7 contains su(8)
as a subalgebra.
E8. Add to the root vectors of D8 the root vectors
!( + e l + e 2 + e 3 + e 4 + e 5 + e 6 + e 7 + e S )
with an even number of positive signs. The E8 algebra is of order 248 and
contains so(16) as a subalgebra.

7
Simple Roots and Dynkin Diagrams
7.1 SIMPLE ROOTS
Van der Waerden's root diagrams give a simple portrayal of the root
vectors for groups of order 1<2, but for I > 2 a two-dimensional represen-
tation is no longer possible. Dynkin 37 has shown that all the essential
information concerning the root vectors of a given semisimple Lie algebra
may be derived from a small subset II of the total set  of the root vectors.
These privileged root vectors are referred to as simple roots. Dynkin's
approach is markedly superior to that previously outlined, as he has
further shown that the simple roots may be represented by two-
dimensional diagrams referred to as Dynkin diagrams, from which the
complete set of root vectors may be readily obtained as well as all
information concerning root lengths and angles.
A root a + is said to be positive if in some arbitrary basis its first
coordinate different from zero is positive. Thus among the eight nonnull
roots
( 1,0); ( I, 1); (0, I); ( - 1, I); ( - 1,0); ( - I, - I); (0, - I); ( I, - 1) (7.1 )
associated with B 2 , there are just four positive roots, namely,
( 1,0); ( I, 1 ); (0, 1 ) ; ( I, - I )
( 7.2 )
In general half the nonnull roots comprising the root diagram are positive.
73

74 SIMPLE ROOTS AND DYNKIN DIAGRAMS
In later work it is important to be able to evaluate the sum of the positive
roots. In the case of B 2 we have from Eq. 7.2
a+=(3,1)
(7.3 )
EXERCISE
7.1 Evaluate a+ for the exceptional Lie algebra F4 to obtain the result 68
a+=(11,5,3,1) (7.4)
We say that a root is simple if it is positive and cannot be decomposed
into the sum of two positive roots. In the case of B 2 we can write
( 1,0) = ( 1, - 1) + (0, 1) ;
( 1, 1 ) = ( 1, 0) + (0, 1 )
and hence (1,0) and (1,1) cannot be simple roots. However, no stich
decomposition can be made for the roots (0, 1) and (1, -1), and hence the
simple roots of B 2 are just
a= (0, 1) and p= (1,-1)
(7.5)
The system of simple roots is designated II. All simple roots are linearly
independent, and as a result every positive root can be represented in the
form
 kaa
aEII
(7.6 )
where the ka are nonnegative integers.
For a semisimple Lie algebra of rank I there are just I simple roots,
which form a basis for the I-dimensional space of the root vectors.
We now state two theorems that lead to a complete classification of all
the simple Lie algebras.
Theorem 7.1
A. If a and p are two simple roots, then their difference is not a simple root,
that is,
if a,p Ell, then a - p ll
B. If a,p Ell, then
2( a,p)
=-p
(a, a)
(7.7)
where p is a positive integer.

EXAMPLES OF B 2 AND B3 75
Theorem 7.2
If a and f3 are two simple roots, then the angle (}a,p is equal to 90°, 120°,
135°, or 150°. If (a,a)(f3,f3), then
1 when (}a,p = 120°
( f3,(3) 2 when (}a,p = 135°
- (7.8)
(a, a) 3 when (}a,p = 150°
undetermined when () = 90°
a,p
7.2 EXAMPLES OF B 2 AND B3
For B 2 we find that the angle between two simple roots a and f3 is given
by
( a, f3 )
cos(}a p =
, v (a,a)( f3,(3)
-1
-
vT2
(from Eq. 6.42)
() = 135°
a,p
and the ratio of the squares of their lengths is
(P,P) =2
(a, a)
Thus we can represent the simple roots of B 2 by the figure
{3
ex
where a is the short root and f3 is the long root.
For B3 we find that there are three simple roots,
a = (0,0, 1 ) , f3 = (0, I, - 1 ) , y = ( 1, - 1,0)

76 SIMPLE ROOTS AND DYNKIN DIAGRAMS
and that
-1
cos()a a =-,
,p V2
i.e. ()a,f3 = 135 0 ,
(fJ,fJ) =2
(a, a)
(y,y) =2
(a, a)
(fJ,fJ) = 1
(y,y)
cos () a, 'Y = 0
i.e. () a, 'Y = 90 0 ,
cos()p,'Y = -!
i.e. (}p,'Y = 120 0 ,
We could plot the simple root diagram for B3 in a three-dimensional
space, but Dynkin has given an easy prescription for portraying the simple
roots for any simple Lie algebra in a two-dimensional space.
EXERCISE
7.2 Establish the following results for simple roots a; and extend your results to
the exceptional groups.
IT (B[) a; = e j - e j + 1 (i= 1,...,/-1), a, = e,
IT( C,) a; = e j - e j + 1 (i= 1,...,/-1), a, = 2e,
IT(D,) aj =e; -e j + 1 (i= 1,...,/-1), a,=e'-l +e,
7.3 DYNKIN DIAGRAMS
Let us associate any simple root with a small circle on the diagram. Join
the circles by one, two, or three lines according as the angle between the
corresponding simple roots is 120 0 , 135 0 , or 150 0 . Circles corresponding to
orthogonal roots remain unjoined. Circles corresponding to simple roots of
the least length are filled, and those of greatest length left open. (N.B. For
any simple Lie algebra there exist simple roots of at most two distinct
lengths.)
Thus for B 2 we obtain the diagram
(3 a
() -

THE CARTAN MATRIX 77
and for B3
'Y
o
(3
)
ex
-
Continuing in this way, we can readily arrive at the Dynkin diagram for an
arbitrary Lie algebra B[ as
a,
o
a2
O- --- --CJ
a
-
Dynkin has shown that a unique diagram may be associated with every
simple Lie algebra, as shown in Table 7.1. No other diagrams are possible.
EXERCISE
7.3 Show that ( ) . 0 cannot be a Dynkin diagram for any Lie
algebra. (Hint: Show that the set of roots determined by this diagram contains
a positive root /3 such that 2/3 is a root.)
7.4 THE CART AN MATRIX
If II = (al,a 2 ,...,a/) is a system of simple roots, then the matrix with
elements
2 ( a., a, )
A..= I J
l} ( a;, a; )
(7.9)
is known as the Cartan matrix for the given Lie algebra. Given any Dynkin
diagram we may re,adily construct its associated Cartan matrix using
Theorems 7.1 and 7.2. The diagonal matrix elements will always be 2, and
the off-diagonal elements are restricted to the values 0, -1, - 2, and - 3.

Table 7.1. Dynkin diagrams and root structure of the classical Lie algebras
Cartan Group
Order label label
Dynkin diagram
Roots
1(1+2) AI SU(/+ 1)
o---<>----<r --<>
a1 a2 a 1
e;-e j (i,j= 1,...,1+ 1)
1(21 + 1) BI SO(21 + 1)
12
-
a1 a2 a 1
:t e; and -z. e;ej(i,j = 1,..., I)
1(21 + 1) C I
13
Sp(21)
. . . --€:::D
a1 a2 a ,
-z.2e; and -z.e;-z.ej(i,j= 1,...,1)
1(21-1) DI
14
SO(21) 0---<>--0--  a'-1 -z.e;:tej(i,j= 1,...,1)
a1 a2 aa,
14 G 2 G 2 ( ) - e; - ej(i,j = 1,2,3; i =l=J)
a1 a2 -z. 2e; =+e j =+ek( i,j,k = 1,2,3 + b,i =l=j=l=k)
52 F4 F4 0  . As for B 4 plus the 16 roots
a1 a2 a3 a4 ! ( -z. el -z. e2 -z. e 3 -z. e 4 )
78 E6 E6 a1 a2 a3 a4 a5 As for A s plus the roots :t v'2  and

!(:te l :t:te3 :te 4 :tes :teJ:t/v'2
a6 (three signs + and three - in first
fraction)
a1 a2 a3 a4 a5 a6 As for A 7 , plus the roots
133 E7 E7 
! ( -z. el -z. e2 -z. e3 -z. e4 -z. es -z. e6 :t e 7 :t e8)
a7 with four signs + and four -.
248 E8 E8 a1 a2 a3 a4 a5 a6 a7 As for D 8 , plus the roots
! ( -z. el -z. e2 -z. e3 -z. e4 -z. e s :t e6 -z. e 7 -z. e 8 )
as with the number of + signs even
78

EXAMPLES OF CARTAN MATRICES 79
7.5 EXAMPLES OF CARTAN MATRICES
(a) SU(3). The Dynkin diagram is
a1 a2
o 0
from which we deduce that
( a], a] ) = ( a 2' a 2) = 1 and (a], a 2) = - !
since the angle between the simple roots a] and a 2 is 120°. Thus the Cartan
matrix for SU(3) (or A 2 ) is
[ 2 -  ] (7.10)
-1
(b) SU(4). The Dynkin diagram is
a1 a2 a3
0 0 0
with (al'al)=(a2,a2)=(a3,a3)= 1, (a 1 ,a 2 )=(a 2 ,a 3 )= -!, and (a 1 ,a 3 )=0.
Thus the Cartan matrix for SU(4) (or A 3 ) is
2 -1 0
-1 2-1
o -1 2
(7.11)
(e) G 2 . The Dynkin diagram is
a1 a2
() -
with (a],a 1 )=3, (a 2 ,a:J= 1, and (a],a 2 )= - 1, since
-V3
cos 150° =
2
(a],a 2 )
-
V (a],a 1 )(a 2 ,a 2 )
(a],a 2 )
-
V3

Table 7.2. Scalar Products (ai,a i )
A,
1
o
a,
1 1
0 -------<>
a2 a l
2 2
o ---- ()
a2 a l-'
1
Bl
2
o
a,
-
a l
C ,
1
.
a,
1 1
. --- -
a2 al-1
2
( )
a l
1
<a l _,
1 1 1
D, 0 2 - - - a l -2 1
a,
a l
3
G 2 ( ) .
a, a2
2
2
o
a4
F4
.
a,
-
( )
a2
a3
1 1 1
E6 0 0 ! 7 0 0
a, a2 a4 as
a6
1 1 1
E7 0 0 I 3 0 0 0
a, a2 a4 as a6
a7
1 1 1 1
Es 0 0 r 3 0 0 0 0
a, a2 a4 as a6 a7
as
80

EXAMPLES OF CARTAN MATRICES 81
Thus the Cartan matrix for G 2 is
[ 2 -1 ]
-3 2
(7.12)
EXERCISES
7 .4 We may label each circle of the Dynkin diagrams given in Table 7.1 with the
scalar product (ai,a i ) which will in every case be an integer I, 2 or 3. Establish
the results shown in Table 7.2.
7.5 Use the results of Table 7.2 together with Theorems 7.1 and 7.2 to establish
the general form of the Car tan matrix for every class of semisimple Lie
algebras, as shown in Table 7.3.
Table 7.3. Cartan Matrices for the Classical Lie Algebras
2 -1 0 0 0
-1 2 -1 0 0 2 -1 0 0
AI: 0 -1 -2 0 0 F4: -1 2 -2 0
0 -1 2 -1
0 0 0 2 -I 0 0 -1 2
0 0 0 -I 2
2 -1 0 0 0
-1 2 -1 0 0
B I , C / : 0 -1 2 0 0
0 0 0 2 -2
0 0 0 -1 2

Table 7.3 ( Continued)
2 -1 0 0 0 0
-1 2 -1 0 0 0
E6: 0 -1 2 -I 0 -1
0 0 -I 2 -1 0
0 0 0 -1 2 0
0 0 -1 0 0 2
2 -I 0 0 0 0
-I 2 -1 0 0 0
0 -1 2 0 0 0
D,:
0 0 0 2 -1 -1
0 0 0 -1 2 0
0 0 0 -1 0 2
2 -1 0 0 0 0 0
-1 2 -1 0 0 0 0
0 -1 2 -I 0 0 -1
E7: 0 0 -1 2 -1 0 0
0 0 0 -I 2 -1 0
0 0 0 0 -1 2 0
0 0 -1 0 0 0 2
G 2 : [ - - ]
2 -1 0 0 0 0 0 0
-1 2 -1 0 0 0 0 0
0 -1 2 -I 0 0 0 -1
E8: 0 0 -1 2 -1 0 0 0
0 0 0 -1 2 -1 0 0
0 0 0 0 -1 2 -1 0
0 0 0 0 0 -I 2 0
0 0 -I 0 0 0 0 2
82

APPLICATION TO G 2 83
7.6 THE CARTAN MATRIX AND THE ENUMERATION OF ROOTS
The complete set of roots of a Lie algebra may be determined from the
system of simple roots II = (a I ,... ,at) and the Cartan matrix, that is, the
sequences (k I ,... ,k t ) such that aEnk;a; are roots can be determined from
the Cartan matrix. In practice we need only determine the positive roots. If
f3 = k;a; is a root, then we define its level as I f31 = lk;l. The level is
always a positive integer, and the simple roots are all of level I.
Suppose we already know the positive roots of level n, where n is a
positive integer. The positive roots of level n + I are all of the form
f3 = a + a} (a) E II). Thus we must determine for a given a > 0 of level n
the a} E II such that a + a} is a root. If a = a}, then a + a} is not a root.
Hence we may assume that a=k;a; and some k;>O for i=l=j. Then the
linear forms a - a}, a - 2a},. .. that are roots are positive and of level less
than n. Hence one knows which of these are roots. Thus the number r such
that the a} string containing a is a - ra}, . . . ,a, . . . ,a + qa} is known, and we
have
2 ( a, a) ) I
q=r- ( . .) =r- . kjA jj
a}, a} 1=1
(7.13)
Hence q can be determined from the Cartan matrix. Since a + a} is a root if
and only if q > 0, this gives a method of determining whether a + a} is, or is
not, a root.
7.7 APPLICATION TO G 2
Let us illustrate the above procedure by enumerating the roots for the
exceptional Lie algebra G2 From the Dynkin diagram
a1 () . a2
we deduced the Cartan matrix
[- -  ]
that is,
2(a I ,a 2 ) 2(a.,a 2 )
= - I and = - 3
(a.,a.) (a 2 ,a2)
Since a l - a 2 is not a root, these relations imply that the a. string contain-
ing a 2 and the a2 string containing a. are respectively
a 2 , a 2 +a I
aI' a. + a2' a. + 2a2' at + 3a 2

84 SIMPLE ROOTS AND DYNKIN DIAGRAMS
where we recall that if a 2 is a root, then so is a 2 - 2a I (a I' a 2 ) / (a I' a I)'
The only positive root of level 2 is a l + a 2 . Since a 2 + 2a l is not a root,
we deduce that the only root of level 3 is a l + 2a 2 . Since 2a l + 2a 2 is not a
root, the only positive root of level 4 is a l + 3a 2 .
Now, noting Eq. 7.13, we have
2 ( a I + 3a 2' a I )
=2-3= -1
(al,a l )
which implies that (a I + 3a2) + a l = 2a l + 3a 2 is a root. Since a l +4a 2 is not
a root, 2a l + 3a 2 is the only positive root of level 5. As (2a l + 3a 2 ) + a l
=3(a l +a 2 ) and (2a l +3a+a2=2(al +2a 2 ) are not roots, there are no
roots of level 6 or higher, and hence the non-null roots associated with G 2
are
+ aI' + a 2 , + (a l + a 2 ), + (a l + 2a 2 ), + (a l + 3a 2 ), + (2a} + 3a 2 ) (7.14)
EXERCISE
7.6 Plot out the roots of G 2 found above and show that they form the customary
"Star of David."
7.8 CONSTRUcnON OF SOME SIMPLE LIE ALGEBRAS
To close this chapter we consider the construction of the Lie algebras
so(3) and su(3).
(a) so(3). The Dynkin diagram for so(3) consists of a single circle 0, from
which we readily deduce the root figure
J- J+
. . .
-1 0 +1
If we label the simple root J + and the negative root J _, we have the
commutator algehra (cf. Eqs. 6.24a-d)
[HI,HI ] =0,
[HI,J:t ] = + J:t'
[J+,J_]=H I
To obtain a realization of the above algebra we may express HI and J:t in
terms of the infinitesimal operators Jx,Jy,Jz of so(3). If we let
HI = Jz
and
J :!c =  (Jx + iJ y )

CONSTRUCTION OF SOME SIMPLE LIE ALGEBRAS 8S
we obtain the well-known commutation algebra
[Jz,Jz] =0
[ Jz, J :t ] = + J :t'
[J+,J_]=J z
(b) su(3). The Dynkin diagram for su(3) is just
o 0
from which we deduce that there are two simple roots a and {3 of equal
length separated by an angle of 120 0 . Clearly - a and - {3 are roots, as
also are + (a + {3). These roots may be plotted out to give the well-known
diagram for A 2 :
-{3
a
-(a + (3)
a + {3
-ex
{3
We may normalize the roots by insisting that
 a.a. = 8..
 I J l}
a
(7.15)
to give
a= I (1,Y3),
2Y3
{3=  (1,- Y3),
2 3
I
a+{3= Y3 (1,0)
(7.16)
Noting Eqs. 6.24a-d, we readily deduce the commutator relations
[H\,E :!:a] = + 1 E:!:a
2Y3
[H\,E:!:,B] = + 1 E:!:,B
2Y3
[H2,E:ta] = + !E:ta
[H 2' E:t 13] = + ! E :t 13
1
[H\,E:!:(a+,B>] = Y3 E:!:(a+,B>
[H 2' E:t (a + 13) ] = 0
HI H 2
[Ea,E-a] = + _ 2
2Y3
HI
[Ep,E_p] =
2Y3
H 2
--
2

86 SIMPLE ROOTS AND DYNKIN DIAGRAMS
HI
[E a + p ,E_(a+J3)] =-
Y3
[Ea,Ea+J3] =0
1
[Ea,EJ3] = -E a + J3
V6
[EJ3,Ea+J3] =0
-1
[Ea,E-(a+,Bd = V6 E_,B
1
[E,B,E_(a+,Bd = v'6- E-a
and of course [Hi'] = O. Our particular choice of normalization has
produced a Lie algebra encumbered by awkward structure constants. In
deriving the above commutation relations we have used Eqs. 6.36-6.38 to
gIve
1
Na.,B = v'6 = - N a .- (a+,B> = N,B.- (a+,B>
EXERCISE
7.8 Construct the commutation relations for the Lie algebras B 2 , C 2 , and G 2 .

8
The Chevalley Basis
8.1 CO-WEIGHTS AND THE CHEV ALLEY BASIS
In 1955 Chevall ey 69 introduced a new basis for the Lie algebras that had
several advantages over the traditional Cartan-Weyl basis and which
resulted in the discovery of new finite simple groups, the first since
Dickson's papers of fifty years earlier. 70
The basic idea is first to associate the commuting Weyl operators Hi
(i = 1,.. ..,l), which characterize the Cartan subalgebra, with the simple
roots a i E II by writing
2a,
H= I
a; ( ai' a i )
(8.1 )
where we certainly have
[Ha;,H]=O
( i,j = 1, . . . , 1 )
(8.2 )
Any root {3 may be written as a linear combination of simple roots, viz.
I
{3=  kia i
j= 1
(8.3 )
87

88 THE CHEV ALLEY BASIS
Let us associate with any root {3 E  the element
2{3
Hp = ({3, {3) -
2 I
k.a.
( {3, {3) ; = 1 I I
 ( ai' a i )
= k, H
i = 1 I ( {3, {3 ) a,
( 8.4 )
The Hp are termed the co-weights attached to the roots {3. It is not difficult
to show that for any Dynkin diagram the coefficients k i (a i ,a i )/({3,{3) are
necessarily integers.
Chevalley then shows that for each root {3 an eigenvector Ep may be
chosen such that
[Ep,E_p] = Hp
and
[Ep,Ey] = NpyEp+y
=0
( {3+yE)
({3+y)
Furthermore,
INpyl =p + 1
(8.5 )
where p is the greatest integer a> 0 for which y - a{3 is a root. The elements
of the Lie algebra constructed in the Chevalley basis then combine
together as
[Hp,Hy] =0
[Ep,E_p] =H p
[Ep,Ey] =0
= + (p+l)E p + y
( {3, y )
[Hp,Ey] =2 ({3,{3) Ey
(8.6a)
(8.6b)
( {3 + y  ) (8.6c )
({3 + y E) (8.6d)
(8.6e)
If {3 and yare simple roots, say a i and ai' then Eq. 8.6e simplifies to
[Ha;,E]= + Ai}E
where Ai} is an element of the Cartan matrix.
(8.7)

PHASES IN THE CHEV ALLEY BASIS 89
8.2 PHASES IN THE CHEV ALLEY BASIS
The phase in Eq. 8.6d must be chosen with some care, especially since
the NaP of the Chevalley basis are not directly equivalent to those of the
Cartan-Weyl basis. The phases may be consistently fixed, and the magni-
tudes related for all NaP by use of two simple theorems.
Theorem 8.1
If a, {3, 'Y E and a + {3+ 'Y =0, then
NapHy + NpyHa + NyaHp =0
and hence
N py
..L..-- =
N ya
( a, a ) and N ya = ( p, p )
( {3, {3 ) NaP ( a + {3, a + {3 )
(8.8 )
Proof: Using the Jacobi identity for the vectors Ea,Ep,Ey, we have
[Ea' [Ep,Ey]] + [Ep, [Ey,Ea]] + [Ey, [Ea,Ep]] =0
From Eq. 8.6d we obtain
N py [Ea' E p + y] + N ya [E p, Ea + y] + Nap [Ey, Ea + p] = 0
But a + {3 + 'Y = 0; hence
NpyHa + NyaHp + NapHy = 0
Equation 8.4 together with the condition a + {3 + 'Y = 0 requires that
-2(a+{3) (a,a)Ha+({3,{3)Hp
H=H =-H = =-
y -a-p a+p (a+{3,a+{3) (a+{3,a+{3)
( a, a ) NaP H a + ( {3, {3 ) NaP H p
NapHa + NyaHp = ( {3 {3 )
a+ ,a+
and upon comparing both sides, remembering that {3 is not proportional to
a, we arrive at the result of the theorem.
Theorem 8.2
If a, {3, 'Y,  E  and a + {3 + 'Y +  = 0 with no pair of roots summing to zero,

90 THE CHEV ALLEY BASIS
then
N yp N a6 N ya N p6 N ap N Y6
+ + =0 (89)
({3+y,{3+y) (y+o:,y+o:) (o:+{3,o:+{3) ·
The proof of this theorem may be obtained in a similar manner to that of
Theorem 8.1 and is left as an exercise.
Theorems 8.1 and 8.2 together with the phase choice
Nap = - N pa = N -a,-p
(8.10)
allow us to determine completely the phases in Eq. 8.6d.
Note: The choice of phases in Eq. 8.10 is consistent with that of Weyl
and Chevalley; however, the convention NaP = -N -a,-p is also common.
EXERCISE
8.1 Prove that the product of any two elements in the Chevalley basis may be
expressed as a linear combination of the basis elements with integer
coefficients. 7 0
8.3 THE ALGEBRA su(3) IN THE CHEV ALLEY BASIS
As an example of working in the Chevalley basis let us derive the
commutator relations satisfied by the elements of the Lie algebra A 2'
From the Dynkin diagram
Ci {3
o 0
we deduce the complete set of nonnull roots as
+ 0:, + {3, + ( 0: + {3 )
with
(0:,0:) = ({3,{3) = 1
and
(o:,{3) = - t
Equation 8.6a gives the trivial commutation relations
[Ha,Ha] =0,
[Hp,Hp] =0,
[Ha,Hp] =0
(8.11 )
Since (a + {3,a + {3)= 1, we deduce that
Ha+p=Ha+Hp

EXERCISES 91
Using Eqs. 8.6d and 8.10 we arrive at
[Ea,Ep] =E(a+p)
Equation 8.8 establishes the relationships
N py =N ya =N a /3 = 1
and thus from Eq. 8.6d we have the nonzero commutators
[Ea,Ep] =E(a+fJ)' [Ea,E+(a+p)] = -E+ p , [Ep,E+(a+P)] =E+ a
(8.11 b )
Equation 8.6b gives
[Ea,E-a] =Ha'
[E p ,E_/3] =Hp,
[Ea+p,E -a-P] = Ha + Hp
(8.llc)
while Eq.. 8.6e gives
[Ha,Ea] = + 2Ea'
[H/3,E/3] = + 2E/3'
[Hp,Ea] = + Ea
(8.1Id)
[Ha,Ep] = + E/3'
and thus we have completed the construction of the commutator re-
lationships for A 2 in the Chevalley basis. Comparison with the results
obtained in Eq. 7.17 for the Cartan-Weyl basis shows the added simplicity
of the Chevalley basis.
EXERCISE
8.2 Construct the Lie algebra G 2 in the Chevalley basis using the content of the
Dynkin diagram and Theorems 8.1 and 8.2 to fix the various NaP'

9
Representations of Lie Groups and Lie
Algebras
9.1 GROUP REPRESENTATIONS
Let us first recall a few elementary notions concerning the theory of
group representations. 5 A group of linear transformations in a vector space
R(N) which is homomorphic to a given group is called an abstract represen-
tation of the group in the representation space R(N). If the representation
space R(N) is N-dimensional, then a matrix representation of the group is a
set of N X N matrices (which themselves constitute a group) onto which the
group to be represented is homomorphic. Thus we may assign a matrix
D(A) to each group element A in such a way that
D(A)D(B) =D(AB)
(9.1 )
for all matrices D.
A representation of an r-parameter Lie group will be determined if we
have r matrices, Dx, such that
[DA,Do] =c{oD"
( 9.2 )
where the c{o are the structure constants of the associated Lie algebra. A
representation of a Lie algebra amounts to a homomorphism of the given
Lie algebra onto a Lie algebra of linear transformations on the vector
space R (N) with the commutation multiplication.
92

REAL AND COMPLEX REPRESENTATIONS 93
If all the matrices assigned to different group elements are different,
then the matrix group is isomorphic to the group it represents and the
representation is said to be faithful. Every Lie algebra has a faithful
representation of finite degree. 7 } The number of rows or columns in a
regular representation matrix is termed the dimension or degree of the
represen ta tion.
We call two representations D(A) and E(A) equivalent if there is a
constant matrix X such that
XD(A)X-}=E(A)
(9.3)
A representation is termed irreducible if there exists no invariant sub-
spaces of R(N) apart from the identity. A representation is termed reducible
if it leaves invariant a subspace R(N 1 ) of R(N). In this case the representa-
tion matrices are equivalent to matrices of the form
( A ( O N I) B )
A (N 2 )
(N=N 1 +N 2 )
(9.4 )
A representation is said to be fully reducible if it can be expressed as a
direct sum of irreducible subrepresentations. In this case the representation
matrix may be transformed into a block-diagonal form. A representation
that is reducible but not fully reducible is called an indecomposable repre-
sentation and cannot be reduced to block-diagonal form or expressed as a
direct sum of irreducible representations.
9.2 REAL AND COMPLEX REPRESENTATIONS
The complex conjugate D* of D is also a representation of a group G. If
D is irreducible and unitary, then so is D*. However, the irreducible
unitary representations D and D* may not be equivalent. 5 If they are not
equivalent, then D is complex. If they are equivalent, then from Eq. 9.3 we
have
D=CD*C- l
(9.5)
and the unitary matrix C is either symmetric or antisymmetric. If C is
symmetric, then a transformation matrix U can be found such that the
representation p = UDU- l is real, that is, p = p*. If C is antisymmetric,
then no matrix U has the above properties. However, we can find a matrix
U such that the representation p = UDU- l satisfies the condition
Zp=p*Z
(9.6)

94 REPRESENTATIONS OF LIE GROUPS AND LIE ALGEBRAS
where Z is a real anti symmetric unitary matrix having nonzero elements
only in the super- and subdiagonals:
0 -1 0 0
1 0 0 0
Z= 0 0 0 -1
0 0 1 0
(9.7)
When D and D* are equivalent, D=CD*C- I ; then D is said to be real
and of positive sign if C is symmetric (i.e., C = C t), and of negative sign if C
is antisymmetric (i.e., C= -C t ). Mehta 72 ,73 and Bose and Patera 74 have
given a systematic classification for all the irreducible representations of
the simple Lie groups.
9.3 CONTRAGREDIENT REPRESENTATIONS
To every representation AD(A) there is a conjugate (or contragredient)
representation defined by the mapping
-I
A[Dt(A)]
(9.8 )
where Dt(A) is the transpose complex conjugate of the matrix D(A). This
representation is sometimes also referred to as a star representation,45 being
designated as D(A)*. Clearly (D(A)*)* = D(A). To avoid confusion we
reserve the asterisk * for complex conjugation.
EXERCISES
9.1 Prove that the above definition of a contragredient representation satisfies the
group multiplication rules.
9.2 Prove that the complex conjugate of a representation D(A) is also a representa-
tion.
9.4 ADJOINT REPRESENTATIONS
We noted in Section 5.7 that the linear mapping
ad(X) :Z[X,Z]
(Z EA)
( 5.17)

UNITARY AND NONUNITARY REPRESENTATIONS 95
of a Lie algebra A onto itself gives a representation known as the adjoint
representation. The Killing form defined earlier in Eq. 5.22 may be reexpres-
sed in terms of the adjoint representations as the trace of the linear
transformation
B(X,X) =Tr{ ad(X o )ad(X A )}
(Xo,X A EA)
- c T c P
- op AT
(9.9)
EXERCISE
9.3 Show that 75 for X, Y,Z EA
B( [X, Y],Z) + B( Y,[X,Z]) =0
9.5 UNITARY AND NONUNITARY REPRESENTATIONS
The representations of continuous Lie groups enjoy a wider range of
diversity than those of finite groups. In the case of finite groups we need
only consider finite-dimensional unitary or antiunitary representations.
Antiunitary representations can arise only for finite groups. In general, the
representations of a Lie group, or of its associated Lie algebra, may be of
finite or infinite dimension, discrete or continuous, unitary or nonunitary,
fully reducible or indecomposable. These differences will be seen with
added clarity when we take up the specific case of the representations of
the groups 80(3) and 80(2,1). Here we simply note a few general
properties of the representations of Lie groups.
In the case of solvable Lie groups we have the important theorem:
Theorem 9.1
Every finite-dimensional irreducible representation of a solvable group is
one-dimensional.
Thus we may immediately conclude that all the finite-dimensional irreduc-
ible representations of the Euclidean group E 2 are one-dimensional.
If we restrict our attention to unitary representations of simple Lie
groups, we find for the case of compact Lie groups:
Theorem 9.2
The irreducible unitary representations of a connected simple compact group
are all finite-dimensional.

96 REPRESENTATIONS OF LIE GROUPS AND LIE ALGEBRAS
We note that the finite-dimensional unitary representations of any group
are fully reducible and discrete. All other representations of a connected
simple compact Lie group are necessarily of infinite dimension and nonun-
itary, and may be indecomposable and discrete or continuous.
In the case of simple noncompact Lie groups we find
Theorem 9.3
The irreducible unitary representations of a connected simple noncompact Lie
group are, apart from the trivial one-dimensional representation, of infinite
dimension.
It follows that the finite representations of a connected simple noncompact
Lie group are necessarily nonunitary and hence may be indecomposable.
Theorem 9.3 must be applied with caution to semisimple noncompact Lie
groups, since clearly in direct-product groups the product of the one-
dimensional trivial representation of the simple noncompact group with a
finite-dimensional unitary representation of a simple compact group yields
a finite-dimensional unitary representation. This is trivially apparent in the
direct-product group 80(2, l)@ 80(3).
Before proceeding further, we review the general problem of the labeling
of the irreducible representations of semisimple Lie groups.

10
Weights and the Labeling of Irreducible
Representations
10.1 WEIGHTS AND WEIGHT SPACES
The concepts of weight and weight space play an important role in the
theory and application of semisimple Lie groups-compact and noncom-
pact. In this chapter we limit our attention to the finite-dimensional
unitary representations of compact semisimple Lie groups.
A given Lie group will have associated with it an infinity of representa-
tions, and it is necessary to develop a systematic procedure for describing
and distinguishing these various representations.
We recall that a linear representation of a Lie algebra g is a
homomorphism of g into the Lie algebra of all linear transformations of
the N-dimensional space R(N). There is a one-to-one correspondence
between this representation and the corresponding Lie group. We desig-
nate the space in which a representation <p operates by l\p, and the
dimension of l\p by N(cp). The representation matrices may be constructed
to satisfy either the Cartan-Weyl or the Chevalley basis. Thus for an
N-dimensional representation of a group of rank 1 we have 1 self-
commuting N X N matrices Bel.; (i = 1,... ,1) with a; E II and a further set of
N X N matrices Ea with a E, all of which satisfy the same conditions as
either the operators Bel.; and Ea in the Chevalley basis defined by Eqs.
8.6a-e or the corresponding ones in the Cartan-Weyl basis defined by Eqs.
6.24a-d.
97

98 WEIGHTS AND THE LABELING OF IRREDUCIBLE REPRESENTATIONS
Since the I matrices Ha; are self-commuting, we may construct a set of
eigenvectors Iu) that are simultaneous eigenvectors for the I matrices in the
space Rcp such that
Ha;lu)=Ailu)
(i=I,...,/)
( 10.1 )
The set of I eigenvectors Ai form the covariant components of a weight
vector A in an I-dimensional weight space cp' The vector A is referred to as
the weight of the eigenket lu).
If lu) is a vector of weight A, then Eplu) is of weight A + {3. (N.B. {3 is
actually an I-component root vector.) This may be seen by the following
argument:
Ha,Eplu A ) = ([ Ha;,Ep] + EpHa, )lu A )
2 ( lXi' {3 )
= (ai' a;) +Aa, EpIUA)
= (Aa; + {3i )Eplu A )
Writing Ha = (Ha;,...,Ha) we have
HaEplu A ) = (A + {3 )Eplu A )
(10.2 )
Since the Ha,'s commute, we also conclude that Halu A ) belongs to the
weight A.
The representation space Rcp may be decomposed into the direct sum of
weight subspaces R such that
l\p=  R
A E cp
( 10.3)
Every vector of R is called a vector of weight A, and from Eq. 10.2 we
have
EplA)ER+P
if A + {3 E cp
( 10.4 )
=0
if A + {3 flcp
A weight A is said to be positive if its first nonvanishing component is
positive, and a weight is said to be higher than another if the difference is
positive. A weight A such that A > M for any other weight M is said to be
the highest weight in cpo

THEOREMS CONCERNING WEIGHTS 99
Let us choose in  a basis t,... 'N of weight vectors of the representa-
tion cp, and let Ai be the weight of the vector i' The basis t'... 'N will be
termed canonical if the weights Ai are arranged so that
AtA2'" AN
( 10.5)
Let a be a root. Then a sequence of weights Mt, M 2"" ,M k such that
M 2 -M I =M 3 -M 2 =... =Mk-Mk-t=a
( 10.6)
and M I - a,M k + a f!.cp is called an a series of weights. If the weights of a
given a series are arranged in an arithmetic progression with a first term
A - ra and last term A + qa, then
2( A, a)
=r-q
(a, a)
( 10.7)
The proof of this statement closely parallels that of Theorem 6.4 and will
not be givenS, 76.
10.2 THEOREMS CONCERNING WEIGHTS
We now consider a number of theorems concerning weights that will be
relevant in developing the theory of representations.
Theorem 10.1
For any weight A and root a, 2(A,a)/(a,a) is an integer and A-2a(A,a)
/(a,a) is a weight.
The proof of this theorem parallels that of Theorem 6.4 and is omitted.
Theorem 10.2
Every representation cp has at least one weight.
Proof: The matrix HI has at least one eigenvalue At; let RI be the
subspace of  spanned by the eigenvectors of Ht belonging to AI' Since
H I H 2 I u A) = H 2HtluA)=AtH2IuA)' it follows that H2Rt = RI and H 2 has at
least one eigenvector in its invariant subspace Rt. Continuing the process,
which is possible because every matrix has at least one eigenvector in every
invariant subspace, we arrive at the subspace R which consists of the
simultaneous eigenvectors of Hi (i = 1,... ,I) corresponding to the weight
A=(AI,...,A[).

100 WEIGHTS AND THE LABELING OF IRREDUCIBLE REPRESENTATIONS
Theorem 10.3
A vector I u> of weight A that is a linear combination of vectors I Uk> of weight
A (k), all different from A, must vanish.
As a consequence vectors with different weights are linearly indepen-
dent, and hence there are at most N different weights associated with a
representation cp of dimension N(cp). We note that distinct vectors having
the same weight are not precluded. The number of times a given weight
occurs in a representation <p is referred to as its multiplicity. If a weight
belongs to just one vector, it is said to be simple.
10.3 THE WEYL REFLECTION GROUP
The weights A and A-2a(A,a)/(a,a) have the same multiplicity. For
every root vector a there is a corresponding transformation in the weight
space
2a(A,a)
Sa:AA/=A- ( )
a, a
( 10.8)
Weyl34 showed that these linear transformations have the form of an affine
reflection which leaves fixed every vector in the hyperplane orthogonal to a
and sends a into - a. These reflections Sa' a a root, generate a finite group
of linear transformations nowadays called the Weyl reflection group.
Weights that are related by a reflection, or a product of reflections, are
said to be equivalent weights. The Weyl reflection group is generated by
the I elements Sa; (i = 1,... ,I), where a I ,... ,a[ are the I simple roots of the
group, together with the identity element . Clearly the same Weyl
reflection group holds for both roots and weights.
Coxeter and Moser 77 have shown that the vector diagrams of all the
semisimple Lie groups are closely related to finite symmetry groups de-
fined by reflections. Lezu0 78 and Alisauskas and Jucys 79 have shown that
the Weyl reflection group plays a fundamental role in yielding re-
lationships between Wigner coefficients for weights that are equivalent
under Weyl reflections.
EXERCISE
10.1. Establish that the elements of the Weyl group for A 2 are
, Sa' S/3, Sa S /3, S/3Sa, Sa S /3Sa
(N.B. S/3SaSp = SaSpSa') See Konuma et a1. so for further examples.

WEIGHTS AND THE CLASSIFICATION OF IRREDUCIBLE REPRESENTATIONS 101
10.4 WEIGHTS AND THE CLASSIFICATION OF IRREDUCIBLE REPRE-
SENT A TIONS
We first state two fundamental theorems: 8
Theorem 10.4
If a representation is irreducible, its highest weight is simple.
Theorem 10.5
Two irreducible representations <PI and CP2 are equivalent if their highest
weights are equal.
Dynkin 35 - 37 has used these two theorems to establish the following key
theorem:
Theorem 10.6
For A to be the greatest weight of some irreducible representation cp of G it is
necessary and sufficient that all the numbers
2( A, a)
A=
a ( a, a)
(a ElI)
( 10.9)
be nonnegative integers. If  is a greatest weight vector of cp and a Ell, then
Ea=FO
( k < Aa )
(10.10)
=0 (k)Aa)
Thus we may designate any irreducible representation cp of a Lie group
by associating the nonnegative integers Aa with the circles of the Dynkin
diagram associated with the corresponding simple roots. Since there are
just I simple roots, there can only be I components Aa of A. The greatest
weight equal to zero corresponds to an irreducible representation of G in a
one-dimensional space.
Examples
A 2 : Typical designated representations of SU(3) are
o
a,
1 1
o and 0
a2 a,
o
a2
(a)
(b)

102 WEIGHTS AND THE LABELING OF IRREDUCIBLE REPRESENTATIONS
Consider (a) first. We have from Eq. 10.9
2(A, a}) 2(A, a 2 )
A = = 0 and A = = I
at (a},a}) a2 (a 2 ,a 2 )
Writing A=aa} +ba2 and recalling that (a},a})=(a 2 ,a2)= I and (a},a 2 )
= - t, we find the greatest weight is
a} + 2a2
A=
3
Continuing in the same way we find for (b) a greatest weight of
2a} + a 2
A=
3
[Note The collection of integers (A},... ,AI) associated with the greatest
weight will not necessarily coincide with those commonly given by Weyl
and will depend on the order of labeling of the simple roots.]
EXERCISES
10.2 Show that the greatest weight of the irreducible representation
of SU(3) is (13a} + Ila2)/3.
5 3
o 0
10.3 If the simple roots of G 2 are designated as a land a2, show that the greatest
weight of the irreducible representation
1
(J _
is al + 2a2'
10.5 COMPUTATION OF THE COMPLETE SET OF WEIGHTS
In some applications it is desirable to be able to calculate systematically
the complete set of weights M of a given representation cp starting from a
knowledge of the expansion of the greatest weight A in terms of the simple
roots.
Let A be the greatest weight and M an arbitrary weight of cpo We now
adopt the notation
(A) =2  Aa
aEII
( 10.11 )

COMPUTATION OF THE COMPLETE SET OF WEIGHTS 103
and write
y(M) = ![(A) -(M)]
( 10.12)
It follows from Theorem 10.1 that Y(M) is always an integer and corres-
ponds to the number of simple roots that have to be subtracted from A in
order to obtain M.
Following Dynkin 3tJ we denote by Ll; the collection of all weights M
for which y(M) = k, and call the subsystems Ll: layers. Obviously
Ll =Ll o ULl 1 ULl 2 U"'ULlT ( 10.13 )
    
The number T= T(cp) gives the number of layers minus one and is termed
the height of the representation cpo The greatest weight A belongs to the
uppermost layer Ll, and the least weight A' to the lowest layer Ll, that is,
y(A) =0 and y(A') = T( cp)
( 10.14 )
Let Sk(CP) denote the multiplicity of the weights belonging to the kth
layer. Clearly
So (cp) + Sl (cp) + · · · + ST( cp) = N( cp)
( 10.15)
The number
III( cp) = maxS k ( cp)
(10.16)
is called the width of the representation.
For the greatest and least weights (A and A', respectively), we have
(A) + (A') =0
(10.17)
and from Eqs. 10.11 and 10.13 we obtain
(A)-(A')=2T(cp)
(10.18)
and thus
 (A) = T( cp )
(10.19)
and
(M) = T( cp) -2y(M)
( 10.20)
Thus, for all the weights M of an irreducible representation cp the
numbers (M) are congruent (mod2). When they are all odd the represen-
tation cp is said to be of odd type, and when even of even type. The evenness
or oddness of a representation cp coincides with that of T( cp).
The formulation of the necessary machinery to compute the complete

104 WEIGHTS AND THE LABELING OF IRREDUCIBLE REPRESENTATIONS
set of weights for a given representation cp from its greatest weight A is
concluded by the following two theorems due to Dynkin. 37
Theorem 10.7
If A is the greatest weight of an irreducible representation cp, then
T( cP ) =  raAa
aEII
(10.21)
Dynkin has given the values of r for all the simple Lie algebras which we
collect together in Table 10.1.
Theorem 10.8
The system Llcp of weights of an irreducible representation cp is spindle-shaped,
that is,
Sk=ST-k and Sr>Sr-I'>'" >S2,>SI
( 10.22)
where
T
r=2+ 1 , Sk=Sk(CP), and T=T(cp).
It follows that if T( cp) = 2r, then
III ( cp ) = Sr ( cp )
( 10.23 )
while if T(cp)=2r+ I, then
III ( cp ) = Sr ( cp ) = Sr+ 1 ( cp )
(10.24 )
We now have sufficient information at our disposal to calculate the
weights M from the highest weight A associated with a representation cpo
We do this by systematic calculation layer by layer. If we already have the
weights for Ll, Ll,... ,Ll-I, then we can find the weights of Ll by finding
for an arbitrary weight M of Ll-I all the simple roots a for which
M-aELlcp' It follows from Eq. 10.7 that M-aELlcp if
Ma+q,>O
( 10.25)
where
M + ka ELlcp
forq>k
f!. Llcp
forq=k-I
( 10.26)

EXAMPLES OF COMPUTATIONS OF WEIGHTS 105
Table 10.1 Values of r for use with Eq. 10.20.
An
Bn
C n
o (n-2) (n+3)
 n2
(n-1) (n+1)
(n-2) (n+2)
l:-1)2
!(n-2)3
" n(n+1 )/2
y (n-1) (n+2)
.
.
.
.
.
.
.
.
.
o (n-k-1 )k
o (n-k+ 1) (n+k)
. (n-k+ 1) (n+k-1)
.
.
.
.
.
.
.
.
.
1:- 1 )2
?(2n-1 )/2
6 2n
I (2n-2)2
2n-1
Dn
G2
n(n-1 )/2
n(n-1 )/2
10 ( ]
- 6
.
F4
.
.
o (n-k+1) (n+k-2)
.
22 42
o ()
30
16
.
.
.
9 (2n-3)2
6 (2n-3)2
E6
16 30 42 30 16
0 0 1 22 0 0
E7
E8
34 66 96 75 52 27 92 182 270 220 168 114 58
49
All the elements M + ka that are weights belong to one of the layers
, , . . . ,- I, and hence the number q is known for each a series.
The total number of weights N ( cp) coincides with the degree or dimen-
sion of the representation cpo
10.6 EXAMPLES OF COMPUTATIONS OF WEIGHTS
Let us first construct the set of weights associated with the representa-
tion

106 WEIGHTS AND THE LABELING OF IRREDUCIBLE REPRESENTATIONS
1
o 0
of SU(3). ex (3
The greatest weight is A=(a+2f3)/3 and belongs to the uppermost
layer Ll. From Eq. 10.21 and Table 10.1,
T( cp) =2Aa +2A{J =2
Hence there are three layers, and the maximum width is
III ( cp ) = S I ( cP )
Now we investigate if A - a is a weight. From Eq. 10.26 we deduce that
A - a E Llcp
forq> -I
f£. Llcp
forq= -2
But Aa = ° and hence Aa + q < ° and thus A - a is not a weight.
Now try A - f3:
A - {3 ELlcp
forq> -}
f£. Llcp
forq= -2
and since A{J=I and hence A{J+q>O for q=-I, we conclude that
M = A - {3 = (a - {3)/3 is a weight belonging to the layer Ll and that
SI(CP) = I.
M - a = - (2a + {3) 13 is the weight belonging to the lowest layer Ll; since
M - a E Llcp
forq> -I
f£. Llcp
forq= -2
2(a - {3,a)
M = =1
a 3(a,a)
and
Ma+q>O.
Thus the 0
1
o representation contains the weights

EXERCISES 107
e (ex +2{3)/3
.(ex-{3)/3
e-(2ex+{3)/3
o
cp
l
cp
2
cp
Since there are three weights, we conclude that the representation IS
three-dimensional.
EXERCISES
10.4 Establish the weight pattern for the
2
o 0
representation of SU(3),
-(2a-2{3)/3.
.(4a+2{3)/3
.(a+2{3)/3
.(a-{3)/3
.-(2a+{3)/3
.- (2a+4{3)/3
10.5. Establish the weight pattern for the
1
o 0
representation of SU(3),
.a+{3
{3. ea
oe eo
-{3. .-a
.-a-{3
[N.B. There are two null weights (0,0). One comes from the weight {3 under a
Weyl reflection, and the other from the weight a.]

108 WEIGHTS AND THE LABELING OF IRREDUCIBLE REPRESENTATIONS
10.6. Establish the weight pattern for the
2
o 0
representation
e(2a+413)/3
e(2a + 13) /3
2(a-I3)/3e e( -a+I3)/3
e- (a+213)/3
e- (4a+213)/3

11
Kronecker Products
11.1 DEFINITION
The Kronecker products (or direct products) of the irreducible repre-
sentations of groups playa very important role in the development of the
representation theory of groups and in practical applications of group
theory.
Let R', R", and R be three linear spaces of dimensions m, n, and mn,
respectively. We say that R = R' X R" is the Kronecker product of R' and
R" if to every pair of vectors  E R' and 11 E R" there corresponds a vector
 ER (= X 11) such that
(a) The operation  X 11 is linear in each argument.
(b) The space R is spanned by all the vectors of the form  X 11. Thus, if
1"" 'm is a basis of R' and 111'"'' 11n a basis of R", then the vectors i X 11k
(i . I,...,m; k= I,...,n) form a basis of R'xR".
Let a{ be a matrix A with respect to some basis 1"" 'm' and bi a matrix
B with respect to some basis 111,"',11n' Then the matrix C=A xB with
respect to the basis i X 11k has the form
C jl = a j b l
ik i k
(11.1)
The Kronecker product of linear transformations has the following
properties:
109

110 KRONECKER PRODUCTS
(i)
(A IA 2) X (B I B 2 ) = (A I X B I ) (A 2 X B 2 )
( 11.2 )
(ii)
( iii )
(AXB)-I=A-IXB- I
'X"=
( 11.3 )
( 11.4 )
(iv) If R' is a subspace invariant under A, and R" a subspace invariant
under B, then R' X R" is invariant under A X B.
11.2 KRONECKER PRODUCT OF REPRESENTATIONS
Let cp' and cp" be representations of a group G, then the representation cp
defined by
cp = cp' X cp"
. ( 11.5)
is called the Kronecker product of cp' and cp" and will be denoted by
cp' X cp". The following properties may be readily established:
(i) If CP'I CP'2 and cp'{ cp', then CP'I X cp'{ CP'2 X cp'.
(ii) cp' X cp" cp" X cp'
(iii) ( cp' X cp") X cp'" cp' X ( cp" X cp"')
(iv) ( «p') -+- «P'2 ) X «p" -«p') X «p" -+- «P'2 X «p"
The representation cp formed in Eq. 11.5 is normally reducible, that is,
, X "  r II'
cp = cp cp =  cp' cp" cp' "cp
( 11.6)
where r cp' cp" cp' I I is the number of times cp'" occurs in the decomposition of
the Kronecker product cp' X cp" into the direct sum of irreducible represen-
tations of G. The problem of determining the decomposition of the
Kronecker product is of fundamental importance.
11.3 THE WEIGHT SPACE FOR KRONECKER PRODUcrS
Let cp' and cp" be representations of a semisimple Lie algebra and let
Rcpl=  RI and ,,=  R;t
A ELlcp' M ELlcp"

DECOMPOSITION OF THE KRONECKER PRODUcr 111
be the decompositions of Rcp' and " into weight subspaces. Then the
decomposition of the space , X Rcp" in which cp' X cp" operates into weight
subspaces is given by
Rcp' x,,=   XR:
A E Llcp' M E LlcpH
(11.7)
The weight subspace R, X R: corresponds to the weight A + M in the
representation cp' X cp", and thus
Llcp' x cp" = Llcp' + Llcp"
( 11.8)
-that is, an arithmetical sum where every element of Llcp' is added to every
element of Llcp'"
It is easily seen that if A and M are the greatest weights of cp" and cp",
then A + M is the greatest weight of cp' X cp", and that
S k ( cp' X cp") =  Sj ( cp') S k - j ( cp" )
i
If T( cp") > T( cp'), then
( 11.9)
III (cp' X cp") > III ( cp") + T( cp')
(11.10)
11.4 DECOMPOSITION OF THE KRONECKER PRODUCT
In principle a knowledge of the weights of the representations of a group
G together with Eqs. 11.7-11.10 gives sufficient information to resolve any
Kronecker product. As an example consider the decomposition of the
Kronecker product 0 b x 0 6 for SU(3). We have from
Eq. 11.8
(a+2fi)/3.
(a-fi)/3.
- (2a + fi) 13.
= (2a + fi)/3.
- (a - fi) 13.
- (a+2fi)/3.
.(a+2fi)/3
X .(a-fi)/3
.-(2a+fi)/3
.(2a +4fi) 13
.(2a+fi)/3
.(2a-2fi)/3 .-(a-fi)/3
. - (a + 2fi) 13
.- (4a+2fi)/3

112 KRONECKER PRODUCTS
The greatest weight is (2a+4fi)/3, and hence by Eq. 10.9 we deduce
that the 0  representation is contained in the Kronecker product.
Removing the weights associated with this representation leaves the residue
of weights
e(2a+p)/3
e-(a-p)/3
e- (a+2fi)/3
which corresponds to the representation with greatest weight (20: + P)/3,
and hence the Kronecker product contains the representation 6 0
and we have
o
1
o x 0
1
o = 0
2 1
o + 0
o
The evaluation of Kronecker products by the method given above
suffers from the need to construct the complete set of weights for several
representations. For representations of large dimension this becomes a
formidable task even with the use of auxiliary theorems. In these cases
alternative methods exist, and extensive tables have been published. 31,32
Unfortunately, no general analytic technique seems to be known that
readily leads to the determination of the multiplicity of a given representa-
tion for a particular Kronecker product. The problem would seem ulti-
mately to resolve itself as a problem in combinatorial theory.
EXERCISES
11.1. Establish the following Kronecker-product decompositions:
(i)
1 1 2 1
0 0 x 0 0 = 0 0 + 0 0
(ii)
1 2 1 2 1
0 0 x 0 0 = 0 0 + 0 0
(iii) 1 1 2 1 1
( ) - x ( -- = ( ) - + ( ) - + ( ) - + (J -

12
Representations, Weights, and Labeling
12.1 BASIC REPRESENTATIONS
We have seen that in principle we may construct from two irreducible
representations cp' and cp" with greatest weights A and M a third irreduc-
ible representation cp' X cp" with a greatest weight A + M. Irreducible repre-
sentations that cannot be built up in this way are referred to as basic
representations and are characterized by the fact that their greatest weight
cannot be factorized. Clearly a representation will be basic only if all the
Aa, (i = 1,... ,I) are zero except for one having the numerical value 1. Thus
for a given group G of rank I there will be just I basic representations. For
example, the basic representations of C 3 would be
1
o
, ]
- , 0
1
, )
- , and 0
{ 1
1
-
and those of D 5 would be
113

114 REPRESENTATIONS, WEIGHTS, AND LABELING
and
12.2 KRONECKER POWERS
The previous section precludes the possibility of constructing basic
representations from the representation of greatest weight derived from the
Kronecker product of a representation. We can, however, sometimes
construct them by symmetrizing the Kronecker square of the basic repre-
sentation 6 0 into its symmetric and antisymmetric parts 81 to give
and
1 1 (){2J 1 112}
0 0 x 0 0 + ( 0--0)
1 0 ) { 2 t 2
( 0 = 0 0
1 11 2 1 1
( 0 0 ) = 0 0
where
We note that in the above example the second basic representation
o b occurs as the highest weight in the antisymmetric part of
the Kronecker square of the first basic representation 6 0 . This sug-
gests that the symmetrization of Kronecker powers could be a profitable
line of investigation.
Let us first examine the above example for A 2 in some detail. The basic
representation 6 0 is three-dimensional and may be regarded as
being spanned by three basis vectors 1'2'3 associated with the weights
(2a+f3)/3,-(a-f3)/3,-(a+2f3)/3, respectively. The construction of the
Kronecker square leads to the formation of nine polynomials of degree 2
in the basis vectors  I'  2'  3' These polynomials may be divided into two
groups, those that are symmetric with respect to interchange of indices:
f, f, f, 12 + 21' I 3 + 31' 23 + 32
( 12.1 )
and those that are antisymmetric:
12 - 21' 13 - 31' 23 - 32
( 12.2 )
The weight associated with a given polynomial is simply that of the sum of
the weights of the basis vectors. Thus the symmetric term gives the

KRONECKER POWERS lIS
following set of "'/eights:
?;  i; ;;  I 2 +  2 I;  I 3 I;  2 3 +  3 2;
4a+2{3
3
- (2a - 2f3)
3
-(2a+4{3)
3
a + 2{3 . a - {3
3 ' 3
- (2a + f3 )
3
while the antisymmetric term gives
12 - 21; 13 - 31; 23 - 32;
a + 2f3
3
a-f3
3
- (2a + f3 )
3
But the weights associated with the symmetric terms are just those that
arise in the  0 representation of A 2 , and those of the antisymmetric
term are those that arise in the 6 0 representation, and hence
o
1
o x 0
2
o = 0
o + 0
1
o
where, as before, we find the second basic representation in the antisym-
metric part of the Kronecker square.
More generally we may investigate the symmetric <p{k} and antisym-
metric <p {I k} terms that arise in the kth Kronecker power of a representa-
tion <po The following results may be readily verified: 3 ?
1. If (I' 2"" 'n) is a canonical basis of <p and the weight of the vector i
is Ai' then the system of weights of the representation <p {I k} is
A. +A. +... +A.
" ' 2 'k
(i k >'" >i 2 >i 1 )
( 12.3)
and the greatest weight is
A+A+...+A
1 2 k
2. The representation <p {lit} is of degree
( 12.4 )
N(cp{lk}) = ( :)
( 12.5)
3. If l' 2"" 'n is a canonical basis of cp and the weight of i is Ai' then
the system of weights of the representation <p{k} is given as
A. +A. +... +A,
"'2 
(i k >'" >;2>;1)
( 12.6)

116 REPRESENTATIONS, WEIGHTS, AND LABELING
and the greatest weight is
kAt
( 12.7)
4. The representation q;{k} is of degree
N(cp{k})=( n:k)
( 12.8)
Note: The above process of forming symmetrized Kronecker powers is
equivalent to Littlewood's construction 26 of compound and induced
matrices A {}k} and A {k}.
We designate the representation of q;{k} and q;{lk} associated with the
highest weights by attaching a subscript >, for example, q;} and q; k}.
EXERCISES
12.1 Verify for A3 that
1 o ) t 1 2 } 1
( 0 0 = 0 0 0
>
1 O ){1 3 } 1
( 0 0 0 0 0
>
1
12.2 If the basis vectors of 0 0 0 are  l' 2' 3' 4' find the basis
vectors corresponding to the irreducible representations
o
1
o
o and 0
o
1
o
12.3 Show that if l"" 'n is a canonical basis of cp, then the vector of greatest
weight in cp{k} is just (l)k.
12.3 ELEMENTARY REPRESENTATIONS
The basic representations of a group G that correspond to a numeral I
on one of the terminal points of a Oynkin diagram are termed elementary
representations of the group G. Thus for C 3 we have
.
( )
and
.
-
1
( )

ELEMENTARY REPRESENTATIONS 117
and for Ds
and
An arbitrary basic representation may be constructed from an arbitrary
elementary representation cp by means of the operation cp k} .
Let a be a terminal point. The branch of a is defined as a sequence of
poin ts
a=a l ,a 2 ,...,a k
with the following properties:
1. Every point a i (i = 1,...,k - 1) is connected with a i - I and a i + I only.
2. The connection between a i and a i + I has the form
a ,
o
or
a
l
.
a , + 1
. or
a , a , + 1
a l + 1
o
- I)
3. The sequence aI' a 2 ,... ,a k cannot be extended by adjoining any
further roots ak + I without violating condition I or 2.
Thus if we adopt Cartan's classification of the simple roots given below,
1 2 n-1 n
An <>---(). .. <>---()
n n-1 3 2
Bn - ( ... 0----<>
n n-1 n-2 2 1
C n (J . ... . .
3 4 n-  n 1
Dn <>---().. .
2
2
G 
2
4 6 5 3
E6
E8
2 4 3 1
F4 0----< ) - .
2 4 6 7 5
E7
3
7
8
6
2

118 REPRESENTATIONS, WEIGHTS, AND LABELING
We obtain the branches for the terminal points as
An: { I, 2, · · · , n B n : { 2, 3, . . . , n
n, n - 1, . . . ,1 1, n
{ 3,4, . . . ,n
C I, 2' n ...,n D
n: n: I,n
2,n
{  { 1,3,4 1, 4, 6
G 2 : F4: E6: 3,5,6
2,4 2,6
2,4,6,7 1,3,5,7,8
E7: 1,5, 7 E8: 2,6,8
3,7 4,8
Let a r be a terminal simple root and let
a=al,a 2 ,...,a k
be its corresponding branch. Then for r= 1,2,...,k we have
m _m{ lr}
.." a,. .." ex >
( 12.9)
-a result that follows directly from Eq. 12.4. Thus any irreducible basic
representation may be constructed from antisymmetrized Kronecker POW-
ers of elementary representations.
12.4 WEIGHTS OF ELEMENTARY REPRESENTATIONS
Dynkin has systematically evaluated the positive weights of the ele-
mentary representations of the Lie groups, giving his results in terms of the
simple roots, to arrive at the following correlations:

N  "t   r--
,<. ,<. ,<.
I I I I I I
,<.   "t It) 
,<. ,<.
II II
N M "t It) (!)
0 0 0 0 0 0
r--

'-v.. +  +  + X LU
N M ,<."t It) ,<.(!)   "t It) (!) r-- eo
,<. ,<. ,<. ,<. ,<. ,<. ,<. ,<.
I I I I I I I I I I I I
 M "t  ,z N M "t   r--
,<. ,<. ,<. ,<. ,<. ,<. ,<.
II II II II II II II
N M "t It) ?J" N M "t It) (!) r--
0 0 0 0 0 0 0 0 0 0 0
(!)  eo
 
9X + 9X + X + = 9U  + LX +  = 8U
s::=
,<.
I
N M
,<. ,<. s::=
I ,<. s::=
N II ,<.
i:: ,.< ,<. N

II II
N s::= i:: N
0 0 0 0 ,<.
. . . . . -=::D I
N ,.< N
e" ,<.
II
?J" N
0
i:: <E3t
,<.
s::=
N ,.[' ,<.
,<. i:::
I I ,<. ,<."t
II s::= s::= I
s::= N ,<.
 ,<. ,<. ,<. 
I II s::=
II II ,<.
N s::= s::= I
0 0 0 0 N II N
0--0  M I ,<.
. . . N 
,<. ,<. s::= I "t I
,<.
I I ,<. s::= I
s::= N II 0 I 
  ,<. V   v
,<. 
II II ...
II II II II
s::= N
0 0 N M OV
,<. 0 0 0
I 0--0 . . . s::=
+ o----a::::::=--
s::= ,<.
M I ,<. +
N
,<. ,<. s::=
I ,<. I
II s::= 
s::= ,.<  ,<.
 ,<.
II II I II II
N s::= s::= s::=
 0 0 0 0
0--0 . . . 
119

120 REPRESENTATIONS, WEIGHTS, AND LABELING
EXERCISE
12.4. Verify some of the above results.
The roots of the groups An' Bn' Cn' and Dn may be expressed in terms of
the weights Ai of the elementary representations
1 1
0---0 ... ; 0---0 ... (] - ;
 . ... III I) ; 6-..-.0... <
respectively, to give
An
: {_Aq}+l
Bn
n
 : {, +  + Aq } 1
C n
n
: { + 2, +  + Aq } 1
Dn
n
: { +  + Aq}l
(p=l=q)
( 12.10)
where the range of p and q is given to the right of the last curly bracket,
and all combinations of positive and negative signs are permitted.
The corresponding positive roots are given by
An  + : {_ Aq }  + 1
n
Bn +:{, + Aq}l
n
C n  + : {2,  + Aq } 1
n
Dn  + : { + Aq } 1 (p <q) ( 12.11 )
Dynkin classifies the roots of the exceptional groups in terms of the
weights associated with the elementary representations of the subalgebras
of the same rank as that of the exceptional group. Thus the roots of G 2 , F 4 ,
E 7 , and E8 are expressed in terms of those associated with the classical

SPINOR REPRESENTATIONS AND THE GROUPS BII AND DII 121
subalgebras A 2 , B 4 , A 7 , and A 8 , respectively, to give
G 2
3
: { + ,\-Aq}1
F4
4
 : { + , + Aq,! ( + Al + A 2 + A3 + A 4 ) } 1
E7
8
: {- Aq,  + Aq + Ar + As } I
E8
9
: {- Aq, + ( + Aq + Ar) } I
( 12.12 )
For the particular case of E6 the sub algebra A 5 + A 1 is used to give
E6
6
: {- Aq, + 2A,  + Aq + Ar + A } I
( 12.13)
where XI'''' ,A6 are the weights of the elementary representations of A 5'
and + A are those of A I'
EXERCISES
12.5 Verify that the weights for the elementary representations of C 3 are
a3
Al =al +a2+ 2'
a3
A 2 =a2+ 2'
a3
A3=-
2
12.6 Obtain expressions for the positive roots of each of the exceptional Lie
groups.
12.5 SPIN OR REPRESENTATIONS AND THE GROUPS Bn AND Dn
One elementary representation Tl of Bn may be obtained by considering
the group O(2n + 1) of all unimodular orthogonal transformations in a
(2n + 1) - dimensional space. The branch of this elementary representation,
1
o
o · .. ()
-
includes n - 1 simple roots of Bn' and corresponding to these we obtain
n - 1 basic representations by the formula
ff" -T {lk}
'k - I
(k = 1, . . . ,n - 1 )
( 12.14 )

122 REPRESENTATIONS, WEIGHTS, AND LABELING
The representations Tllk} are found to be irreducible for k = 1,...,n - 1.
The second elementary representation
o
o ... ()
-
of Bn cannot be obtained from TI by the operation of anti symmetrization.
This is the so-called spinor representation and will be denoted by (J.
The weight system of TI has the form
0, + AI' + A 2 , . . . , + An
while that of (J is given by
(12.15)
1 ( +' +' +... +' )
2 -1\1 - 1\2 - - I\n
( 12.16)
where every combination of signs must be taken. The greatest weight for (J
will be
l ( A+A+"'+A )
2 1 2 n
(12.17)
The group Dn may be realized by the group of unimodular orthogonal
matrices of order 2n, that is, O(2n). The branch of the elementary
representation T 1 contains n - 2 roots, and the associated n - 2 basic
representations are found by
Tk=TI{lk}
(k = 1, .. .,n - 2)
( 12.18)
There are two spinor representations (for n >4), (J2 and (J2:
0 o ... <1 0 o ... <1
°1 °2
which are carried into one another by an inner automorphism. The weight
system is again given by Eq. 12.17, one representation being associated
with an odd number of plus signs, and the other with an even number.
In the special case of D 4 we have
a 2
a,
a 3

LABELING OF IRREDUCIBLE REPRESENTATIONS 123
The three elementary representations are carried into one another by
automorphisms of D 4 , and we must identify one of them with 'rt and the
other two with the spinor representations 0t and ° 2 ,
We note that the Kronecker product of two spin or representations gives
true representations, while the Kronecker product of a spinor representa-
tion with a true representation gives spin or representations. In the simple
case of 0(3), the true representations are all of odd degree, while the spinor
representations are all of even degree.
12.6 LABELING OF IRREDUCIBLE REPRESENTATIONS
Several schemes exist in the literature for labeling the irreducible repre-
sentations of the Lie groups. In each case use is made of the greatest
weight A of the irreducible representation. In Dynkin's notation we may
simply attach the components A to the kth vertex of the associated
Dynkin diagram.
Cartan 33 makes frequent use of the I basic representations Ai of the
group by writing
I
A=  n.A.
 I I
i= 1
( 12.19)
while Dynkin 37 writes the highest weight A of an irreducible representation
in ,terms of the weights A; of the elementary representation specified in
Section 12.4 to give
A=  /.h.
 I I
(12.20 )
If we define the set of numbers a k by
2( A, a k )
a =A =
k CXk ( )
ak,a k
(12.21 )
we may use the results of Section 12.4 to express the coefficients Ii of Eq.
12.20 in terms of the a k , which are the numbers used by Dynkin to give the
diagrammatic labeling of the irreducible representations. Direct evaluation
gives the results
Bn:
a n-l
I k = -.!!.. +  a i
2 ;=k
(12.22 )

124 REPRESENTATIONS, WEIGHTS, AND LABELING
Cn:
n
I k =  a;
i=k
Dn:
a -a n-2
n-l n 
I k = + £.J a;
2 i=k
F4:
11 = a 1 + 2a 2 + fa 3 + a 4
a 3
1 2 =a 1 +a 2 + T
a 3
1 3 =a 2 + T
a 3
1 4 =-
2
(12.23 )
( 12.24)
( 12.25)
The numbers (11"" ,In) are in this scheme identical to the customary
Cartan-Weyllabeling scheme.
EXERCISES
12.7. Write the following representations in the Cartan-Weyllabeling scheme:
1 2
0 [ ) - , 0 ( ) - , 0 ( ) -
1 1 2
. ( ) , . - ( ) , . - ( )
3 4
3 5
1 1 1 1 1 2
0 . ) . , 0----< ) . , 0----< ) .
12.8. Show that for F4 we have
11 > 1 2 + 13 + 14 and 1 2 > 13> 14

EXERCISES 125
For the groups An' G 2 , E 7 , and E8 the vector A expressed in the form
n+1
A=  /.'A.
 I I
i= I
( 12.26)
is not unique. However, if the numbers 1 1 ,/ 2 ,,,, ,I n + 1 are chosen such that
n+1
 1;=0
i= I
( 12.27)
then Eq. 12.26 becomes unique.
It then follows from Section 12.4 and Eqs. 12.26 and 12.27 that we have
An:
n
I k = I n + 1 +  a;
i=k
-1 n.
In + 1 = 1  la;
n + i= I
(12.28 )
G 2 :
a 2
11 =a 1 + 3
a 2
1 2 =-
3
2a 2
I = -a --
3 1 3
( 12.29)
"
E7:
6
I k =/ 7 +  a;
i=k
/7= a 7 -a 4 -:a s -3a 6 (12.30)
9a 4 +6a s + 3a 6 + 7a 7
1 8 =-(a 1 +2a 2 +3a 3 )- 4
E8:
7
I k = 18 +  a;
i=k
a 8 - 2a 2 - a 6
/8= 3 (12.31)
3a 1 +6a 2 +9a 3 + 12a 4 + 15a s + 10a 6 + 5a 7 + 8a 8
1 9 = -
3
For the group E6 we look for a representation in the form
6
A=  /''A,+/'A
 I I
i= I
(12.32 )

126 REPRESENTATIONS, WEIGHTS, AND LABELING
where
6
 Ij=O
;= 1
(12.33 )
Thus we have
E6:
5
I k =/ 6 +  a j
i=k
5 ia,
16 = -  -2.
;= 1 6
1= a l + 3a 2 + 5a 3 + 5a 4 + 5a S +a 6
(12.34 )
The numbers II'..' ,I n + I of An defined above correspond to the usual
Cartan-Weyl labeling of SU(n + 1). This notation, involving fractions, is
somewhat awkward and may be simplified by using the n integers I{,... ,/,
where
n
/"=  a,
I :J
j=i
( 12.35)
and
I{  I  . ..  I  0
( 12.36)
The integers I{,... ,/ correspond to a partition of the integer
n n
N= - ia j =  I j
;=1 ;=1
( 12.37)
The partitions (I{,... ,/) are normally used in labeling the irreducible
unitary representations of the unitary group U(n + 1).
12.7 A MATTER OF NOTATION
I t is desirable to develop a systematic notation labeling the irreducible
unitary representations of the semisimple Lie groups. In the case of
U(n + 1) we enclose the n integers in curly braces { }. The n integers or half
integers for O(2n+ 1) or O(2n) are enclosed in square brackets [], while
those for the symplectic group Sp (2n) are enclosed in angular brackets < >.
Finally, for the exceptional groups the numbers are enclosed in
parentheses (). We note that this final notation is unambiguous, as each
exceptional group involves a different number of numerical labels.
The spinor representations of O(2n) and O(2n + 1) will all involve n half
integers, while the true representations involve only integers. In the par-
ticular case 26 ,82 of the special orthogonal group SO(2n), the representa-
tions having In =1=0 separate into two conjugate representations, [II"" ,In]
and [11"'" -In]'

13
The Exceptional Groups
13.1 BASIC REPRESENTATIONS OF THE EXCEYfIONAL GROUPS
The highest weight A of an irreducible representation of a Lie group of
rank I may be written as a linear combination of the highest weights M(i)
of the I basic representations:
I
A=  m;M(i)
i= 1
I
=  Pkak
k=l
(13.1)
where
I
M(i) =  n.a.
 :J "}
}=l
(a j Ell)
( 13.2 )
Here mi and Pk are nonnegative integers. If the basic representations and
simple roots of the exceptional groups are labeled as in Section 12.3, we
may readily evaluate Eq. 13.1 using Eq. 10.9 to obtain the results
CX 2 CX 1
G 2
()
-
M (1)-2a +a M(2)=3a +2a
- 1 2' 1 2
127

F4
a 2
o
a 4
[ )
a 3 a l
- e
M( 1) =2a 1 + 3a 2 + 3a 3 + 2a 4
M(2) =2a 1 + 2a 2 +4a 3 + 3a 4
M(3) = 3a + 2a + 6 f\/ + 4f\/
1 2 3 4
M(4)=4a +3a +8a +6a
1 234
E6
a l a 4 a 6 a 5 a 3
0 0 I 0 0
a 2
M(l) = 40: 1 + 3a 2 + 2a 3 + 5a 4 +4a 5 + 6a 6
3
M(2) = a 1 + 2a 2 + a 3 + 2a 4 + 2a 5 + 3a 6
M(3) = 2a 1 + 3a 2 +4a 3 +4a 4 + 5a 5 + 6a 6
3
M(4)= 5a 1 +60: 2 +4a 3 + 10a 4 +8a 5 + 12a 6
3
M (5) = 4al + 6a 2 + 5a 3 + 8a 4 + 10a 5 + 12a 6
3
M(6) =2a 1 + 3a 2 + 2a 3 +4a 4 +4a 5 + 6a 6
E7 a 2 a 4 a 6 a 7 a 5 a l
0 0 0 I 0 0
a 3
128

M(}) = 2a} + 0: 2 + 20:3 + 2a 4 + 3a 5 + 3a 6 +4a 7
M(2) = 2a} + 3a 2 + 30: 3 +4a 4 +4a 5 + 5a 6 + 6a 7
2
M(3) = 4a} + 30: 2 + 7a 3 + 6a 4 + 8a 5 + 9a 6 + 12a 7
2
M(4) =2a} + 2a 2 + 3a 3 +4a 4 +4a 5 + 5a 6 + 6a 7
M(5) = 3a} + 2a 2 +4a 3 +4a 4 + 6a 5 + 6a 6 + 8a 7
(6) _ 6a} + 5a 2 + 9a3 + IOa 4 + 12a 5 + 15a 6 + 18a 7
M - . 2
M (7) = 4a} + 3a 2 + 6a 3 + 6a 4 + 8a 5 + 9a 6 + 12a 7
E8 a 1 a 3 as a 7 as as a 2
0 0 0 0 I 0 0
cx 4
M(l) =2a} + 2a2 + 3a 3 + 3a 4 +4a 5 +4a 6 + 5a 7 + 6a 8
M(2) = 2a} +4a2 +4a 3 + 5a 4 + 6a 5 + 7a 6 + 8a 7 + IOa 8
M(3) = 3a} +4a 2 + 6a3 +6a 4 + 8a 5 + 8a 6 + IOa 7 + 12a 8
M(4)=3a} +5a 2 +6a 3 +80: 4 +9a 5 + IOa 6 + 12a 7 + 15a 8
M(5) =4a} + 6a 2 + 8a 3 + 9a 4 + 12a 5 + 12a 6 + 15a 7 + 18a 8
M(6)=4a} +7a 2 +8a 3 + IOa 4 + 12a 5 + 14a 6 + 16a 7 +20a 8
M (7) = 5a} + 8a 2 + IOa 3 + 12a 4 + 15a 5 + 16a 6 + 20a 7 + 24a 8
M(8) = 6a} + IOa 2 + 12a 3 + 15a 4 + 18a 5 + 20a 6 + 24a 7 + 30a 8
129

130 THE EXCEPTIONAL GROUPS
13.2 LABELING OF REPRESENTATIONS FOR THE EXCEPTIONAL
GROUPS
To proceed further with the labeling of irreducible representations of the
exceptional groups, a suitable vector basis for defining the roots is now
chosen. In the case of G 2 we may put
a 1 = (0, 1 ) and a 2 = ( 1, 2 )
to give
M ( 1) = ( 1, 0) and M ( 2) = (2, I )
The appearance of negative integers may be avoided by writing
(U 1 , u 2 ) = (m 1 + m 2 , - m 2 )
( 13.3)
to reproduce the standard labeling adopted by Racah. 9 In the case of
2
(] -
we have from Eq. 13.1
A = 2M(2) + M( 1) = (4, - 2) + (1,0) = (5, - 2)
which, with the help of Eq. 13.3, becomes
A= (3,2)
In Dynkin's scheme we have the correspondence
(U 1 ,U 2 ) = (a 1 +a 2 ,a 1 )
( 13.4)
For the group F4 we may write the simple roots in terms of the unit
vectors e k (k = 1, . . . ,4) to give
e 1 -e 2 -e 3 -e 4
a = = ( -!-_1_.1_1 )
1 2 -  2 2 2.
a2=e 2 -e 3 (01-10)
a 3 =e 4 = (OOOI)
a 4 =e 3 -e 4 = (OOI-I)
We then obtain the highest weights of the basic representations of F4 as

LABELING OF REPRESENTATIONS FOR THE EXCEPTIONAL GROUPS 131
M(l)= (1000); M(2)= (1100); M(3)= (t!t!) M(4)= (2110)
(13.5)
If we represent the highest weight of an arbitrary representation of F4 by
A, we have from Eqs. 13.1 and 13.5
Al =m l + m 2 + tm 3 + 2m 4
m 3
A 2 =m 2 + T +m4
m 3
A 3 = T +m4
m 3
A =-
4 2
(13.6)
from which we deduce that
A2>A3>A4 and AI>A2+A3+ A 4
(13.7)
We note that Dynkin labels the representations of F4 with the positive
in tegers
Q1
o
Q2
( )
Q3
-
Q4
.
whereas we have used Cartan's order 83
m 2 m 4
o )
m 3
m 1
.
It is readily seen that Dynkin's numbers Ii' 1 2 , 1 3 , 14 defined in Eq. 12.25 are
identical to the numbers AI' A 2 , A 3 , A 4 , respectively.
The choice of a suitable basis for labeling the irreducible representations
of the groups E 6 , E 7 , and E8 has been considered by a number of
authors. 33 ,48,5o,73 In each case the weights of the group EI span an
(I + I)-dimensional weight space, and it is possible to project them onto a
suitable I-dimensional subspace in much the same manner as is done for
the groups AI' A detailed account has been given by Mehta and Sri-
vastava. 73
However, the Cartan-style labels for the exceptional groups E 6 , E 7 , and
E8 tend to be clumsy, and in most cases the collection of non-negative
integers (a l ,... ,a l ) given in Section 12.6 forms the simplest consistent
method of labeling the irreducible representations.

132 THE EXCEPTIONAL GROUPS
EXERCISE
13.1. Show that the basic representations of B I , C I , and DI may be labeled, after
the manner of Cartan, as
B . M (I)= [ 11...1 ]
I' 22 2
M(2)= [10,...,0]
M(/)= [11,...,1]
C / : M ( I) = < 1 0.. . 0)
M(2) = <11. ..0)
M(I) = <11. ..I)
D I : M(I)=[tt...t-t]
M(2)= [ 11...11 ]
22 22
M(3)=[10...0]
M(4)= [11...0]
M(/)= [11...100]
where each pair of brackets encloses I integers or half integers.

14
Dimensions of Irreducible Representations
14.1 SCALAR PRODUCTS OF BASIC WEIGHTS
Each semisimple Lie group is characterized by one or more elementary
representations corresponding to the terminal points of its associated
Dynkin diagram, from which the complete set of basic representations may
be constructed by the process of anti symmetrization of Kronecker powers.
The roots of the associated Lie algebra may be expressed in terms of the
weights A; of the elementary representations as in Section 12.4. The scalar
product (A;,Aj) of two weights of the elementary representations may be
evaluated by first expressing the weights in terms of the roots and then
using the properties of the Cartan matrix. When this is done we find for
the groups Bn, Cn' Dn, and F4
( \, A j ) = 0
(i=l=j)
( 14.1 )
and
(\,A i ) = K
(i = 1, . .. ,n )
( 14.2 )
where K is a constant.
If
n n
hI =  b;A; and h 2 =  CiA;
i=1 i=1
( 14.3)
133

134 DIMENSIONS OF IRREDUCIBLE REPRESENTATIONS
then the scalar product
n
(h l ,h 2 ) =K  b;c;
i= I
( 14.4 )
an important result we shall require for later developments.
For the groups An' G 2 , E 7 , and E8 we have
n+l
 A;=O
i= I
and
("A;, A;) = nK ( i = 1, . . . ,n + 1)
( 14.5 )
("A;,A j ) = - K (i=l=j)
 14.6)
from which we find that if
n+l n+l
hi =  b;A; and h 2 =  CiA;
i=l i=l
( 14.7)
then
n+l [ n+l ][ n+l ]
(h),h2)=(n+l)Kj) bjcj-K j) b j j) c;
( 14.8)
As usual, the group E6 proves to be different, and we have, in addition
to Eqs. 14.1 and 14.2,
("A;,A) =0
(i= 1,...,6)
( 14.9)
and
(A,A) =3K
( 14.10 )
Thus if
6 6
hi =  b;A;+bA and h 2 =  CjAj+CA
i=l i=l
( 14.11 )
then
(h),h 2 ) = 3Kbc + 6K jt) bjc; - K[ jt) b j ] [ j) C j ]
(14.12)

DIMENSIONS OF IRREDUCIBLE REPRESENTATIONS 135
The above results are completely determined once the constant K is
fixed. Usually we fix K by requiring that the roots form an orthonormal
set, that is,
 (a;, a j ) = ij
a
( 14.13)
and hence
 (a,a) =n
a
( 14.14)
where n is the number of components in a. This summation may be readily
evaluated in terms of K by expressing the roots in terms of the weights A;
of the elementary representations using Eqs. 12.10, 12.12, and 12.13, and
then using Eqs. 14.1-14.14 to give the left-hand side of Eq. 14.13 as a
multiple of K. Doing this, we readily establish the following results:
Cn:
K= 1
2(n+l)2
K= 1
4( n + 1)
Bn:
K= 1
2(2n - 1)
K= 1
4( n - 1)
An:
Dn:
G 2 : K- 1
- 24
E6: K--L
-144
E8: K- 1
- 540
F . K - 1
4' - 18
E . K - --L
7 . - 288
(14.15)
14.2 DIMENSIONS OF IRREDUCIBLE REPRESENTATIONS
If cp is an irreducible unitary representation of a compact semisimple Lie
group characterized by its maximal weight A, then Wey134,84 has shown
that the dimension N(cp) of cp is given by
- II (A+g,a)
N(cp)- ()
g,a
aE+
( 14.16)
where
g= t  a+
aE+
(14.17)

136 DIMENSIONS OF IRREDUCIBLE REPRESENTATIONS
That is, g is half the sum of the positive roots a + .
We may determine g by noting that 37
2( g, a)
ga= ( ) =1
a, a
(aEII)
( 14.18 )
a result that follows directly from the properties of the scalar products of
roots. We then write
g=  k;A;
i
( a; E II)
( 14.19)
and with Eq. 14.18 we can set up a system of equations to solve for the k;.
Alternatively, we may write
g= g;A;
( 14.20)
where the A; are the weights of the fundamental representation of the
appropriate groups, which in turn may be related to the simple roots of
Section 12.4. When this is done we obtain the results
An: g. = n _ i + 1 Bn: . I
I 2 g; = n - I + 2"
Cn: g;=n-i+ 1 Dn: g; = n - I
G 2 :
I
g2= J
g3 = - i
F4:
g _ II
1-2
g _ 5
2-2
g -1
3- 2
g _ I
4-2
gl=
4
"3
7 . (i<5) { _ 23-4i
g; = 2 - I (i<7)
E6: E7: g;- 4
g --
6- 2 g _ _ 49
g= 11 8- 4
E8:
{ = 22-3i
g; 3
g --
9- 3
(i<8)

DIMENSIONS OF IRREDUCIBLE REPRESENTATIONS 137
We are now in a position to evaluate N(cp) for each of the semisimple
Lie groups. Consider the case of An' We have
IT (g,a) = IT [gi(A;,a) ]
aE+
= IT [gi(Ai'\, -Aq) ]
(p>q)
( 14.21 )
since from Eq. 12.11 each positive root of An may be expressed in the form
\, - Aq (p > q). Noting Eqs. 14.5 and 14.6, we find for An
II ( g, a) = nK II ( gp - gq) (p >q) (14.22 )
aEY+
Similarly
IT (A+g,a)= IT[ mi(A;,a)]
aE+
= nKII (m p - m q ) (p >q) (14.23 )
where
m.=/'+g. (14.24 )
I I I
Hence for An
N ( cp ) = II ( mp - mq )
gp - gq
Continuing in this manner we find the results given below:
An: N(cp) = II( mp=m q ) ( 14.25)
gp gq
Bn' C n : N ( cp ) = II ( mp ) II ( mp = mq ) II ( mp + mq ) ( 14.26)
gp gp gq gp + gq
Dn: II ( mp - mq )( mp + mq ) ( 14.27)
N( cp) =
gp - gq gp + gq
G 2 : II ( mp ) II ( mp - mq ) (14.28 )
N(cp) = -
gp gp - gq

138 DIMENSIONS OF IRREDUCIBLE REPRESENTATIONS
F4: N( cp) = II ( m p ) II ( mp = mq )( mp mq ) II ( m)  m2 m 3 : m 4 )
gp gp gq gp gq gl-g2-g3-g4
(14.29 )
E6:
m II ( mp - m q ) II ( mp +m q +m r +m/2 )
N(cp)=-
g gp-gq gp+gq+gr+g/ 2
( 14.30)
E7:
N( cp) = II ( mp - m q ) IT ( mp +mq+mr+ms )
gp - gq gp + gq + gr + gs
1
( 14.31 )
E8:
N(cp) = II ( mp - m ) II ( mp +mq+m r )
gp - gq gp + gq + gr
(14.32 )
The indices in the above formulas range over all possible values, subject
to the conditions that (i) the indices denoted by distinct letters have
distinct values, and (ii) of all permutations of a combination of values,
only one is to be counted. An extensive tabulation of N(cp) for An' Bn' Cn'
D n' and G 2 has been given by Butler 31 .
EXERCISES
14.1. Show that for the group B 2 ,
(II + 1 2 + 2) (II -/ 2 + 1) (21 1 + 3) (21 2 + 1 )
N[ 1./ 2 ] = 6
14.2. Show that for the group C 2
(II + 2) (1 2 + 1) (II -/ 2 + 1) (II + 1 2 + 3)
N -
</1/2> - 6
14.3. Use the above two results to prove that
N -N
<II + 1 2 ,11 -1 2 > - [/./ 2 ]
(N.B. The groups SP4 and R5 are locally isomorphic.)
14.4. Show that for the group G 2 in Racah's (UIU2) scheme
N(UIU2) = (UI + U2 +3) (UI + 2) (2uI + U2 +5) (UI +2U2 +4)
x (u I - U 2 + 1 ) ( U2 + I ) /120

15
The Casimir Invariants
15.1 EIGENVALUES OF THE QUADRATIC CASIMIR OPERATORS
The quadratic Casimir operator C was introduced in Section 5.18 and
defined as
C= g P(JX X
P (J
which in the standard Cartan- Weyl basis may be written as
C=gikHiHk+ EaE-a
a
( 15.1 )
If we let A be the highest weight of an irreducible representation and lu A >
be a vector of this weight, remembering that Ealu A> = 0 for positive roots,
we find
CIUA>=gikAiAkluA>+  [Ea,E-a]lu A >
aE+
= {(A,A) + (A,2g)} lu A >
= (A,A+2g)lu A >
( 15.2 )
where g is half the sum of the positive roots, as in Eq. 14.17. The
eigenvalues of the quadratic Casimir operator, (A, A + 2g), may be
139

140 THE CASIMIR INVARIANTS
evaluated using Eqs. 12.20 and 14.20 to give
(A, A + 2g) =  Ij(  + 2) (Aj,A j )
;,j
( 15.3)
The values of  and gj may be deduced from Eqs. 12.28-12.34 and the
results of Section 14.2, while the scalar products (A,-, A j ) were evaluated in
Section 14.1, leading to the results
1 n+l .
An: (A,A+2g)= ( )  li(li-21)
2 n+l ;=1
( 15.4)
1 n
Bn: (A,A+2g)= ( _)  li(li+2n-2i+l)
2 2n 1 ;= 1
( 15.5)
1 n .
Cn: (A,A+2g)= ( )  li(li+2n-21+2)
4 n+l ;=1
( 15.6)
Dn:
1 n
( A, A + 2g ) = ( )  Ii (Ii + 2n - 2i)
4 n - I ;= 1
( 15.7)
G 2 :
( A, A + 2g ) = 2 [ I} (3/} + 8) + 1 2 (31 2 + 2) + 13 (31 3 - 10) ]
( 15.8)
F 4: ( A, A + 2g ) = l8 [ I} (I} + II ) + 1 2 ( 1 2 + 5) + 13 ( 13 + 3) + 14 ( 14 - 1 ) ]
( 15.9)
E6: (A,A+2g) = 18 [1(1+22) +2 it\l i (li- 2 i) ]
(15.10)
E7: (A, A + 2g) = 3 [ i/i(li - 2 i ) - 20/ 8 ]
(15.11)
E8 : (A, A + 2g) = 2JO [ it\l;( Ii - 2i) - 42/ 9 ]
(15.12)
15.2 GENERALIZED CASIMIR INVARIANTS
The generalized Casimir invariants were defined earlier by the equation
I = C {32 C {33 ... C {31 X {3 IX {32. . . X {3n
n a I {31 a 2 {32 a" {3n
(15.13)

GENERALIZED CASIMIR INVARIANTS 141
where /0 and /1 are null and
/ = g XlXIXlX2
2 lXIlX2
(15.14)
which is trivially different from the quadratic Casimir operator defined in
Section 5.18.
If we evaluate /3 for the compact group SO(3) we find
/3 =X I X 2 X 3 - X I X 3 X 2 +X 2 X 3 X I - X 2 X I X 3 + X 3 X I X 2 - X 3 X 2 X I
= XI [X 2'X 3 ] + X 2 [X 3'X I ] + X 3 [X I ,X 2 ]
=X+X;+X;
a:./ 2
Indeed, for any value of n > 2 we find that the invariant is simply
proportional to /2' Thus the invariants found from Eq. 15.13 are not
necessarily independent.
Racah 85 has shown that for a semisimple Lie group of rank I it is
possible to construct a set of I independent invariants whose eigenvalues
completely specify the irreducible representations of the semisimple group.
In particular he showed that the orders of the independent invariants were
A,: /2' / 3' . . . , /, + I
B,: /2'/4"" '/2'
Ci: /2'/4""'/2'
D,: /2' /4' . .. '/2'- 2' /,
G 2 : /2'/6
F4: /2'/6'/8'/12
E6: /2'/5'/6'/8'/9'/12
E7: [2' /6' /8' /10' /12' /14' /18
E8: /2'/8'/12'/14'/18'/20'/24'/30
The problem of constructing the individual invariants and then computing
their eigenvalue spectrum has been studied on many occasions, and the
reader is referred to the literature for specific details. 86 - 98

142 THE CASIMIR INVARIANTS
15.3 INV ARIANTS FOR NONSEMISIMPLE LIE GROUPS
N onsemisimple groups, such as the Euclidean and Poincare groups, find
considerable application in theoretical physics. In these cases we cannot
form the reciprocal of the tensor gOA and construct Casimir invariants in
the ordinary sense. This does not, however, preclude the construction of
invariants that commute with all the infinitesimal operators of a nonsemi-
simple Lie group, as was seen in Section 5.18 in the case of the Euclidean
group E3'
The problem of constructing invariants for the so-called inhomogeneous
groups, which involve the semidirect sum of the group of translations in n
dimensions with an n-dimensional semisimple group, has been examined in
detail by Rosen 98 and by Nagel and Tahir Shah. 99
The number v of independent Casimir invariants associated with any Lie
algebra has been determined by Beltrametti and Blasi,97 who give the
important theorem:
1beorem 15.1
The number v of independent Casimir operators associated with any r-
parameter Lie group is equal to
v = r - rank II c: 1' a p II ( (1, p, 'T = 1, . .. , r ) ( 15.15 )
where the c: 1' are the structure constants of the associated Lie algebra, and a p
the r group parameters.
The quantity rank II c:1'apll is to be interpreted as the maximum rank of
the r X r matrix II c:1'apll, with the a p treated as independent variables. In the
particular case of the Euclidean group E3 we have six parameters
associated with the six infinitesimal operators JI,J2,J3,PI,P2,P3 that satisfy
the commutation relationships given in Eq. 5.52, which are displayed as a
tableau below:
J I J 2 J 3 PI P 2 P 3
J I 0 J 3 -J I 0 P 3 -P
2 2
J 2 -J 0 J I I -P 0 PI
3 3
J 3 J -J o I P -P 0
_ _ 2_ _ _ _I _ _ p- -I - -0 2 - _ _ 1_ - --
PI 0 P 3 0 0
2
P 2 -p 0 0 I 0 0 0
3
P 3 P 2 -P 0 I 0 0 0
I

CASIMIR OPERATORS FOR SO(3) AND SO(2, l) 143
Inspection shows that rank IIcJ.ba(P1)11 is equal to 4; hence by Eq. 15.15
there must be two independent Casimir invariants. Explicit construction
gives these as p2 and P'J.
15.4 CASIMIR OPERATORS FOR SO(3) and SO(2, 1)
The eigenvalues of the quadratic Casimir operator given in Section 15.1
were derived for the case of the compact semisimple Lie groups. In these
cases the eigenvalues were all found to be real and discrete. For noncom-
pact groups the situation is rather more complex. As an example we now
compare and contrast the situation for SO(3) with that of the much-
studied 60 - 62 ,100-102 noncompact group SO(2, 1). In the process we obtain a
classification of the unitary irreducible representations of SO(3) and
SO(2, 1).
Consider first the group SO(3) and its compact subgroup SO(2). Let us
designate, as in Section 4.9, the infinitesimal group generators of SO(3) as
J I' J 2' J 3' with
[J 1 ,J 2 ] = iJ 3 ,
[J 2 ,J 3 ]=iJ 1 ,
[J 3 ,J 1 ] =iJ 2
(15.16)
where J 3 will be identified also as the generator of the SO(2) subgroup.
Following Section 7.8 we write
Jz=  (J\ + iJ 2 )
(15.17)
to obtain the standard form of the commutation relationships:
[J +, J _ ] = J 3 and [ J 3' J :t ] = + J :t
(15.18)
Suppressing a factor of ! in Eq. 5.48, we obtain the Casimir operator J2 of
SO(3) as
J2 =J? +Ji +Jf =J +J _ +J _J + +J;
( 15.19)
Comparison with Eq. 15.18 gives
2J+J_=J 2 -J 3 (J 3 -1) and 2J_J+=J 2 -J 3 (J 3 +1) (15.20)
We now seek to determine the eigenvalue spectra of J2 and J 3 . Let us
label the representations of SO(3) by the eigenvalues X of J2. The
eigenvectors IXa) spanning the space of a particular representation are
constructed to be simultaneous eigenvectors of J2 and J 3 , and will be

144 THE CASIMIR INVARIANTS
labeled by their associated eigenvalues X and a.
Since J2 is the sum of positive-definite Hermitian operators, it must itself
be a positive-definite Hermitian operator. Thus for a unitary represen-
tation the ei genvalues of J2 must be real and positive. Likewise, J 3 is
a Hermitian operator and must have real eigenvalues. Hence we may
wri te
J 2 \Xa)=XIXa)
(X >O,X ER)
( 15.21 )
and
J 3 I Xa )=aIXa)
(aER)
( 15.22)
Using Eq. 15.20, we have
2J _J +IXa)= [X -a(a+ 1) ] IX a)
2J +J _IXa)= [X -a(a-l) ]IXa)
{15.23a)
(15.23b)
In a unitary representation we must have from Eq. 15.17 that
J =J_
( 15.24)
and hence the eigenvalues of J +J _ or J _J + must be positive definite.
Thus from Eqs. 15.23a and b we must have for unitary representations
X-a(a + 1»0
( 15.25)
We now attempt to determine the permissible values of X and a that
satisfy Eq. 15.25 and that at the same time are consistent with the
commutation relationship of Eq. 15.18. Using Eq. 15.18, we obtain
<Xa'\[J 3 ,J +] IXa) = (a' - a )<Xa'\J +\Xa) = <Xa'\J +IXa) (15.26)
from which we may conclude that successive eigenvalues a of J 3 must
differ by unity, that is,
a' - a = 1
(15.27)
For a given finite nonnegative value of X, it is possible to satisfy Eq.
15.25 with real values of X and a only if a has an upper positive bound a+
and a lower negative bound a _, with a + - a _ an integer. Solution of Eq.
15.25 for a+ and a_ gives
a = -! + ! V I +4X
( 15.28)

CASIMIR OPERATORS FOR SO(3) AND SO (2, I) 145
and hence
X=a+(a++l)
and a = - a-I
- +
( 15.29)
Since a + and a _ differ by an integer, 2a + must be a positive integer, and
hence a + is limited to the field of positive integers or half odd integers.
Let us put} = a + and replace a by m. It follows from the above that a
given unitary representation of SO(3) may be labeled by the upper bound}
with the eigenvectors designated in the familiar angular-momentum ba-
SiS 103 ,104 as I}m), where for a given value of} we have the 2}+ 1 values of
m
m = },} - 1, . . . , - } + 1, - }
( 15.30)
The range of m is bounded above and below, and hence the unitary
representations of SO(3) are all of finite dimension equal to 2} + 1.
Noting Eqs. 15.23a-15.24, we may readily deduce that
J;t!jm)=  W U+l)-m(m + l) lim + l)
( 15.31 )
where the arbitrary phase factor has been fixed as positive. The operators
J + and J _ have the property of stepping up or down the value of m by
one unit. In the language of weight vectors, m distinguishes the different
weight vectors associated with a representation Dj of SO(3), and the
operators J + and J _ allow us to pass from one weight vector to another
belonging to the same irreducible representation. The representations
characterized by integer values of } are termed true representations of
SO(3), and those by half-integer values of}, the spin or representations.
We note that the preceding equations are left invariant under the
substitution 105,106
}-}-1 and mm
( 15.32)
and yield an equivalent representation D-j-I of SO(3).
The weights associated with the unitary representations Dj of SO(3)
may be conveniently displayed by a plot of} against m as in Fig. 15.1. The
weights of the Dj representations are shown above, and those of the
equivalent D-j-I representations below. The operators J:t allow us to pass
from one weight to another along the horizontal solid lines (or dashed lines
for the spin or representations). The extremum values of m are bounded by
the inclined lines.

146 THE CASIMIR INVARIANTS
J
Vi
-m
m
v- } - 1
Fig. 15.1. The weights of the unitary representations of SO(3). The weights associated with
true representations are marked by blackened circles . joined by solid lines. The weights
associated with spinor representations are marked by blackened squares. joined by dashed
lines.
Let us now obtain a classification of the unitary representations of
SO(2, 1) in terms of the eigenvalue spectra of its Casimir operator J,2 and
the generator J 3 ' of its compact subgroup SO(2). The group generators
J 1',J 2',J 3' of SO(2, 1) satisfy the commutation relationships
[J{,J] = - iJ,
[J,J]=iJ{,
[J,J{] = iJ
( 15.33)
If we write
J   (u; + JD
(15.34)
we obtain the standard commutation relationships
[J,J] =J and [J,J] = + J
(15.35 )
which are identical to those found in Eq. 15.18 for SO(3). Suppressing a
factor of - t in Eq. 5.48 gives the Casimir operator of 80(2, 1) as
J,2=J,2+J,2_J,2= -J' J' -J' J' -J,2
1 2 3 + - - + 3
( 15.36)

CASIMIR OPERATORS FOR SO(3) AND SO(2, 1) 147
We note that J,2 is no longer a sum of positive-definite Hermitian
operators, and hence its eigenvalues may range over the domain of real
positive and negative numbers.
Again we write
J,2IXa>=XIXa> X ER
( 15.37)
and
JIXa>=aIXa> aER
( 15.38)
where IXa> is a simultaneous eigenvector of J,2 and J 3"
Since for unitary representations of 80(2, 1) we must have
J't = -J'
+ -
( 15.39)
it follows that the eigenvalues of J +' J _' and J _' J +' must be real and
negative definite, and hence from Eq. 15.36 it is necessary that
X+a(a + 1»0
( 15.40)
We may now obtain a complete classification of the unitary representa-
tions of 80(2,1) in terms of the eigenvalues of J,2 and J 3 ' by determining
the real values of X and a that satisfy Eq. 15.40, remembering that
successive values of a must differ by unity. We find that the representa-
tions may be divided into two distinct series, a continuous series C
associated with continuous eigenvalues of J,2 and a discrete series D
associated with discrete eigenvalues of J,2. The 80(2) content of these
representations follows directly from consideration of Eq. 15.40 to yield
A. Continuous Series
(a) 0 <X < 00. Here we have X + a(a + 1) >0 for all
a = 0, + 1, + 2,.. .
(15.41 )
and the representations C are unbounded from above and below.
(b) t <X < 00. Here we have X + a(a + 1) > 0 for all
a =+.l +J. +2
- 2' - 2' - 2""
( 15.42)
I
and the representations C X 2 are unbounded from above and below.

148 THE CASIMIR INVARIANTS
B. Discrete Series
In these cases the eigenvalues of X may all be written in the form
k(l- k), where k is a positive integer or half integer. Inspection of Eq.
15.40 shows that a has either an upper bound with no lower bound or a
lower bound with no upper bound. We distinguish these two possibilities
as follows:
(i) D:. Here X = k(l- k) with
k -.l 1 3
- 2' '2""
and
a = k, k + 1, k + 2,...
(ii) D;. Again ..-¥ = k(l- k) with
k -.l 1 1
- 2, ,2""
( 15.43)
but now
a = - k, - (k + 1), - (k + 2),. . . ( 15.44)
The weights of the D: and D; representations are displayed in Fig. 15.2.
We note that whereas for SO(3) the unitary irreducible representations
are uniquely labeled by the eigenvalues of the Casimir operator, in the case
of SO(2, I) different unitary irreducible representations may possess the
k
3
Dk
4
Dk
+ ------
2
---.----.----
----.----.---
--_tl----.----.----
---- .----.----.- --
-m
m
-4
-3
-2
-1
o
2
3
4
Fig. 15.2. Weights associated with the discrete representations Dk+ and D k - of SO(2, 1). The
weights associated with true representations are marked by blackened circles. and joined by
solid lines. The weights associated with spinor representations are marked by blackened
squares . joined by dashed lines.

EXERCISES 149
same eigenvalue of the Casimir operator. These representations are dis-
tinguished by their different eigenvalue spectra of J 3'
The unitary representations of SO(3) are all of finite dimension, while
those of SO (2, 1) are all of infinite dimension. The finite-dimensional
representations of SO(2, 1) are all nonunitary. Sannikov l07 ,108 has shown
how to construct infinite-dimensional representations of the Lie algebra
so(3), but these representations are all nonunitary.
Finally, we note that in deriving the representations of SO(2, 1) we chose
to diagonalize the compact generator J 3" Had we chosen to diagonalize
the noncompact generators J I ' and J 2 ' of the SO (1, 1) subgroups, we would
have been led to the need for a continuous basis. 60,62,102,109,110
EXERCISES
15.1. The 15 generators of the noncompact group SO(4,2) are labeled by the
antisymmetric tensor Lab = - L ba with
0 L I2 L I3 L I4 L I5 L I6
0 L 23 L 24 L 25 L 26
Lab = 0 L34 L35 L36 (15.45)
0 L45 L46
0 L56
0
and satisfy the commutation relation
[Lab' Lcd] = - i ( gacLbd - gadLbc - gbcLad + gbdLac)
( 15.46)
where the metric tensor gab is diagonal with elements ( - 1 - 1 - 1 - 1 + 1 + 1).
Show that the Lie algebra has three independent Casimir invariants 15, III
II = LabLab
( 15.47)
I = E: Lab Lcd L ef
3 abcdef
14 = Lab L be Led La d
15.2. Construct the two independent Casimir invariants 1 2 and 14 for the de Sitter
group SO(4, 1) and use these to analyze the SO(4) content of the representa-
tions1l2-116 labeled in terms of the eigenvalues of 1 2 and 14'

16
Some Global Properties of Lie Groups
16.1 TOPOLOGICAL NEIGHBORHOODS
In much of our discussion in Chapter 4, and indeed in later chapters, we
were primarily concerned with infinitesimal transformations in the neigh-
borhood of the identity element of the group of interest. We have seen that
different Lie groups may have the same Lie algebra but have properties
that are very different in the large. To proceed further we need to look at
some of the global properties of Lie groups. To this end we first look at
some elementary aspects of topology that will at the same time sharpen our
understanding of the distinctive differences that arise between compact
and noncompact Lie groups.
The study of topology is intimately concerned with the concepts of
"nearness" and "continuity." The. idea of a neighborhood plays a key role
in what follows. Consider a point p in an n-dimensional Euclidean space
8n' A neighborhood of p is a set of points near p entirely surrounding p.
More precisely, we define a neighborhood U of p to be any set U such that
U contains an open solid sphere or ball with center p. (N.B. In this context
an open sphere. means the set of all points of the sphere except those on the
surface, i.e., only interior points are included.) The set U Fig. 16.1a
constitutes a neighborhood U of the point p in a plane, since we may draw
an open disk in U with p at its center. This is clearly impossible in Figs
16.1b and c, as any disk with center p will contain points outside of U.
150

TOPOLOGICAL SPACES 151
u
u
u
(a)
(b)
(c)
Fig. 16.1. Neighborhoods of a point in a plane.
In a Euclidean space S Ifl we readily find that: 44,50,117,118
1. A point p belongs to any neighborhood of p.
2. If U is a neighborhood of p and V:J U, then V is a neighborhood of p.
3. If U and V are neighborhoods of p, then so is U n v.
4. If U is a neighborhood of p, then there is a neighborhood V of p such that
V c V and V is a neighborhood of each of its points.
16.2 TOPOLOGICAL SPACES
A topological space {M, U} may be defined as an abstract set M along
with the assignment to each p EM of a collection { U} of subsets U i of M,
called neighborhoods of p, such that:
1. If U i is a neighborhood of p, then p E Ui'
2. Any subset of M containing a neighborhood of p is itself a neigh-
borhood of p.
3. If U i and  are neighborhoods of p, then so is U i n .
4. If U i is a neighborhood of p, there is a neighborhood  of p such that
U i is a neighborhood of every point of .
Any system of subsets {U} is called a topology for M, and the set M
together with the topology for M will be called a topological space T
= {M, U}. A set M in a topological space T - {M, U} will be called an
open set if for any point p EM there is a neighborhood U of p such that
U eM. If the topology for M is introduced by taking the empty set 0, M,
and all open subsets of M, we obtain the discrete topology for M.
Suppose T is a topological space and TI is a subset of T. If p is a point
in T I , then the subset U of TI is called a neighborhood of p in TI if
U = TI n V, where V is a neighborhood of p in T. It may be readily verified
that the neighborhoods in TI so defined satisfy the requirements for
forming a topological space, and thus the neighborhoods in TI of the

152 SOME GLOBAL PROPERTIES OF LIE GROUPS
points of TI define a topology on TI known as the topology induced on TI
by T. Thus TI is a topological subspace of T. TI is said to be closed in T if
T - TI is open.
If there are two topologies TI and T 2 on T and if TI C T 2 , then TI is said
to be coarser than T 2 and T 2 finer than TI' If, in a topological space
T = {M, U}, U consists of M and the empty subset 0 of M, then the
resultant topology is referred to as the coarsest topology on T.
16.3 EXAMPLES OF TOPOLOGICAL SPACES
The Euclidean space S n is obviously a topological space. Consider the
surface S of a unit sphere in S 3 with its center at the origin. If we define
the neighborhood U of any point p on S as containing all points at a
distance < E from p, then the points p clearly form a topological space
which in turn is a subspace of 8 3' Likewise the interior points in an open
n-sphere form a topological space which is an open subspace of 8 n' If we
include the points lying on the surface of the sphere, then the subspace
becomes closed in 8 n'
The above examples all have a geometrical association. However, we
may on occasion wish to consider a topology in a purely algebraic context.
F or example, consider the infinite set M of integers (positive, negative, or
zero). It is possible to endow M with many topologies. Consider a neigh-
borhood of an arbitrary integer n EM as being the set uq (n, r) of integers
n+mqr, where m=O, + 1, + 2,...,q is a fixed prime number, and r is an
integer. A given choice of q will define a particular topology for M. For a
given neighborhood of n the integer r will be fixed and m will vary.
Different values of r will correspond to different neighborhoods of n. The
set {uq} of neighborhoods Uq(n,r) of the points nEM will form a
topology for M, and hence we may define a topological space T
= {M,U q }.
The set of real numbers likewise can be endowed with a topology,
namely that of the real line m. The set of real numbers falling in an
interval, say a < x < b for some a and b, will form an open set on the real
line.
The natural topology of the real line (R may be formed from the
collection of sets U C <R where for every x E U there exists a number E > 0
such that the interval {x - E < X < x + E} C U. The collection of sets U will
then contain all open intervals with rational end points together with their
intersections and unions. The topology of the real line is equivalent to that
of S I'

PRODUCTS OF TOPOLOGICAL SPACES 153
16.4 HOMEOMORPHISMS
Let Tt and T 2 be two topological spaces. We may make a mapping t of
Tt into T 2 by assigning to every pointp of Tt a well-defined point t(p) of
T 2 . This mapping is said to be continuous at p if for every neighborhood U
of t(p) in T 2 there is a neighborhood V of p in Tt such that t( V) c U. If
the mapping is continuous at all points in Tt, then it is said to be
continuous.
A mapping which establishes a one-to-one relation between all points of
one topological space and all points of a second topological space such
that the open sets of the two spaces are also in one-to-one correspondence
is said to establish a homeomorphism. Spaces which admit a
homeomorphism are said to be topologically equivalent, and the mappings
t and t - t are continuous.
16.5 PRODUCTS OF TOPOLOGICAL SPACES
We may construct a new topological space either by forming a subspace
of a given topological space or by forming the product of two or more
topological spaces.
Suppose Tt and T 2 are two topological spaces. Define the set Tt X T 2 to
be the set of pairs (p, q) where p E Tt and q E T 2 . If (p, q) E Tt X T 2 , then a
neighborhood of (p,q) is any set U X V such that U is a neighborhood of p
in Tt and V one of q in T 2 . If Tt X T 2 is made into a topological space,
then Tt X T 2 is known as the topological product of Tt and T 2 .
We have already mentioned that the real line forms the Euclidean space
8 t . If we form the topological product 8t x8 t , we clearly obtain the
topology of the plane, which is just the topology of the Euclidean space.
Indeed, the topological product space X 8n defines a product topology
on the space 8 m+n' We may of course form the product topology for any
finite number of topological spaces.
EXERCISES
16.1. Show that the topological product of two circles generates a torus.
16.2. Show that the topological product of a real-line interval with a circle
generates a cylinder.

154 SOME GLOBAL PROPERTIES OF LIE GROUPS
16.6 HAUSDORFF SPACES
A sequence {Pn} of points Pn EM is said to converge to p EM or have a
limit p EM if for every open set U of p there is an integer N such that
Pn E U for all n > N.
The range and variety of topological spaces is vast, and it is desirable to
be able to separate particular classes of topological spaces by imposing one
or more restrictive conditions. For an arbitrary topological space there is
no guarantee that the limit of a sequence of points, if it exists, is unique. It
is convenient to restrict our attention to those topological spaces where the
limit is unique. Such a restriction can be made by considering only those
topological spaces that satisfy the Hausdorff separation axiom, which we
now state:
Hausdorff Separation Axiom
If for every pair of distinct points p, q in a topological space T there is a
neighborhood U of p and V of q such that U n V = 0, then the space is said to
be a Hausdorff space and to satisfy the Hausdorff separation axiom.
If T is a Hausdorff space and a sequence {Pn} has a limit p, then the
limit is unique. Every subspace of a Hausdorff space is a Hausdorff space,
and every product of Hausdorff spaces is a Hausdorff space. A Euclidean
space 8 is obviously a Hausdorff space, since any two distinct points can
be encompassed by nonoverlapping spheres of sufficiently small radius.
Likewise the real line CR with the natural topology is a Hausdorff space.
16.7 METRIC SPACES
A nonempty set M of points will be called a metric space if to every pair
of p, q E M there is associated a nonnegative real number d(p, q), the
distance from p to q, such that
1. d(p,q)=O if and only if p=q.
2. d(p,q)=d(q,p).
3. d(p, q) + d(q,r) d(p,r) for all p, q,r E M.
The distance function d(p, q) is known as the metric. A typical metric
space will be written as (M,d), where M is the nonempty set of points and
d the distance function. A metric space may be shown to be necessarily a
Hausdorff space, though the converse need not hold.

COMPACT SPACES 155
EXERCISE
16.3. Verify that the real line <R can be considered as a metric space with the
metric d(p,q) = Ip - ql.
16.8 CONNECTED SPACES
It is important to know, in the discussion of the properties of a given
topological space, whether it consists of a single connected piece or
whether it can be decomposed into several disjoint pieces.
A topological space T is said to be connected if it cannot be represented
as the union of two disjoint nonempty sets. A subspace Tl of T is said to
be a connected subspace of T if it is itself connected as a topological space.
If Tl and T 2 are connected spaces, then so is their product Tl X T 2 .
Furthermore, if t: TlT2 is a continuous mapping t of a connected space
Tl onto a space T 2 , then T 2 is necessarily a connected space. It follows
that if Tl and T 2 are homeomorphic topological spaces, then Tl is
connected if and only if T 2 is connected.
The real line (R provides an example of a connected topological space.
The intervals on the real line are connected subspaces, and in fact are the
only connected subspaces of the real line. The subspace of the real line
consisting of all rational numbers is clearly a disconnected space, since
given any two distinct rationals x and z with (say) x <z, we can always
find an irrational number y such that x <y < z. In this case the space is
totally disconnected.
All finite-dimensional Euclidean gn and unitary spaces C n are connected.
We may build the Euclidean space S.n from the n-fold product of the real
line, which is known to be connected, and hence & must be connectd.
The connectedness of C n can be demonstrated by showing that C n and 8 1n
are homeomorphic; since 8 2n is connected, C n must be also.
A topological space T is said to be a locally connected space if every
point pET has a connected neighborhood. The union of two disjoint open
intervals of the real line provides an example of a locally connected space.
16.9 COMPACT SPACES
Closed surfaces in 8 3 such as the sphere and the torus are contained in a
finite portion of the space and may be described as closed and bounded.
In an arbitrary metric space fil" a subspace X having an induced metric

156 SOME GLOBAL PROPERTIES OF LIE GROUPS
and a topology will be bounded if there is a number k such that the
distance function d(p,q)<k for allp,qEX.
The concept of the boundedness of closed sets in Euclidean spaces 8h
can be related to systems of open sets, known as open coverings, via a
generalization of the Heine-Borel theorem. This has the advantage of not
then requiring the concept of boundedness, which is not a topological
property. The concept of topological compactness then follows.
A system of subsets {U} of a topological space T is said to cover T if
their union contains T, so that every point pET is contained in some set of
the system. The system of sets is known as a covering of T. Thus the
collection of all vertical lines in8 2 can be said to form a covering of 8 2 and
will also form a covering of any subset of 8 2 , If each set is open, then the
covering is said to be open. If the covering involves a finite number of sets,
then the covering is said to be a finite covering.
We now have, from a generalization of the Heine-Borel theorem to
Euclidean spaces, the theorem
Theorem 16.1
If X is a closed bounded subspace of 8n then every open covering {U} of X
contains a finite open covering.
As a consequence of Theorem 16.1, we define a topological space T, not
necessarily a Euclidean space, to be compact if any open covering of T
contains a finite open covering. Thus a closed bounded subspace of 8" is
by definition compact. Indeed, it may be shown that any compact sub-
space of a metric space is closed and bounded. Furthermore, a closed
subspace of a compact space is necessarily compact.
The topological product Tl X T 2 will be compact only if Tl and T 2 are
compact. Thus the product of two circles generates the torus in 8 3 , which is
obviously compact, whereas the product of the real line at wIth a circle
clearly yields a noncompact topological space, namely that of an open
cylinder. We say that a space is locally compact if every point lies in an
open set whose closure is compact. Thus 8" is a locally compact space
since any open sphere centered on any point forms the neighborhood of a
point whose closure is compact.
16.10 HOMOTOPIC PATHS
The concept of homotopic paths plays an important role in the further
development of the idea of connectivity in topological spaces and particu-
larly in the discussion of the global properties of groups.

HOMOTOPIC PATHS 157
Consider some topological space T, and let / denote the unit interval
O<s< I, which may be regarded as a subspace of the space of real
numbers. A path in T joining two points p and q of T is defined as a
continuous mapping t of / into T such that t(O)=p and t(I)=q.
If for every pair of points p and q of T there is a path in T joining p and
q, then the space T is said to be arcwise connected. As an example consider
a two-dimensional Euclidean space S 2' Any pair of points p and q in 8 1
may be designated by the coordinates (x},y}) and (X2'Y2)' respectively. A
mapping t of / into 8'). may be made by putting t(s) equal to the point with
coordinates (f}(s),g}(s)), where f}(s)=(I-s)x} +sy} and g}(s)=(I-s)y} +
sx 2 . Clearly, the mapping is continuous, and t(O)=p and t(l)=q. Thus
there is a path t from p to q in 0:2' and since the points p and q are
arbitrary points in 8 2 , we conclude that 8 2 is arcwise connected.
EXERCISES
16.4. Verify that any Euclidean space S n is arcwise connected.
16.5. Verify that the circumference of a circle is arcwise connected.
A mapping t: /--+T is said to form a curve. If the mapping t: /T is such
that t(O)=t(I)=P o (the base point), then the curve is said to be closed.
Clearly we may have whole families of curves that have the same begin-
ning and end points or that share a common base point. The mapping t(s)
of the unit interval/into the topological space T constitutes the parame-
terization of curves in such a way that to every point (apart from the base
point) on a given curve there corresponds a unique value of the parameter
s.
Consider two curves t} = t}(s) and t 2 = t 2 (s) that are continuous functions
of a parameter s (O<s< I) such that they have common end points, that is,
t } ( 0) = t 2 ( 0 ) and t } ( 1 ) = t 2 ( 1 )
( 16.1 )
The curves t}(s) and t 2 (s) are said to be homotopic with fixed endpoints (or,
more loosely, simply homotopic) if there exists a function t(r,s) (0< r,s < I),
continuous in both rand s, such that
t ( 0, s ) = t } ( s ) and t ( 1, s ) = t 2 ( S )
( 16.2)
We note that rand s define points in the unit square /2 lying in the (r,s)
plane, and thus we may say that curves or paths in a topological space T
will be homotopic if there is a continuous mapping t of /2 into T such that
t(O,s) = t}(s) and t(l,s)= t 2 (s) for all s E/.
If two curves are homotopic, then we can pass continuously from one to

158 SOME GLOBAL PROPERTIES OF LIE GROUPS
the other by deformation of the parameter r. The set of all curves
homotopic to a given curve t is here designated as [t] and is said to form a
homotopy class.
The homotopy classes of closed paths are of particular interest. Consider
a point p in a topological space T. The path I p = t p (s)=p (O<s< 1) is
called the constant path or null path at p. The set of all curves homotopic to
the null path Ip at a point p is [lp]' If tp(s) is a closed path, not necessarily
a null path, at a point pET with t p E[l p ], then the homotopy from tp(s) to
Ip must satisfy the conditions:
t ( 0, s ) = t ( s ) ) (O<s<l) ( 16.3 )
t(l,s)=p
t(r,O) =p ) (O<r<l) ( 16.4 )
t(r,I)=p
The homotopy from tp(s) to Ip amounts to the continuous deformation of
tp(s) into the point p. This is possible only if the curve tp(s) does not
enclose any holes in the space T.
16.11 SIMPLY CONNECTED AND MULTIPLY CONNECTED SPACES
A topological space T is said to be simply connected if for each point
pET there is only one homotopy class of closed paths. If there are m
homotopy classes associated with each point pET then the space is said to
be m-fold connected.
Paths on a disk are simply connected, since for any point p on the disk
we can construct only one homotopy class of paths that can be deformed
into the point p. If the disk is transformed into an annulus by the removal
of a central inner disk, then there will be, for any point in the annulus, two
distinct types of homotopy classes, namely, those paths that do not enclose
the central hole and are thus deformable to the chosen point, and those
paths that enclose the hole and cannot be deformed to the chosen point.
For these latter paths, a distinction must be made between curves that
circumscribe the center different numbers of times, since these cannot be
deformed into one another, and hence there is an infinite number of
homotopy classes, so that the space of the annulus is infinitely connected.

THE FUNDAMENTAL GROUP 159
EXERCISES
16.6. Show that for a torus there is an infinite number of homotopy classes, and
hence the torus is infinitely connected.
16.7. An n-sphere S n is the set of all (n + I)-tuples of real numbers (XI,X2"'"
X n + I) such that
xi + x + . .. + x; + I = 1
( 16.5)
It forms a subspace in 8 n+ I' Show that for n> 1 the n-sphere is simply
connected. (N.B. The I-sphere is the circle, which is infinitely connected.)
16.12 THE FUNDAMENTAL GROUP
We may use the concept of homotopic paths to define a finite group
known as the fundamental group or the homotopy group which bears an
intimate relationship to the connectivity properties of a topological space.
To construct the fundamental group in terms of homotopic paths, we must
first define what is meant by the inverse of a path and the product of two
paths.
Consider a path t(s) (O<s< I) in a topological space T. The inverse path
t-I(s) is defined by the same set of points traversed in the opposite order.
Clearly
t-I(s) =t(l-s)
( 16.6)
Formally, we write the set of paths homotopic to t as [t], and that of the
inverse paths as [t- I ].
Suppose the end point of a path coincides with the beginning of a
second path. We define their product path as the first path followed by the
second. The function associated with the product path t 12 = tIt 2 can be
written as
t - ( t I (2s )
12 -
t 2 (2s - I)
(O<s<t)
(-!<s<l)
( 16.7)
In terms of sets of homotopic paths we have
[t I2 ] = [t l ][t 2 ]
( 16.8)
Furthermore,
[t][t-I]=[I]
( 16.9)

160 SOME GLOBAL PROPERTIES OF LIE GROUPS
where [1] is, as before, the set of paths homotopic to the null path, and [t]
and [t- I ] are closed paths having a common base point.
It is apparent that closed curves or paths that possess a common base
point can be multiplied together to yield a path belonging to the same set.
The set [1] plays the role of an identity element. Furthermore, it can be
shown that the product (t l t 2 )t 3 is homotopic to t l (t 2 t 3 ) and hence the
product [t 1][ t 2 ][ t 3] is associative. Hence the set of homotopic paths begin-
ning and ending at a base point p may be used to define a group known,
following the early work of Poincare, as the fundamental group and
designated as III(T,p).
While the fundamental group is defined via a particular point pET, we
may show that the fundamental groups of any two points of an arcwise
connected space are isomorphic, and thus the fundamental group is
essentially independent of the base point.
The fundamental group of a simply connected space is just the identity
element [1]. The fundamental group of the circle will be an infinite cyclic
group consisting of the identity and the integral powers of the class of the
path that goes around the circle just once. Similarly the fundamental group
of the torus will contain an infinite number of elements. Spaces that are
m-fold connected will be just those spaces having fundamental groups of
order m. Topological spaces having the same fundamental group and the
same connectivity may be mapped into each other in a continuous one-to-
one correspondence.
16.13 UNIVERSAL COVERING SPACES
We have seen that the I-sphere S 1 or circle is infinitely connected. The
fundamental group of the circle is isomorphic to the additive group of
integers. Each nonnegative integer n corresponds to a loop wound coun-
terclockwise about the circle n times, whereas - n corresponds to a loop
would clockwise about the circle n times. If S 1 is a unit circle and CR I the
real line, then we may make a covering of S I with CR I by means of the
mapping p:lR\lS I, where p(s) = e 2 'lTis. In this case <R 1 provides a covering
space for S I. In a sense, the mapping just indicated may be regarded as
wrapping the real line around the circle, covering the circle an infinite
number of times.
In general, if W is a covering space of a sapce T and if W is simply
connected, then W is called the universal covering space of T. The real line
CR \ is simply connected and hence forms the universal covering space for
the circle S I. Schreier l18a has shown that for any locally connected and
locally compact space there is a unique universal covering space. The

TOPOLOGICAL GROUPS 161
importance of the universal covering space becomes apparent in Section
16.24, where we discuss Lie groups that are locally isomorphic.
16.14 TOPOLOGICAL GROUPS
So far in this chapter we have sketched some of the broad properties of
topological spaces. Now we attempt to combine the concept of a topologi-
cal space with that of an abstract group to introduce the idea of a
topological group.
It is possible for a set {G} of elements g to define at one and the same
time both an abstract group {G,m} and a topological space {G,T}. The set
{G} is supplied with a group structure {G, m} by taking it together with a
function m: GX GG called the group product, which defines the law of
combination of the group elements. The same set {G} may be equipped
with a topological structure by specification of a topology T on G.
The triple {G, m, T}, where {G, m} is a group G and {G, T} is a topologi-
cal space 9, is said to define a topological group if the group operations in
G are continuous in the topological space 9 .
The continuity requirement will be met if both the following conditions
hold:
1. If gl and g2 are two elements of {G} then for every neighborhood U I2 of
g I g2 there exist neighborhoods U I and U 2 of g I and g2 such that U I U 2 C U 12 .
2. If g is any element in {G}, then for every neighborhood V of g -I there
exists a neighborhood U of g such that U - leV.
These two requirements may be combined into the single condition that
3. If gl,g2 E {G}, then for every neighborhood U;2 of glg2- 1 there exists
neighborhoods U I and U 2 of gl and g2 such that U I U 2 - 1 c U;2'
In practice we designate the topological group associated with the triple
{G,m,T} simply as G. The properties earlier associated with topological
spaces may be carried over to topological groups. Thus if a topological
group G considered as a topological space is connected, then so is the
topological group.
A set {G} may be an abstract group and a topological space without
necessarily being a topological group. The formation of a topological
group is possible only if the aforementioned continuity requirement is
satisfied.
A topological group may be continuous or discrete. As a trivial example,
consider the finite group G with elements e,a,b which multiply as shown in
Table 16.1.

162 SOME GLOBAL PROPERTIES OF LIE GROUPS
Table 16.1. Group Multiplication table for G.
e a b
e e a b
a a b e
b b e a
The group G is an abstract group, and its elements may be equipped with a
variety of topologies. The open sets (or neighborhoods)
{0}
U I
{e}
U 2
{a}
U 3
{ e, a}
U 4
{e,a,b}
Us
( 16.10)
define a particular topology which satisfies the requirements for a topologi-
cal space. We now ask, "Does G equipped with this topology form a
topological group?" Consider the group-element product aa. Then a-I = b,
and hence aa- I = e exists in the neighborhoods U 2 , U 4 , and Us. The
element a exists in the neighborhoods U 3 , U 4 , and Us' while a-I exists
only in Us. Consider the neighborhood U 2 of e. Clearly U 3 U S ' U 4 U S ' and
UsUs cannot be contained in U 2 , and this requirement 3 is not satisfied.
Hence the group structure and the topological space are inconsistent, and
thus, given the topology defined by the open sets of Eq. 16.10, G cannot be
a topological group.
Let us equip G with a topology defined by the open sets
{0}
U I
{e}
U 2
{a}
U 3
{b}
U 4
{e,a,b}
Us
( 16.11 )
It is now trivial to see that requirement 3 is satisfied, and thus if G is
equipped with the topology of Eq. 16.11, then G forms a topological group.
In general, a discrete group can always be made into a topological group
by taking the neighborhood of every element as the element itself.
The additive group of real numbers may be equipped with the topology
of the real line to form an example of a continuous topological group. As
noted in Chapter 4, in the case of the group GL(n, C) we may regard the n 2
elements of the matrices as labeling points in a complex Euclidean space
C n2 or in a real Euclidean space R 2n \ that is, we may equip GL(n, C) with
a topology so that GL(n, C) becomes a topological group. Again we have
an example of a continuous topological group. Throughout this book, our
primary interest is in the properties of continuous topological groups rather
than those of discrete topological groups.

TOPOLOGICAL SUBGROUPS 163
16.15 PRODUCTS OF TOPOLOGICAL GROUPS
Let G 1 - {G1,ml,'T 1 } and G 2 - {G 2 ,m 2 , 'T 2 } be two topological groups
having associated group structures {G1,m l } and {G 2 ,m 2 } and topological
spaces {G1,'T 1 } and {G 2 ,'T2}' The product G 1 X G 2 is a product topological
group if it is equipped with the group structure {G1,m l } X {G 2 ,m 2 } and the
topological space {G 1 , 'T 1 } X {G 2 , 'T 2 }.
In the case of GL(n l , C) X GL(n 2 , C) the product group structure may be
formed by taking the direct products of the elements of GL(n l , C) with
those of GL(n 2 , C). The resulting matrices are of rank n l + n 2 . The ele-
ments of these matrices will be expressible in terms of ni + n complex or
2(ni + n) real parameters. The topological product space may be formed
2 2 2 2
from the complex product space cnl+n2cnl X cn 2 or the real product
space R2(n?+n)R2n? X R2n. Thus GL(n l , C) X GL(n 2 , C) forms a topologi-
cal group.
16.16. ISOMORPHISM OF TOPOLOGICAL GROUPS
A given abstract group {G, m} may be associated with different topolo-
gies leading to different topological groups. Likewise different topological
groups may have homeomorphic topological spaces. Thus care must be
taken in applying the concept of isomorphism to topological groups.
Two topological groups are said to be topologically isomorphic if their
abstract groups are isomorphic and their associated topological spaces are
homeomorphic.
16.17 TOPOLOGICAL SUBGROUPS
A topological group G = { G, m, 'T} admits a topological subgroup H
= {H, m, 'T} only if simultaneously:
1. {H, m} is a subgroup of the abstract group {G, m }.
2. {H, 'T} is a closed subspace of the topological space {G, 'T}.
An abstract subgroup of a topological group need not be a topological
group. Thus the abstract additive group of integers is obviously a subgroup
of the abstract additive group of real numbers, but the set of real integers
cannot form a closed subspace of the natural topology of the real line, and
hence we cannot form a topological subgroup out of the set of integers.

164 SOME GLOBAL PROPERTIES OF LIE GROUPS
16.18 INVARIANT TOPOLOGICAL SUBGROUPS
A topological subgroup N of a topological group G is termed a normal
or invariant subgroup if
g-INg=N
(forallgEG)
(16.12)
The center Z of an abstract group G consists of all the elements z E G
that commute with every element g E G. It follows that the center Z is
necessarily an Abelian invariant subgroup of G. The center of the abstract
group G is also termed the center of the topological group G when G is
endowed with a topology.
Since for a continuous semisimple group there can be no continuous
Abelian invariant subgroups, the center of a continuous semisimple group
is necessarily discrete and cannot contain more elements than its rank I.
Thus in the case of SU(2) the center Z2 comprises just the two elements,
(  )
and ( - I 0 )
o -1
( 16.13)
In the case of the nonsemisimple group GL(n, C), any multiple A of the
identity matrix will commute with all the elements of the group, and the
center will comprise the infinite set Z = {A }.
16.19 COSET SPACES AND FACTOR GROUPS
Let G be an abstract group and H a subgroup of G. The sets glH and
g2H with gl'g2 E G either coincide or are completely disjoint. Coincidence
occurs if, and only if, gllg2 E H. The sets g;H are termed left cosets of H,
and the sets Hg; right cosets of H. The set {gH} of disjoint left co sets of H
will be designated as G / H.
The cosets gH and Hg need not necessarily coincide. However, if
gH = Hg for all g E G, we have
g-IHg=H
and hence H is then an invariant subgroup. In this case we may construct
a new group G / H known as the factor or quotient group. The elements of
G / H are the cosets of H, and multiplication in G / H is defined in terms of
the products of cosets. If cp( g) = gH defines a mapping of G into G / H,
then the resulting homomorphism is called the natural or canonical

COSET SPACES AND FACTOR GROUPS 165
homomorphism of G onto G / H. In the case of SU(2) the mapping
SU(2) SU(2)/Z2
will be two to one, the matrices A and - A of SU(2) being mapped onto a
single element of SU(2)/ Z2'
Table 16.2. Group Multiplication for V 4
e a b e
e e a b e
a a e e b
b bee a
e e b a e
As a sitnple example consider the four-group V 4 , whose multiplication
table is given in Table 16.2. There are three Abelian invariant subgroups
comprising the elements {e,a}, {e, b}, and {e, c}. Suppose we consider
the subgroup H = { e, a}. The two distinct left cosets of Hare e { e, a} and
b{ e,a}, where
G  {e,a} +b{e,a} = {e,a} + {b,c}
Let us designate the two cosets as E = { e, a} and A = { b, c} . We then have
for the factor group
G/H=E+A
where EA =A and A 2 =E. A two-to-one homomorphic mapping of
G  G / H can be made by mapping e and a onto E, and band c onto A,
namely,
e
aE
b
A
c
(16.14)

166 SOME GLOBAL PROPERTIES OF LIE GROUPS
For a factor group G / H to become a topological group, it must be
endowed with a topology. If G is a space, then the set of cosets G / H forms
a coset space. Let us now topologize this coset space. Let cp be the natural
mapping of G into G / H; then we can define a quotient topology on G / H
by requiring that a set U in G / H be open if, and only if, cp - I( U) is an
open subset of G. In this way we ensure that G / H is a topological space.
The topological space G / H endowed with a quotient topology is called a
quotient space.
In the preceding example of V 4 we may equip V 4 with a simple topology
defined by the open sets
{ 0} { e } { a } { b } { c } { e, a, b, c }
It follows from the mapping cp of Eq. 16.14 that the open sets
{ 0} {E} {A } {E,A }
of G / H form the quotient topology for G / H, since the inverse mapping of
each set in G / H is an open subset of G.
16.20 HOMOGENEOUS SPACES
Consider a topological group G with elements g. For each gEG the
mappings glggl and g2gl g are termed left (Tg) and right (Tg) transla-
tions respectively. Any element gl E G can be carried into any element
g2 E G by a left translation TgL, where
g2 = TgLgI = ggl (16.15)
in which
-I
g=g2g1
(16.16)
-and similarly for right translations. The uniqueness and continuity of
group multiplication ensures that the left or right translations by a con-
stant element are homeomorphisms of G onto G.
A topological space {G, T} is said to be a homogeneous space if each
point may be mapped into any other point by an element of G. It follows
that any topological group is necessarily homogeneous. The homogenity of
a group G ensures that any local property determined in the neighborhood
of one point in G can be translated to any other point in G, justifying our
earlier concentration on the local properties in the neighborhood of the
identity element.

REAL SIMPLE LIE GROUPS AND LIE ALGEBRAS 167
EXEROSE
16.8. Show that every quotient space G / H, where G is a topological group and H
a closed subgroup, is a homogeneous space.
16.21 MANIFOLDS AND LIE GROUPS
. The Euclidean space 8n is a particular example of a Hausdorff topologi-
cal space that has been equipped with a system of real coordinates
(x l' X 2' .. . ,x n ). Euclidean spaces of different dimension cannot be
homeomorphic. A topological space Tn of fixed dimension n in which
every point has a neighborhood that is homeomorphic to an open set in the
Euclidean space 8n will be said to be a locally Euclidean space. If Tn is a
connected locally Euclidean space, then Tn is said to form a topological
manifold. A topological manifold is necessarily locally compact and locally
connected.
If it is possible to introduce differentiable coordinates into the topologi-
cal ma.nifold Tn' then Tn is said to be a differentiable manifold. This means
that if in any two overlapping neighborhoods of Tn there exist two
coordinate systems, then these two coordinate systems, in the region of
overlap, are differentiable functions of one another. If the coordinate
transformations are described by n functions with continuous partial
derivatives of degree x, then the differentiable manifold is said to be of
class ex. If the manifold is differentiable to all orders, it is said to be of
class e 00 .
Likewise, if in the region of overlap the n functions associated with the
coordinate transformation are analytic functions, then the topological
manifold is said to be an analytic manifold.
A complex manifold of dimension n is formed by the replacement of the
real Euclidean space (Rn by the complex space en.
A topological group whose topological space is a topological manifold is
called a Lie group. It follows that a Lie group is necessarily locally compact
and locally connected. The group is termed a real Lie group if the
topological manifold is real, and a complex Lie group if the topological
manifold is complex.
16.22 REAL SIMPLE LIE GROUPS AND LIE ALGEBRAS
A complete classification of the complex semisimple Lie algebras was
given in Chapter 6. We found that there were four infinite sequences of

168 SOME GLOBAL PROPERTIES OF LIE GROUPS
complex semisimple Lie algebras An' Bn' Cn' and Dn' as well as the five
exceptional Lie algebras G 2 , F 4 , £6' E 7 , and Eg. We noted in Section 5.8
that several real simple Lie algebras may have complex extensions that are
isomorphic to a single complex Lie algebra. For example, the real Lie
algebras so(3) and so(2, 1) both have A 1 as their complex extension. Among
the nonisomorphic real simple Lie algebras whose complex extensions are
isomorphic to a single complex simple Lie algebra, only one is a compact
Lie algebra. We recall that a Lie algebra A over the field of real numbers is
compact if, and only if, its Killing form is negative definite.
The task of supplying a complete classification of the real simple Lie
algebras was first undertaken by Cartan,33, 119, 120 and later by Lardy;21
who obtained a complete classification. A purely algebraic solution was
given by Gantmakher 122 and later simplified by Yen Chih-ta 123. Reviews
have been given by Barut and Raczka 64 and by Sirota and Solodovni-
kOV 124 . Here we are content to sketch the principal ideas and then present
the complete classification.
Any complex simple Lie algebra Ac can be developed in a basis where
the structure constants c:>.. are all real and a real simple Lie algebra Ar
isomorphic to Ac can be taken as a real Lie algebra of twice the dimension.
The algebra Ar is necessarily compact. For example, from the complex
simple Lie algebra so(3, C) we may obtain the real simple Lie algebra
so(3). For future reference we designate the real compact form of a
complex simple Lie algebra Ac as Arc.
Let e 1 , e 2 ,... ,en be a basis for Arc. We can transform the basis of Arc into
a new basis (see Section 5.2) by making the linear transformation
_lDk
a. -\r ,e k
I I
(i= l,...,n)
(16.17)
Not every transformation <Pwilllead to a nonisomorphic Lie algebra.
Consider the case of so(3, C). The real compact form of so(3, C) is the
Lie algebra so(3) with basis elements X 1 ,X 2 ,x3 which satisfy the commuta-
tion reI a tions
[X 1 ,X 2 ] = x 3 ,
[X 2 ,X3] =x 1 ,
[X 3 ,X 1 ]=X 2
( 16.18)
We can obtain a new algebra with basis elements Yl,Y2'Y3 by making the
transformation
Yl I Xl
Y2 - I X 2
Y3 1 x 3
(16.19)

REAL SIMPLE LIE GROUPS AND LIE ALGEBRAS 169
The Yi now satisfy the commutation relations
[ Y I'Y 2] = - Y 3'
[y 2'Y 3] = Y I'
[Y3'YI] =Y2
( 16.20)
which is nonisomorphic to so(3), being in fact the noncompact simple
algebra of so(2, 1). Had we chosen
1
I
(P=
I
or (P=
1
I
I
we would simply have obtained two algebras isomorphic with the so(2, 1)
already found. Applying (P twice restores the original algebra and thus
constitutes an involutive automorphism of so(3).
Gantmakher l22 has shown that any transformation (P of the basis of Arc
will lead to an inequivalent set of structure constants if, and only if, (P is an
n X n matrix such that
(P=vs = l-i S + l+i 
2 2
(16.21 )
where  is the n X n unit matrix and S is an involutive automorphism of
A:. For S to be an involutive automorphism we necessarily have that
S2= 
(16.22)
A basis for Arc can always be chosen so that S is diagonal, and hence
diagonal elements of + 1 and hence (P must be of the form
1
1
(P=
1
(16.23)
I
I
I

170 SOME GLOBAL PROPERTIES OF LIE GROUPS
Cartan 33 , 119, 120 has shown that the complete set of real forms of a given
complex semisimple Lie algebra Ac may be found by first forming all the
nonequivalent involutive automorphisms S of Arc in a basis that diagona-
lized S. Now multiply all basis vectors of Arc associated with the - I
eigenvalues of S by i and leave the remaining basis vectors unchanged.
The new basis then corresponds to that of the real simple Lie algebra
associated with the involutive automorphism S.
We now give a listing of all the nonexceptional real simple Lie algebras.
Complete listings, including the exceptional algebras, have been given by
Helgason,46 Barnt and Raczka,64 and Sirota and Solodovnikav}24 Acces-
sible derivations are given by Sirota and Soiodovnikov I24 and by Hausner
and Schwartz. 52
A. Real Forms of sl(n, C)
These algebras all have An-I (n> 1) as their complex extension. We list
first the real algebra and then its matrix realization.
1. su(n)-All skew Hermitian matrices Z of order nand TrZ=O. This
algebra is the real compact form of sl(n, C).
2. sl(n, R)-All real matrices X of order n with Tr X = O.
3. su(p,q) (p+q=n withp>q)-All matrices of the form ( ZI Z2 ) ,
tZ* Z
2 3
in which ZPZ3 are skew Hermitian of order p and q, respectively, and
Tr ZI +Tr Z3 =0 with Z2 arbitrary.
4. su*(2n)-All complex matrices of the form ( ZI
-Z*
2
and Z2 are n X n complex matrices and Tr Z I + Tr Z t = o.
Z2 ) , where ZI
Z*
1
B. Real Forms of so(2n + 1, C)
These algebras all have Bn as their complex extension.
1. so(2n + I)-All real skew symmetric matrices of order 2n + I. This
algebra is the real compact form of so(2n + I, C).
2. so(p, q) (p + q = 2n + I with p > q)-All real matrices of order 2n + I of
the form ( XI X 2 ) , where X 2 is arbitrary and Xl and X 2 are skew
tX 2 X 3
symmetric of order p and q, respectively.

REAL SIMPLE LIE GROUPS AND LIE ALGEBRAS 17]
c. Real Forms of sp(2n, C)
These algebras all have C n as their complex extension.
1. sp(2n)-All skew-Hermitian matrices of order 2n of the form
( ZI 2 ) , where the Z; are complex matrices of order n with Z2 and
Z3 - ZI
Z3 symmetries. This algebra is the real compact form of sp(2n, C) and
corresponds to the intersection of sp(2n, C) with su(2n).
2. sp(2n,R)-AlI real matrices of order 2n of the form ( XI 2 ) ,
X 3 - XI
where all X; real matrices of order n with X 2 and X 3 are symmetric.
3. sp(p,q) (p+q=2n withp'>q andp,q both even integers)-All
complex matrices of order 2n and form
Zl1 ZI2 ZI3 ZI4
IZ* Z22 IZ 14 Z24
11
-Z* ZI Z -Z*
13 12
IZ. -Z. _IZ12 Z2*2
14 24
where all Zij are complex matrices with Zl1 and ZI3 of order p, ZI2 and ZI4
p X q matrices, ZII and Z22 skew Hermitian, and ZI3 and Z24 symmetric.
D. Real Forms of so(2n, C)
The algebras all have Dn as their complex extension.
1. so(2n)-All real skew symmetric matrices of order 2n. This algebra is
the real compact form of so(2n, C).
2. so(p, q) (p + q = 2n,pq)-All real matrices of order 2n of the form
( Xl X 2 ) , with XI and X 3 skew symmetric of order p and q, respective-
IX 2 X 3
ly, and X 2 arbitrary.
3. so*(2n)-All complex matrices of order 2n of the form ( ZI Z2 ) ,
-Z. Z
2 I
where ZI and Z2 are complx matrices of order n, with ZI skew Hermitian
and Z2 Hermitian.

172 SOME GLOBAL PROPERTIES OF LIE GROUPS
Knowing the real forms of the complex semisimple Lie algebras leads to
a classification of the real Lie groups.
The classical real simple Lie groups are listed in Table 16.3. The first group
in each class is compact, all others being noncompact. Listings of the
exceptional real simple Lie groups may be found in Helgason,46 Hausner
and Schwartz, 52 and Barut and Raczka. 64
Table 16.3. The Classical Real Simple Lie Groups
Complex
extension
Real
group
Dimension
Invariant form
An-l
SU(n)
SU(p, q)
SL( n, R)
SU*(2n)
n 2 + 1
n 2 + 1
n 2 + 1
n2+ 1
Bn
SO(2n + 1) 2n 2 + n
SO(p, q)
C n
Sp(2n)
Sp(2n, R)
Sp(p,q)
Dn
SO(2n)
SO(p. q)
SO*(2n)
2n 2 + n
2n 2 + n
2n 2 + n
2n 2 + n
2n 2 - n
2n 2 - n
2n 2 - n
XIXr+ ... + xnx:
P n
-  x;x;*+  XkX:
i-I k-p+1
Unimodular group
(Z l' . . . , Zn' Zn + l' . . . , Z 2 n)
q> ( . . * - * )
 Z n + l' . . . , Z(211 , - Z I , . . . , Zn
2n+ I
 xl
i= I
P 2n + I
 xl-  XTc
i=1 k=p+1
X 1 Y2 - X2YI + ... + X 2n - 1 Y2n
XIY2 - X2YI + ... + X2n-IY2n
XIY2 - X2YI + ... + x 2n - 1 Y2n and
. . . + + *
-x1YI- ... -xpYp+xp+IYp+1 ... x 2 nY2n
2n
 xl
i= I
P 2n
-  xl +  xlc
i=1 k-p+l
2n
 xl and
i-I
2n-1
 (X;Xf+ 1 - X;+ lxf
i-I

ISOMORPHISMS OF LIE GROUPS AND LIE ALGEBRAS 173
Table 16.4. Isomorphisms oj Complex Semisimple Lie Algebras
A1--B1--C 1
B 2 --C 2
D 2 --A 1 EBA 1
A3-- D 3
16.23 ISOMORPHISMS OF LIE GROUPS AND LIE ALGEBRAS
An inspection of the Dynkin diagrams associated with the complex
semisimple Lie algebras readily reveals that for the lowest-order algebras
isomorphisms can occur. These are gIven in Table 16.4. These isomorphisms
imply the existence of isomorphisms for the corresponding real simple
Lie algebras. These are given in Table 16.5. In addition to the iso-
morphisms given in this table, there is 46
so( 6,2) so*(8)
which was missed in Cartan's original classification.
Table 16.5. Isomorphisms of Real Simple Lie Algebras
Standard form Isomorphic Lie algebras
A 1-- B 1-- C 1 su(2)--so(3)--sp(2)--su*(2)
su(l, 1)--so(2, 1)--sp(2,R)--sl(2,R)
B 2 --C 2 so(5)--sp(4)
so(4,1)--sp(2,2)
so(3,2)--sp(4,Jl)
D 2 --A 1 EBA 1 so(4)--su(2) EBsu(2)--so(3) EBso(3)--sp(2) EBsp(2)
so*(4)--su(2)EBsl(2, R)
so(3, 1 )--sl(2, C)
so(2, 2)--s/(2, R)EBsl(2, R)
A3--D3 su(4)--so(6)
su(3, 1 )--so*(6)
su*(4)--so(5,1)
sl( 4, R)--so(3, 3)
su(2,2)--so(4,2)

174 SOME GLOBAL PROPERTIES OF LIE GROUPS
The corresponding Lie groups are locally isomorphic but generally not
globally isomorphic. A list of the isomorphisms for the exceptional Lie
groups has been given by Helgason. 46
16.24 UNIVERSAL COVERING GROUP
It can be shown 44 that for any multiply connected group_ G there exists a
unique (up to an isomorphism) simply conneted group G, known as the
universal covering group Of G, such that G can be homomorphically
mapped onto G. The group {; contains a discrete invariant subgroup K
such that G is locally isomorphic to G j K. Associated with a given simply
connected Lie goup G there is a set r of connected, locally isomorphic Lie
groups having G as their universal covering group. Any other member of
the set r may be obtained from (; as the factor group G j K, where K is a
discrete invariant subgroup contained in the center i of G. The factor
group G j K has as its center the group i j K with its fundamental group
isomorphic to K.
We have already noted that the center of SU(2) is just Z2' and thus
SU(2)jZ2 must be locally isomorphic to SU(2). Indeed,
SU(2)jZ2SO(3)
and SU(2) is the covering group of SO(3).
EXERCISE
16.9. Verify that the covering group for SO(2) is the group of real numbers under
addition such that
SO(2)-.;R/Zoo
and hence SO(2) is infinitely connected.
A list of covering groups and local isomorphisms for real semisimple Lie
groups having isomorphic Lie algebras is given in Table 16.6. Among the
classical compact Lie groups, S U (I + I) and Sp (n) are simply connected
and thus form their own covering group, while SO(n) with n > 2 is doubly
connected and has a twofold covering by the simply connected spinor
group designated Spin(n) by Chevalley.45 We shall have more to say of the
spinor groups when we take up the construction of the representations of
SO(3). A list of the sets of locally isomorphic compact Lie groups and their
universal covering-groups is given in Table 16.7.

EXERCISES 175
Table 16.6. Covering Groups and Local Isomorphisms
Group
Covering group
Isomorphism
SO(2)
SO(3)
SO(4)
SO(5)
SO(6)
SO(2,1)
SO(3, 1)
SO(2,2)
SO(4,1)
SO(3,2)
SO(3,3)
SO(4,2)
R
S U(2)
S U(2) X S U(2)
Sp(4)
SU(4)
SU(I,I)
SL(2, C)
SU(I, I) X SU(I, 1)
Sp(2,2)
Sp( 4, R)
SL(4,R)
SU(2,2)
SO(2)R/Zoo
SO(3)SU(2)/Z2
SO(4)[SU(2) X SU(2)]/ Z2
SO(5)Sp(4)/Z2
SO(6)SU(4)/Z2
SO(2, 1)SU(I, 1)/Z2
SO(3, 1)SL(2,C)/Z2
SO(2,2)[SU(I, l)x SU(I, 1)]/Z2
SO(4, 1)Sp(2,2)/Z2
SO(3, 2)Sp(4, R)/ Z2
SO(3, 3)SL(4, R)/ Z2
SO(4,2)SU(2,2)/ Z2
The importance of the universal covering group G associated with a set
r of locally isomorphic Lie groups lies in the fact that all of its irreducible
representations are single-valued, whereas among those of the m-fold
connected groups G there will be some that are m-valued. However, every
irreducible representation of G is a single-valued representation of G. Thus
all the representations of G may be found from a study of the single-valued
representations of its universal covering group G.
Table 16.7. Sets of Locally Isomorphic Compact Lie Groups
Universal
covenng
group
Center
Factor groups
G
SU(I+ 1)
Sp(21)
Spin(21 + 1)
Spin(21)
Z
Z/+1
Z2
Z2
j Z 4 (I odd) t
1 Z 2 X Z 2 ( I even) f
e/K
SU(l + 1)/ Z/+ 1; SU(l + 1)/ K
Sp(21)/Z2
Spin(21+ 1)/Z2--S0(21+ 1)
Spin(21)/Z2--S0(21); SO(21)/Z2

17
Representations of Some
Three-Parameter Lie Groups
17.1 THE THREE-PARAMETER LIE GROUPS
The three-parameter Lie groups and their Lie algebras find important
applications in contemporary theoretical physics. Here we take up the
explicit construction of their representations. The three-parameter Lie
algebras have as their complex extensions the isomorphic simple complex
algebras A 1'-- B I......... C I' The real forms of these algebras may be divided into
the three compact isomorphic Lie algebras
so(3).........su(2).........sp(2)
and the four noncompact isomorphic Lie algebras
so(2, 1) .........su( 1, 1) .........s/(2, R) .........sp (2, R)
In addition we can consider the three-parameter Euclidean algebra in two
dimensions (£2)' which is made up of the semidirect sum
E2.........T2EB s so(2)
where T 2 is the Abelian Lie algebra associated with the group of transla-
tions in two dimensions.
Barut l25 has shown that the construction of the representations of the
176

THE STANDARD FORM 177
eight Lie algebras mentioned above may be given a unified treatment.
Here we first put the various Lie algebras into their standard form, and
then construct the basic representations in terms of suitably defined boson
annihilation and creation operators. Having obtained the basic two-
dimensional representation, we classify and construct the higher-
dimensional representations. Finally we investigate the construction of
coupling coefficients for the reduction of the direct product of representa-
tions.
17.2 THE STANDARD FORM
The compact group SO(3) is the group of transformations in a real
three-dimensional Euclidean space that holds invariant the definite form
X2+X2+X2= g JLpX x
I 2 3 JLP
(17.1)
where
gll =g22=g33= 1
( 17.2 )
while the noncompact group SO(2, 1) is the group of transformations that
holds invariant the indefinite form
X2+X2_X2= g JLpX X
I 2 3 JLP
( 17.3)
where
gll = g22 = - g33 = 1
If the generators of the Lie algebras are written as
(17.4 )
LI2,L23,LI3
with
Lp.v = - LpJL
( 17.5)
then the commutation relations for both Lie algebras may be written in the
form
[LJLp, LpA] = ig JLJL 41'
( 17.6)
The isomorphic compact Lie algebras are defined using the metric given in
Eq. 17.2; the noncompact Lie algebra, using that given in Eq. 17.4.
The above algebras may be cast into the standard form by first choosing
the appropriate Weyl self-commuting operator to be diagonalized in the

178 REPRESENTATIONS OF SOME THREE-PARAMETER LIE GROUPS
standard basis. Having made this selection, we may then construct raising
and lowering operators from the remaining two generators that will raise
or lower. the eigenvalues of the diagonalized operator by one unit. In the
case of the compact Lie algebras so(3)su(2)sp(2), the choice of the
operator to be diagonalized is arbitrary, and each generator has a discrete
eigenvalue spectrum. However, in the case of the noncompact Lie algebras
so(2, 1)su(l, 1)s/(2,R)sp(2,R), only L I2 has a discrete spectrum; the
other two generators L 13 , L 23 have a continuous spectrum. Here L I2 forms
the generator of the compact Lie subalgebra so (2). If we choose to
diagonalize L I3 or L 23 , we find that the corresponding raising or lowering
operators change the eigenvalues of L I3 by + i, and the generator L I3 or
L 23 forms the noncompact Lie subalgebra so(I,I). In general the
diagonalization of the noncompact generators requires the construction of
a continuous basis. Here we limit our attention to the diagonalization of
the compact generator L 12 , referring the reader to the litera-
ture 60 , 102, 109, 110 for the added complexities that arise in the diagonaliza-
tion of the noncompact generators L I3 and L 23 .
Making the choice of diagonalizing the generator L 12 , we are led directly
to the standard form
[L +, L - ] = g 3 3 L 12'
[L I2 , L:t ] = + L:t
(17.7)
where
L.z=  (L I3 + iL 23 )
( 17.8)
and
g33= + 1 for so(3)su(2)sp(2) (17.9)
g 33 = - 1 for so (2, I)  su ( 1, I)  s/ (2, R )  sp (2, R) ( 17.10 )
g33 = 0 for £2' (17.11)
For a unitary representation we must have
Lt =L and Lt =L
#LV #LV +-
(17.12)
17.3 THE CASIMIR INVARIANTS
The three-parameter groups are of rank one and thus possess just one
independent Casimir invariant, which may be readily constructed using the

THE ELEMENTARY REPRESENTATIONS 179
results of Section 5.18 to give (apart from the suppression of an overall
phase and a factor of !)
C=g33Lf2+Lf3+L3
=g33LI2(LI2+ 1) +2L_L+
and hence
C=2L_L+ +L I2 (L I2 + 1)
[so(3)su(2)sp(2) ]
(17.13)
C=2L_L+ -L I2 (L I2 + 1)
[so(2, l)su(l, 1)sl(2,R)sp(2,R)]
(17.14)
C=2L_L+
[E 2 ]
(17.15)
17.4 THE ELEMENTARY REPRESENTATIONS
We saw in Chapter 12 that there are I basic representations associated
with a Lie group or Lie algebra of rank I, and that all other representations
may be constructed by the formation of Kronecker products of the basic
representations. Furthermore, the basic representations can themselves be
constructed from the anti symmetrized Kronecker powers of the elementary
representations, which are a subset of the set of basic representations being
associated with the terminal points of the corresponding Dynkin diagram.
Thus our first objective must be to construct the elementary representa-
tions of the group or algebra of interest and then to obtain the rest of the
representations.
The Lie algebras of immediate interest are all of rank 1 and hence
possess just one basic representation, which, of course, is also an ele-
mentary representation. The dimension of this basic representation is
found from Chapter 14 to be just 2. Thus we seek a representation of the
Lie algebras defined by Eq. 17.7 in terms of rank-2 matrices.
Any rank-2 matrix can be expressed as a linear combination of the Pauli
spin matrices 126
a l = ( 
1 ) = ( 0 -i ) _ ( 1 0 )
, (12 , (13 -
o i 0 0-1
(17.16)

180 REPRESENTATIONS OF SOME THREE-PARAMETER LIE GROUPS
which satisfy the commutation relationship
[ 0i' OJ] = 2if.ijk O k
(17.17)
Putting
0 3
L 12 = 2'
L+ = I ( +. )
0 1 _10 2
- 2V2
(17.18)
leads immediately to the commutator relationships satisfied by the com-
pact Lie algebras of rank 1. Hence the elementary representation of the Lie
algebras so(3)su(2)sp(2) may be constructed in terms of the Pauli spin
matrices as in Eq. 17.18 and corresponds to the well-known spinor repre-
sentation of so(3).
If we put
°3
L 12 = 2"'
L+ = A ( + )
0 1 - io 2
- 2V2
(17.19)
and demand that the commutation relationship in Eq. 17.7 be satisfied, we
are required to write
A= + 
( 17.20)
Choosing the positive value for A, we obtain the elementary representations
for the three classes of Lie algebras.
For g33 = + I we obtain the elementary representation for so(3)su(2)
sp(2), which is unitary and irreducible, while for g33 = - I we get the
elementary representations for so(2, 1)su(l, 1)sl(2,R)sp(2,R), which
are irreducible but nonunitary. Finally, with g33 = 0 the elementary repre-
sentation of the Euclidean Lie algebra £2 reduces to two one-dimensional
representations.
17.5 BASIS FOR THE SPINOR REPRESENTATION
Having constructed the elementary spinor representation, we must as-
sociate it with a suitable basis. Since the representation is in terms of
two-dimensIonal matrices, the basis vectors must have just two com-
ponents, say the spinors
l = ( ) and 2 = (  )
( 1 7.21 )

REALIZATION IN TERMS OF BOON OPERATORS 181
Noting Eqs. 17.19 and 17.20, we are led immediately to the results
L121 = !l L122 = - t2 ( 1 7.22 )
and
Ii L  = yg33  ( 1 7.23 )
L+2= T l - 1 2 2
The highest weight of the spin or representation is t, corresponding to the
usual Cartan-Weyl labeling. The eigenvalue of the Casimir invariant is
readily found to be
3g 33
c=-
4
( 1 7.24 )
17.6 REALIZATION IN TERMS OF BOSON OPERATORS
We may also obtain a realization of the basic commutators in terms of
differential operators acting on functions f(1'2) constructed from mono-
mials of the spinors l and 2' a typical monomial being written as
la,b>=N(a,b)ff
( 17.25)
where N(a,b) is a suitable normalization constant. If we write
1 ( a a )
L 12 = 2 l al -2 a2
../i;; a
L+=VT l a2
L = yg33 t. 
- 2 S2 al
( 17.26)
we readily reproduce the results of Eqs. 17.22 and 17.23, and thus the
differential operators in Eq. 17.26 form a realization of the three-element
Lie algebras.
An equivalent realization in terms of boson annihilation and creation

182 REPRESENTATIONS OF SOME THREE-PARAMETER LIE GROUPS
operators 1 26 may be found by putting
a J = t. and a a
,s, ; = a;
( 17.27)
to give
L 1 2 = t ( a r a 1 - a 1a 2 )
_. rg;; t
L+ - VT a 1 a 2
1f 33 t
L = -a a
- 2 2 1
( 17.28 )
The aJ and a; are readily seen to satisfy the basic commutation relation
[a7,a j ] = ij
(17.29)
associated with boson annihilation (a;) and creation (aJ) operators, and
thus Eq. 17.28 gives a realization of the Lie algebra in terms of boson
operators.
17.7 CONSTRUCTION OF OTHER REPRESENTATIONS
Having obtained the elementary representation of the Lie algebra, we
may seek to construct the other representations. These other representa-
tions could be expected to be constructed from Kronecker powers of the
elementary representation, and thus to involve basis states constructed
from monomials involving I and 2 as in Eq. 17.25. For maximum
generality we suppose that the exponents a and b are not necessarily
integers and possibly even complex numbers. Using the realization given
by Eq. 17.26 we find
L I2 Ia,b) = t (a - b )Ia,b)
. 1f 33 1V(a,b)
L+la,b)= _ 2 ( ) bla+ l,b-l)
1V a+l,b-l
1f 33 1V(a,b)
L_la,b)= _ 2 ( ) ala-l,b+ I)
1V a-l,b+l
( 17.30)

CONSTRUCTION OF OTHER REPRESENTATIONS 183
where we see that L+ raises the eigenvalue of L l2 by 1, and L_ lowers it
by - 1.
The eigenvalues of the Casimir invariant follow from Eqs. 17.13-17.15
and 17.30 as
Cla,b)=g33<P(<I>+ 1)la,b)
= Qla,b)
( 17.31 )
where
<I>=t(a+b)
( 17.32)
Clearly, <I> and - <I> - 1 correspond to the same eigenvalue of C. Within a
given irreducible representation the eigenvalue Q of C is constant.
The classification of the irreducible representations proceeds much as in
Section 15.4, though with the important difference that here we do not
restrict the eigenvalues of L l2 to just integers or half integers.
Each irreducible representation is characterized by an eigenvalue Q of
C. Since the eigenvalues of L l2 can change only by multiples of unity, we
can write
t(a-b)=Eo+x
( 17.33)
where x is an integer and Eo is the fractional part of t(a - b). The
introduction of Eo allows for the possibility of multivalued representations.
Within a given irreducible representation Eo will have a fixed value, and
hence each irreducible representation (<I>,Eo) may be specified by giving
the values of the two invariants <I> and Eo. The basis vectors for a given
representation may then be uniquely labeled using the notation 1<1>, Eo + x).
The possible classes of representations may be determined by consider-
ing the permissible ranges of the values of a and b associated with fixed
values of <I> and Eo. In all, we find four distinct classes of representations.
A. Representations Unbounded from Above and Below
If a and b are not integers, then starting with an arbitrary eigenvector
lab) we may produce any other eigenvector la' b'), with a - a' and b - b'
integers, by repeated application of the raising and lowering operators L + .
Thus for fixed values of Eo and <P, the eigenvalue spectrum of L l2 is
unbounded from above and below, and the representations D(<P,E o ) will
be irreducible and of infinite dimensionality. Since we are interested in
enumerating inequivalent representations, there is no loss of generality in
imposing the restriction
- t  ReE o < t
( 17.34 )

184 REPRESENTATIONS OF SOME THREE-PARAMETER LIE GROUPS
In this case the representations D(<P, Eo) and D( - <P - I, Eo) are readily
seen to be equivalent, and hence the inequivalent representations may be
labeled as D(Q,E o )' The eigenvalue spectrum of L 12 in D(Q,E o ) may be
portrayed as
o 0 - - - -
a-b+1 a-b+2
2 2
- - - - 0 0
a-b-2 a-b-1
2 2
o
a - b
2
with
a=Eo+x+<P,
b=<P-E -x
o
(for D( <P, Eo)
(17.35)
and
a=E +x-<P-I
o ,
b=-<P-I-E -x
o
(for D( -<P-I,E o )
( 17.36)
B. Representations Bounded Below
If a takes on integer values while b is arbitrary, and there exists an
eigenvector la',b') such that L_la',b')=O, then from Eq. 17.30 we must
have a' = 0, and the lowest eigenvalue of L 12 must be - b' /2. The operator
L + may then be used to produce eigenvectors whose associated L 12
eigenvalue may be increased without limit by multiples of unity to give the
eigenvalue spectrum
P- b'
0 ------ 0
- b ' + 2
2
:::-
-
2
o
- b ' + 1
2
with
t(a-b)=Eo+x
(x = 0, 1,2,. . . )
(17.37)
and
b'
<P=-
2
(17.38)
where Eo is the fractional part of b' /2. Thus the representation is of
infinite dimensionality and bounded from below. For a> 0 the representa-
tion is irreducible and is denoted as D+ (<P).
We note that if a> 0, then there is no operator to transform an
eigenvector from the subspace with a> 0 into the subspace with a < 0, since
L_IO,b')=O. Nevertheless, we can transform an eigenvector la,b) with

CONSTRUCTION OF OTHER REPRESENTATIONS 185
a < 0 into the subspace with a> 0 using the operator L +. Thus if we restrict
our attention to the subspace with a> 0, we obtain an irreducible repre-
sentation. Without this restriction on a we obtain a representation that is
reducible but not fully reducible, that is, an indecomposable representa-
tion.
c. Representations with an Upper Bound
If b is an integer and a is arbitrary, we obtain for b > 0 an irreducible
representation denoted D-(<I», where
t(a-b) =Eo+x
(x = 0, - 1, - 2,... )
(17.39)
and
a'
<1>= -
2
(17.40)
with Eo the fractional part of a' /2. The eigenvalue spectrum of L 12 in this
representation is of the form
<: 0 ------ 0
a' - 2
2

a
o
a' - 1
2
-
2
Again, if we admit b < 0, we obtain an infinite-dimensional representation
that is reducible but not fully reducible. The representation D- (<I» is of
infinite dimensionality, irreducible, and bounded from above.
D. Finite-Dimensional Representations
If a and b are integers, then there exists the possibility of finding
representations that are of finite dimensionality. If a + b < 0 with b < 0 in
the subspace with a>O, we obtain a special case of D+(<I», while if a<O in
the subspace with b>O, we obtain a special case of D-(<I». Both of these
representations are of infinite dimensionality and have <I> < O.
However, if a+b>O, the only invariant subspace will be where a>O and
b>O, with the representation D(<I» characterized by vlue of O. In this
case there will be eigenvectors la',b'> and la",b"> such that L+a',b'=O
and L_a",b'=O with b'=O and a"=O. The eigenvalue spectrum of L 12
will thus be bounded above by a' /2 and below by - b" /2. Since from Eq.
17.32
2<1> = a' = b"
( 17.41 )

186 REPRESENTATIONS OF SOME THREE-PARAMETER LIE GROUPS-
we must have a finite-dimensional representation D(<I» with
a-b
2 = -<P, -<P+.1,...,<I>
( 17.42)
of dimensionality 2<1> + 1. The eigenvalue spectrum of L I2 may be dis-
played as

0 - - - - - 0
-<1>+1 <1>-1

Since a and b are nonnegative integers, it follows that <I> will be a
nonnegative integer or half integer. For <I> = 0 we obtain the identity
represen ta tion.
I t follows from the above that for the infinite-dimensional representa-
tions D:!:(<I», all values of 2<1> are permitted-apart from those involving
nonnegative integers, which give rise to the finite-dimensional representa-
tions D(<I».
17.8 THE UNITARY REPRESENTATIONS
So far we have not imposed any requirement that the representations be
unitary, and thus the representations obtained need not all be unitary. We
can determine the unitary representations by demanding that the eigen-
a
values of L I2 be real and that those of L + L _ and L _ L + be real and
positive definite. As a consequence the eigenvalues of the Casimir operator
must also be real.
The reality of the eigenvalues of L I2 requires
Eo real,
Ima=lmb=p
( 17.43 )
The eigenvalues of the Casimir operator are of the form g33<1>(<I> + 1). If we
put
<I> = <1>1 + i<l>2
( 17.44 )
where <1>1 and <1>2 are real, then <1>(<1> + 1) will be real if and only if
<1>2 = 0 or <1>1 = - 1- (<1>2 real and arbitrary)
( 17.45)
Furthermore, it follows from Eqs. 17.32 and 17.43 that
a+b
<1>2 = 1m 2 = P
( 17.46)

THE UNITARY REPRESENTATIONS 187
Thus for a unitary representation we have either
 real or  = - -!- + i p ( p real)
( 17.47)
Equation 17.30 leads to the condition
g 3 3 a ( b + 1 ) > 0 and g 3 3 b ( a + 1 ) > 0
( 17.48 )
if the eigenvalues of L+L_ and L_L+ are to be real and positive definite.
Use of Eqs. 17.32 and 17.33 in Eq. 17.48 then leads to the requirement that
g33 (+ Eo+x) (- Eo- x + 1) >0
( 17 .49a )
and
g33 (<I> - Eo- x) (+ Eo+ x + 1) >0
(17.49b)
These two equations must be simultaneously satisfied. For the compact
groups, g33 = + 1, and clearly only finite-dimensional unitary representa-
tions are possible with Eo=O if 2+ 1 is odd ( real), and with Eo= -!- if
2+ I is even. For the noncompact group, g33= -1, and only infinite-
dimensional unitary representations are possible. These latter representa-
tions may be conveniently divided into four distinct classes:
(a) Continuous Principal Series The so-called continuous-series representa-
tions are characterized by continuous eigenvalues of the Casimir invariant.
The principal series is designated Dp(Q,Eo) with
= -t+ip
(O<p<oo)
( 17. 50a )
and
Q>t
(17.50b)
The eigenvalues of L l2 are unbounded above and below, with
X =Eo,Eo + I,E o + 2,...
(Eo real)
(17.50c)
(b) Continuous Supplementary Series The supplementary series is desig-
nated Ds(Q,Eo) with
1+tl<t-IEol
(, Eo real)
( 17.51 a)
and
Q<t
(17.51b)

188 REPRESENTATIONS OF SOME THREE-PARAMETER LIE GROUPS
Again the eigenvalues of L 12 are unbounded above and below, with
x=Eo,Eo + I,E o + 2,...
(17.51c)
(c) Discrete Series D+() Here we find
<o and Eo=-
( real)
( 17 .52a )
and the eigenvalues of L 12 are bounded below, with
x= -, -+ 1, -+2,...
( 17 .52b )
(d) Discrete Series D-() Here
<o and Eo=
( real)
( 17 .53a )
and the eigenvalues of L 12 are bounded above with
x=,-I,-2,...
( 17 .53b )
17.9 MATRIX ELEMENTS OF L 12 AND L:!:
The matrix elements of L 12 and Lj: will be fully determined in the
unitary representations if the normalization factors N(a,b) appearing in
Eq. 17.30 are established. The eigenvectors of L 12 may be specified as
(  ) EO+X
la,b>=N(a,b)ff=NA'2)<I> 
=1,Eo+x>
( 17.54 )
to give
LI21,Eo+x>= (Eo+x)I,Eo+x>
( 17 .55a )
L:!:.I<P,EO+X>= vg3 [<P + (Eo+x)] ::, I<P,Eo+X + I> (17.55b)
The inner product of the basis vectors may be defined as
<, Eo + xl, Eo + x'> = x,x'
(17.56)

MATRIX ELEMENTS OF LI2 AND L 189
Use of the unitarity condition L  = L _ then gives
*
Vg3 (-(Eo+x» N:: 1 =( Vg3 ) (+Eo+X+I)* N;t
Recalling the particular form of g33' we obtain
2
N x
N X + 1
(+Eo+x+ 1)*
=g33 -Eo-x
( 17.57)
In the case of the compact Lie algebras, g33 = + 1, and the recursion
relationship of Eq. 17.57 can be satisfied by writing
.1
N x = [( + Eo+ x)! (- Eo- x)!] 2
(17.58)
where n! = f(n+ 1), with n not necessarily an integer.
With g33 = - 1 we have the noncompact Lie algebras. In the case of the
representations Dp(Q,Eo), Eq. 17.57 can be satisfied by putting N x = 1. In
all other cases we may put
N=
x
(Eo+x--I)!
(Eo+x+)!
(17.59)
with the understanding that the ranges of Eo, , and x are appropriate to
the particular unitary representations under discussion.
Once the normalization is fixed, Eqs. 17.55a and 17.55b may be
employed to give the matrix elements of the infinitesimal operators as
<,Eo+xIL121,Eo+x')= (Eo+x)xx'
( 17. 60a )
1
<,Eo+ xIL:!:I,Eo+ x + 1)= [ g3 ( + Eo + x)( + Eo + x+ I) r
(17 .60b )
where again the ranges of Eo, x, and  must be appropriate to the unitary
representations under discussion.
In the particular case of so(3) we have
<jmILI2Ijm') = mmm'
( 17.61 a)

190 REPRESENTATIONS OF SOME THREE-PARAMETER LIE GROUPS
<jmILljm + I)= v t(j + m)(j + m+ I)
(17.6Ib)
where j = <I> is an integer or a half integer and
m=Eo+x= -j, -j+ I,...,j-I,j
(17.62)
17.10 FINITE TRANSFORMATIONS
So far our discussion has concerned the construction of representations
of the Lie algebras. Now we must make the transition to the representa-
tions of the Lie groups. To do this we first construct the spinor representa-
tion and thence obtain the matrix elements appropriate to finite-group
transformations.
The elementary representation of the three-parameter Lie group may be
written in the general form
( a
(1-
-g33{3*
)
(17.63)
where
det (1 = aa * + g 33 {3 * {3 = 1
(17.64)
and a and (3 are complex numbers. For the groups 80(3) and 80(2, 1), (1
will correspond to a spinor representation. The complex numbers a, (3
admit various real parameterizations in terms of three real parameters. For
example, in the case of 80(3) we might employ the Cayley-Klein parame-
terization. For 8U(I, I) we could consider the matrices
k( 0) - (e-;/2 ei/2 )
a(E)= ( coshE/2 -iSinhE/2 )
i sinhE/2 coshE/2
a ( 'T ) = ( cosh 'T / 2 - sinh 'T / 2 )
- sinh 'T /2 cosh 'T /2
11( v) = ( 1- iv /2 - iv /2 )
iv/2 l+iv/2
where k(O) is of the elliptic class, a(E) and a('T) of the hyperbolic class, and

FINITE TRANSFORMATIONS 191
71(V) of the parabolic class. In this case we could typically have for SU(I, 1)
a(O,E,'T) =k(O)a(E)a( 'T)
a(O,'T,v) =k(O)a( 'T)l1(v)
a(O,'T,cp) =k(O)a( 'T)k(cp)
and so on. Clearly a variety of one-parameter subgroups will exist-those
of the elliptic class being compact, and those of the hyperbolic and
parabolic classes being noncompact. Here we shall consider only the
elliptic class of subgroups.
Now to obtain the matrix elements for a finite transformation. The
spinors I and 2 form basis states for the spinor representation a, and
under the operation of a we have the transformation
I' = a1 + {32
;= -g33{3*I+a*2
(17.65)
The basis for an arbitrary representation may be constructed from mono-
mials in the spinors I and 2' a typical basis state being designated as
I  E +x ) =N t.at.b=N t.<I»+Eo+xt.<I»-Eo-x
, 0 x S IS2 xSI S2
( 1 7.66 )
If 1> (a) denotes an element of the spinor group, then 5
1> (a )f( 1'2) = f( a-I (1'2))
(17.67)
and since
( *
-I a
(J = g33f3*
f3 )
( 17.68)
we have
1>(a)I,Eo+x)=Nx(a*I- (32 )<I»+Eo+x( g33{3*1 +a2 )<I»-Eo-x
(17.69)
We may now use the binomial expansion to collect together the terms in I
and 2' remembering that the expansion is for complex numbers and hence
all factorials will normally be taken as gamma functions [i.e.,p!r(p + 1)],

192 REPRESENTATIONS OF SOME THREE-PARAMETER LIE GROUPS
to give
I (J) 1<1>, Eo + x>
=N x  (_l)k (+Eo+x)!(-Eo-x)!
k,;O (+ Eo+ x - k)! (- Eo- x - k') !k!k'!
X a*+EO+x-k( g33 (3 *) -EO-X-k'ak'{3k-k-k';+k'
( 17.70)
We would like to be able to express the right-hand side of Eq. 17.70 as a
linear combination of monomials of l and 2 so as to give
1> (J) 1<1>, Eo + x> = 1>, x (J) 1<1>, Eo + x'>
x'
(17.71)
This is possible only if we can put
k + k' = <I> - E - x'
o
( 17.72 )
But k, k', and x' are necessarily integers, and thus a group representation,
as opposed to the algebraic representations found earlier, is possible only if
<I> - Eo is an integer
( 17.73 )
In that case we may then write
 ( Nx (<I> + Eo + x)! (<I> - Eo - x)!
I> x' x (J) = N x ' ( x' - x) !
X a*+Eo+xa-Eo-x' (g33 {3 *)X'-X
00 k
L (-g33f3f3*jaa*) (x'-x)!
X k! (+ Eo+ x - k)! (- Eo- x' - k)! (x' - x + k)!
k=O
{17.74)
In the particular case of SU(2) we have g33 = + 1 with
j=<I> and m=Eo+x
(17.75)
where j and m are together integers or half integers and m ranges over the
2} + 1 values
m = - j, - j + 1, . . . ,j - 1,j
(17.76)

FINITE TRANSFORMATIONS 193
Using in Eq. 17.74 the normalization obtained in Eq. 17.58, we obtain for
SU(2) the well-known result 5
j ( ) =  oo (- 1 )k[ (j + m)! (j - m)! (j + m')! (j - m')!] t
Um'm U
k!(j+ m - k)!(j - m' - k)! (m' - m + k)!
k=O
x a.1 - m' - ka *) + m - k{3 k{3 *m' - m + k
( 17.77)
where u is an element of the group of two-dimensional unimodular
matrices specified in Eq. 17.63.
As noted in Chapter 16, the group 8U(2) is the covering group of 80(3),
and thus every representation ui of 8U(2) will be simultaneously a repre-
sentation I) of SO(3). The group 80(3) is doubly connected, and hence
we anticipate that some of the representations so obtained will be double-
valued. In terms of the usual Cayley-Klein parameterization 5
a = e - i<p / 2 cos  e - i>¥ / 2 , f3 = - e - i<p / 2 sin  e - i>¥ / 2 ( 17.78 )
with Euler angles (<p,(J,\¥), we have for SO(3)
" ( -1 )k[ (j + m)! (j - m)! (j + m')! (j - m')!] t
1",'m( cp,O,I[;) =  k!(j + m- k)!(j - m' - k)!(m' - m+ k)!
k
, , 2 " + ' 2k (J . 2k + ' (J, ./,
X e'm cp CoS 'J m-m - - SIn m -m _ e'm,#,
2 2
( 17.79)
The so-called spin representations of 80(3) occur for half-integer values of
j and are double-valued, as may be readily seen from the behavior of 5))
with respect to rotations through 0 and 2'lT about a fixed axis. The
representations characterized by integer values of j are termed the "true"
representations of 80(3) and are single-valued.
For the group SU(I, 1) we have g33 = -1, and the true or single-valued
unitary representations follow directly from Eq. 17.74 under the normaliza-
tion derived in Eq. 17.59. The requirement that the representations be
single-valued and that Eq. 17.73 be satisfied limits us to
<I>=j and m=Eo+x
where j and m are together integers or half integers.
The unitary representations associated with continuous values of <I> and
Eo give multivalued representations of SU(I,I) and correspond to the
single-valued representations of the universal covering group S U (1, 1) of

194 REPRESENTATIONS OF SOME THREE-PARAMETER LIE GROUPS
SU(I,I). Whereas SU(2) is its own universal covering group, SU(I,I) is
not simply connected and hence cannot be its own universal covering
group. Representations of the universal covering group that are not true
representations of the covered group are referred to as projective represen-
tations of the covered group. Bargmann 61 has shown that the topology of
the group manifold for SU(I, 1) is the direct product of the circle with a
two-dimensional Euclidean plane, and hence the topology of the covering
group SU (1, 1) is that of the three-dimensional Euclidean space, which
covers SU(I, 1) infinitely many times. Thus we can expect SU(I, 1) to have
infinitely many-valued projective representations.
The group SU(I, I) gives a twofold covering of the group 80(2, I), and
hence the unitary representations of SO(2, 1) consists of the true represen-
tations having integer j,m and the spin representations having half-integer
j, m. These representations may be obtained directly from Eq. 17.74 once
the basic spinor representation has been given an appropriate parametric
form.
EXERCISE
17.1. Obtain a suitable parametrization of the matrices appropriate to the group
Sp(2), and hence obtain an explicit form for the unitary representation
matrices.
Finally we note that the general matrix element ':'x(o) given in Eq.
17.74 may be recast in terms of the hypergeometric functions
00
 f(a+k)f(b+k)f(c) Zk
2 F )(a,b,c;z) = £.J r(a)r(b)r(c+k) k!
k=O
(17.80)
by the application of the identity
-x!
-y!
y-l
(-1) (y-I)!
(-1) x-I (x-I)!
( 17.81 )
to yield
N a-Eo-x'a*+Eo+X ( g {3 * ) X'-X
q> ( 0 ) -  33
x'x - N x , (x' -x)!
x F(x' + Eo-' - x - Eo- ,x' - x + 1, - g33 {3{3*)
( x' > x )
(17.82)
with a similar expression for x' < x.

COUPLING COEFFICIENTS 195
17.11 DIAGONALIZATION OF A NONCOMPACf GENERATOR
In the case of the compact group 80(3) the generators L 12 , L 23 , and L 13
can be physically identified with spatial rotations, and thus it matters little
which of the three generators we choose to diagonalize. For the noncom-
pact group 80(2,1), only L 12 generates a spatial rotation, while L 23 and
L 13 generate accelerations. The generator L 12 generates a compact sub-
group of 80(2,1), while L 23 and L 13 generate noncom pact subgroups.
Whereas the real eigenvalues of L 12 form a discrete set with normalizable
eigenvectors, the real eigenvalues of L 23 or L 13 form a continuous set, and
great care is required in constructing a continuous set of basis states. A
complete solution for the group 8U(I, 1) has been given by Lindblad and
Nagel, 110 who have shown that diagonalization of J 23 or K + = J 12 + J 13
yields the following eigenvalue spectra:
J 23
Continuous series: the real line with multiplicity 2
Discrete series Dj': the real line
K+
Continuous series: the real line
Discrete series Dj+: the positive real line
Dj -: the negative real line
For details of the explicit construction of the appropriate continuous basis
states, the reader should refer to the original article. 110
17.12 COUPLING COEFFICIENTS
The reduction of the Kronecker product 1>1 2 of two irreducible
representations  1 and 2 of a group G into a sum of irreducible
representations of G is a problem of fundamental importance in applica-
tions of both compact and noncompact groups in physics. In view of the
Wigner-Eckart theorem, the keystone of all practical calculations, it could
be considered the central problem.
Three principal problems arise: (1) the construction of a basis for the
Kronecker product; (2) the determination of the representations that arise
in the reduction of the Kronecker product, the so-called Clebsch-Gordan
series in the case of 80(3); (3) the determination of the elements of the
transformation matrix that reduces the reducible representation  11&2
into its irreducible representations, the so-called Clebsch-Gordan or
Wigner coefficients in the case of 80(3).
The solution to the first problem is most readily carried out using a basis

196 REPRESENTATIONS OF SOME THREE-PARAMETER LIE GROUPS
constructed from monomials in the spinors that characterize the funda-
mental representation as developed by Weyl,4,84 van der Waerden,7 and
Eckart 127 and outlined by Bargmann 128 in his survey of the rotation group.
The Clebsch-Gordan series for compact groups may be developed either in
terms of weights (as in Chapter 11) or by the use of Schur functions. 31 , 129 A
universal method for calculating the coupling coefficients of compact
semisimple Lie groups using the methods of Dynkin has been developed
by Patera,I30 though evidently not extended to noncompact groups. Here
we construct an invariant coupling of three representations and then
identify the coupling coefficients with coefficients, a method developed by
Bargmann 128 and stemming from Wigner's original unpublished
manuscript. I3I (Later published in the reprint collection edited by Bieden-
ham and van Dam. I32 ) This method was later extended to noncompact
groups by Barut and Fronsda1. 62
In general the Clebsch-Gordan series problem amounts to solving the
equation
1) 1 @2= gI231>
3
(17.83 )
where gI23 is the number of times the representation 3 is contained in the
reduction of the Kronecker product 5)1 @5)2' These numbers are equiva.-
lent to the number of times the identity representation o appears in the
reduction of the triple Kronecker product.
tD2
( 17.84 )
where   is the representation contragredient to  (cf. Section 9.3). By
definition of the contragredient representations, 1) 3   must be invariant
under the group operations, and hence the triple product in Eq. 17.84 must
be associated with an invariant I such that
f 1 21> )I = I
(17.85)
The representations 5) l' 1> 2' and   can be realized in terms of mono-
mials in the basic spinors using Eqs. 17.25 and 17.32 to give
1) 1:
1 <1> m > = N t1 +m"nI-ml
1 1 ml'i;I 'll
(17.86a)
(17.86b)
2:
1 <1> m > = N t2+m1tn2-m2
2 2 m2'i;2 '12
 *.
oU 3 .
 -m  
<<I>3 m 31 = ( -1) 3 3Nm33'1!3-m:rq33+m3
( 17. 86c )
where
m=Eo+x.

COUPLING COEFFICIENTS 197
Let us write the invariant I in terms of an invariant coupling
1=  F(<P1<P2<P3)C'::::NmlNm2NmJ( _1)4>J- m J
mlm2 m 3
x rl +mIrJrl-mli2+mHrJ!2-m2jJ-m:trJJ+mJ
( 17.87)
where I is invariant in the space of the polynomials rriia;ll/i, F(<P 1 <P 2 <P 3 ) is
a function of the <P;'s only, and C':l m 4> 2 m 4>J are the Clebsch-Gordan coupling
1 2 J
coefficients that reduce the Kronecker product  1 J)2'
It follows from Eq. 17.65 that since the group matrices are unimodular,
the only invariants that can be formed from the three sets of spinor
functions are 128
1 = 21l3 - 31l2'
2 = 31l1 - 11l3'
3 = 11l2 - 21l1 (17.88)
and every monomial In a' Thus we may write, using the bin011!-ial
expansion for the ik"
I = fl;;J
=  ( -1 t+ q + r ( :1 ) ( ; ) ( ; )t3-r+q;I-p+rf2-q+P
X 1l2-q+rll;o -r+ P ll ;l-P+Q
(17.89)
where the binomial coefficients are expressible in terms of gamma func-
tions,
( a ) r(a+l)
b = r(b+ l)r(a-b+ 1)
( 17.90)
If a and b are nonnegative integers, we have 133
( a ) a!
b = b!(a-b)!'
( -a ) = (-I)b(a+b-l)!
b b!(a-l)!'
( -a ) (_I)b-a(b_l)!
-b = (a-l)!(b-a)!
(17.91)
and
( _:)=0

198 REPRESENTATIONS OF SOME THREE-PARAMETER LIE GROUPS
Comparison of Eq. 17.89 with Eq. 17.87 implies that
k} = <P 2 + <P3 - <p},
k 2 = <P3 + <p} - <P 2 ,
k3 = <p} + <P 2 - <P3 (17.92)
Equating powers of the spinor functions then yields
r-p =<P}-<P 3 + m 2 ,
P - q =<P -<P -m
2 } 3'
q-r=<P 3 -<P 2 +m}
(17.93)
from which we deduce the familiar weight addition rule
m} +m 2 =m 3
( 17.94 )
Use of Eq. 17.93 to eliminate p and q in Eq. 17.89, together with Eq. 17.92,
leads to the result
-}
C:::: = [F( <P}<P 2 <P 3 )N ml N m2 Nm 3 ]
x L ( - 1) <1>1-<1>2+<1>3-' ( <P2 + <P3 - <p} ) ( <p} + <P3 - <P 2 ) ( } + <P 2 - 3' )
r 3-<P}-m2+r 4>3-<P 2 +m}+f r
(17.95)
17.13 SPECIALIZATION TO 80(3)
Let us now specialize to the case of the covering group of SO(3), making
the identifications <Pi = ii' where the ii are nonnegative integers or half
integers and the m i their associated projections. The normalization factors
N fnt follow directly from Eq. 17.58 to give Eq. 17.95 as
. . . -1
CfJ{J",3 = F(jlj2j3)
x [ (j 1 + m 1 ) ! (j 1 - m 1) ! (j 2 + m2) ! (j 2 - m2) ! (j 3 + m3) ! (j 3 - m3) ! ] 1/2
X (jI + j2 - j3)! (jI - j2 + j3)! ( - jI + j2 + j3)!
( _1)Ja-h+h- r
XL r!(j\-m\-r)!(h+ m 2- r )!(j3-},-m2+ r )!
r
x (j3 - j2 + ml + r)! (j 1 + j2 - j3 - r)!
( 17.96)

SPECIALIZATION TO SO(3) 199
The unitary properties of the coupling coefficients require that
 C J .J2i3 CJI2i = 8 ,8 ,
 m 1 m2 m 3 ml m2 m3 mlm} m 2 m2
)3' m3
( 17.97)
and
 CJ.J2i3 CJIJ2i3' , = 8. .,8 ,
 mlm2 m 3 m}m2 m 3 JY3 m3 m 3
ml,m2
(17.98)
which, in view of Eq. 17.84, implies that
 CJ.J2i3 CJ.J2i3 = 1
 mlm 2 m 3 mlm2 m 3
ml
( 17.99 )
If we choose j3 = m 3 and consequently m 1 + m 2 = j3' we find that Eq.
17.96 is satisfied only if r= jl - m 1 , and hence Eq. 17.99 allows us to fix
F(jlj1i3) as
F(jlj2.i3) = ( -1) -J}+J2-J3
x
(jl +j2-j3)!(jl-j2+j3)!( -jl +j2+j3)!(jl +j2+j3+ I)!
2j3 + 1
(17.100)
Using this result in Eq. 17.96 then yields the familiar result for SO(3),
1/2
. . , [ (2j3 + 1) (jI + j2 - j3)! (jI - j2 + j3)! ( - jI + j2 + j3)! ]
C hJ2I3 =8 ( m +m m )
m)m2 m 3 1 2' 3 ( . + . + . + 1 ) '
11 12 13 .
x [(jI + mI)! (jI - mI)! (j2 + m2)! (j2 - m2)! (j3 + m3)! (j3 - m3)!] 1/2
( _ l)r
XL r!(j\-ml-r)!(h+ m 2- r )!(j3-h+ m \+r)!
r
x (j 3 - j 1 - m2 + r) ! (j 1 + j2 - j3 - r) !
(17.101)
Although the above formula is cumbersome in practical use, numerous
tabulations of special cases have been given, 104,105 and extensive numerical
tables exist. 134

200 REPRESENTATIONS OF SOME THREE-PARAMETER LIE GROUPS
EXERCISES
17.2. Show that Eq. 17.101 may be equivalently expressed as 104, 105
1/2
,.. [ (2i3+1)(il+i2-i3)! ]
C/r":J",3 = 8 ( m I + m2, m3) ( . + . + . + 1 ) , ( . _ . + . ) , ( _ . + . + . ) ,
JI J2 J3 . JI J2 J3' JI J2 J3 .
1/2
X [ (il - ml)! (i2 - m2)! (i3 + m3)! (i3 - m3)! ]
(il + ml)! (i2 - m2)!
x  (-1)r+il-ml(il+ml+r)!(i2+i3-ml-r)!
£.J r! (i I - m I - r) ! (i 3 - m3 - r) ! (i2 - i 3 + m 1 + r) !
r
17.3. The generalized hypergeometric series 3F2(abclef) may be defined b y 135-137
3 F 2 (abcl ef) = lim 3F2 (abcl ef; z)
zl
where
( I f(e)f(f)  f(a+r)f(b+r)f(c+r)zk
3F2 abc ef;z) =
f(a)f(b)f(c) r!f(e+r)f(f+r)
r=O
( 17.102 )
Show that Eq. 17.101 may be recast as l05
1/2
. . ' ( [ (2i3 + 1) (il - i2 + i3 + 1) (i2 + i3 - il + 1) ]
C ltJ213 = m +m m )
m I m2 m 3 I 2' 3 ( . + . . + 1 ) ( . + . + . + 2 )
JI J2-J3 JI J2 J3
1/2
[ f(il +ml + l)f(i2- m2+ l)f(i3+ m3+ l)f(i3-m3+ 1) ]
x f (i I - m I + 1 ) f (i 2 + m2 + 1 ) [ f (i 3 - i2 + m I + 1 ) f (i 3 - i I - m2 + 1 ) ] 2
x 3 F 2(ml -iI' - m2 -}2,i3 - il- i21i3 - i2 + ml + l,i3 - il - m2 + 1) (17.103)
upon use of the well-known properties of gamma functions,
'IT
f ( z + 1) = z f ( z ) and f ( z ) f ( 1 - z) = .
sin 'lTZ
( 17.104)
(or equivalently, for integer arguments, use of Eq. 17.81).
17.4. Show that the coupling coefficients of SO(3) satisfy the following symmetry
relationships:
ChIJ,h = ( _1 ) h+h-hC h i2 i3
mlm2 m 3 -m) -m2 -m3
Chi])3 = ( _1 ) h-ml
m)m2 m 3
ci 3 i d2
m3 - m)m2

COUPLING COEFFICIENTS FOR SO(2, 1) 201
17.5. Show that the Wigner 3 -j symbol 131
( jl j2 j3 ) = ( _I ) i1-h- m 3 ( 2 i +1 ) -t/2CJJi2 J3
:13 m.m2- m 3
m l m2 m3
( I 7.105 )
is invariant with respect to any even permutation of its columns and is
multiplied by only a phase factor (- I)h + J2 + J3 for any odd permutation or
for the complete reversal of the signs of the numbers mi'
17.6. Show that under the Legendre reflection j] = - j - I we have l05
C JlJ2J3 = ( _ 1 ) i1+h-hCJJi3
m.m2 m 3 m.m2 m 3
C Jlhh = ( 1- I ) jl-h+mIChJ3
m.m2 m 3 m.m2 m 3
C Jlh h = ( -I ) h+m2CJJi3
m.m2 m 3 m.m2 m 2
17.14 COUPLING COEFFICIENTS FOR SO(2, I)
The corresponding calculation of the coupling coefficients for SO(2, I)
and its covering group is complicated by the need to consider both discrete
and continuous representations as well as unitary and nonunitary repre-
sentations. These various difficulties are covered in the literature. 13 8-144
Here we are primarily concerned with the coupling coefficients required
for the coupling of a finite-dimensional nonunitary representation D(<P 1 ) to
an infinite-dimensional unitary representation D+(<P 2 ), which is of consid-
erable practical interest.145-148 However, let us first look at the coupling
coefficients for reducing the Kronecker product D+(<P 1 )XD+(<P 2 ).
The Clebsch-Gordan series for D+Ul)XD+(j2) is readily seen to be
-00
D+(jl)XD+(j2)=  D+(j3)
J3 = Jt + J2
( 17.106 )
The coupling coefficients may then be found by an invariant coupling of
the triple Kronecker product D + (j I) X D + (j 2) X D + *(j 3) to give
(m l +jl)!(m 2 +j2)!(m 3 +j3)!
(ml-j.-I)!(m2-j2-1)!(m3-j3-1)!
X(_lrl-h+h-k ( .hh-jl )( .jlj3-h )( jl+j2-j3 )
r J3-JI- m 2+ r J3-J2+ m .+ r r
. , , -I
C;':{=:"3 = F(jlj03)
( 17.107)

202 REPRESENTATIONS OF SOME THREE-PARAMETER LIE GROUPS
where - i3;) - i I - i2' m i ;) - ii' and ii < 0, and we have used the normaliza-
tion factors given by Eq. 17.59 with i; = <P; and m; = Eo + x;. The first two
binomial coefficients involve negative arguments, and the last, positive
arguments. Noting this, we obtain
C jd .JJ3 - F ( . . . ) - I
m.m2m3 - JIJ2J3
( m I + i I ) ! ( m 2 + i2) ! ( m 3 + i 3) !
(m l - il -I)! (m 2 - i2 -I)! (m 3 - i3 -I)!
X [ (il+i2-i3)! ]
( - i l + i2 - i3 - I)! (il - i2 - i3 - I)!
( 1 ) - j. + j2 - m3 + r ( " 1 ) , ( . 1 ) ,
- JI-J3+ m 2- r - . ml-JI+ r - .
X  '( . + ) ,(. . + ) '(' . . ) , ( 1 7.108)
r. J2 m 2 -r. J3-J2 ml+r. JI+J2-J3- r .
r
The factor F(jli3) may be fixed by demanding that the coupling coef-
ficients satisfy the unitarity condition of Eq. 17.99 and choosing m 3 = - i3'
The use of the identity
 -I (n+m)!
 [p!(n-p)!(r-p)!(m-r+p)!] =
p m!n!r!(n+m-r)!
(n,m;>O)
( 17.109)
then finally gives for D+(il)XD+(i2)
C jd2i3 =
m.m2 m 3
( - 2i3 - 1) (il + i 2 - i3)! ( - i l - i2 - i3 - 2)!
x ( m 1 + i 1 ) ! ( 111 2 + i 2) ! ( m 3 + i 3) !
(il - i2 - i3 -I)! ( - il + i2 - i3 -I)! (m l - il -I)!
x (m 2 - i2 - I)! (m 3 - i3 - I)!
( 1) -j.+j2-m3+r ( .. 1)'( . 1)'
- m 2 +JI-J3- r - · ml-JI+ r - .
X  '( . ),(. . + + )'( . +. . )' (17.110)
 r. m 2 +J2- r . J3-J2 m l r. JI J2-J3- r .
r
EXERCISE
17.7. Remembering that for D-(}I)XD(}2)XD-*(}3) we have}j<O and mj<}j,
show that the Clebsch-Gordan coefficients for coupling two negative dis-
crete representations may be expressed as

COUPLING COEFFICIENTS AND ANALYTIC CONTINUATION 203
C Jd2i3 =
mlm2 m 3
(-2i3- 1 )(it +i2-i3)!( -it-i2-i3- 2 )!( -mt +it)!
X( -m2+i2)!( -m3+i3)!
(it - i2 - i3 -I)! ( -it + i2 - i3 -I)! ( - mt -it - I)!
x (-m2-i2- 1 )!( -m3-i3-1)!
 (_I)iI-h+ m 3- r ( -m2-i2+r-I)!(i2-i3-mt-r-I)!
X £.J (17.111)
r! (it - mt - r)! (i3 -it - m2 + r)! (it + i2 - i3 - r)!
r
The coupling of a positive discrete representation to a negative discrete
representation results in a Clebsch-Gordan series that involves not only the
positive and negative discrete representations, but also the continuous
series of representations. These problems, together with those associated
with the couplings of members of the continuous series, are covered in the
literature and wi!. not be treated here. 138 , 141-144
The coupling coefficients for coupling the finite nonunitary representa-
tions present no problems. The Clebsch-Gordan series is identical to that
of SU(2). Since the nonunitary representations of SU(I,I) differ from
those of SU(2) only by the substitution of hyperbolic functions for
trigonometric functions, the functional forms of the representations are
unchanged, and hence the coupling coefficients may be taken as identical
to those found for SU(2) and are given by Eq. 17.100.
17.15 COUPLING COEFFICIENTS AND ANALYTIC CONTINUATION
If we compare the formula for the coupling coefficients for D + (j I) X
D+(j2) of SO(2, 1) given in Eq. 17.110 with that for D(jl)XD(j2) of SO(3)
given in Eq. 17.101, we note a striking similarity. Indeed, if we express Eq.
17.101 in terms of gamma functions and make the analytic continuation
under ji - ji' using the identities of Eq. 17.104 (or alternatively, in terms
of factorials, using Eq. 17.81), we find, apart from a trivial phase, the result
of Eq. 17.110. Biedenharn and Holman 138 ,142 have used the concept of
analytic continuation to show that the coupling coefficients of the covering
group of SO(3) and SO(2, 1) are derivable from a single entity.
We now evaluate some of the coupling coefficients for the coupling of a
finite nonunitary representation D (j I) to a positive discrete unitary repre-
sentation D+(j2)' The Clebsch-Gordan series for D(jl)XD+(j2) can be
determined by considering the weight diagram formed by the weights of
the two representations, as can be readily seen from the particular case
shown in Fig. 17.1. The discrete representations D+(j3) are represented by
infinite positive weight towers, and those of the finite representations by
finite weight towers. The latter weights have been boxed in. The Clebsch-
Gordan series for D(jl)XD+(j2) divides into two cases: (a)jl < -j2 and
(b) j 1  - j 2' For these two cases we obtain

204 REPRESENTATIONS OF SOME THREE-PARAMETER LIE GROUPS
Infinite towers
\ \ \ II \lJ \lJ \ \ 1/ I
.
.
.
.
.
.
4 .
3 . 3 .
2 . 2 .
1 . 1 .
0 . X m 2
-1 .
-2 .
-3 .
m 1
4...... .
3..... .
2 . . . . .
o .
G . ·
. [!]
.
> 1 .
-1 .
-2 .
m 3
Fig. 17.1. Weight diagrams for resolution of the Kronecker product D(3)X D+( -1).
il + i2
(a) D(il)XD+(i2)=  D+(i3)
)3= -)1 +)2
(il < -i 2 )
(17.112)
(b)
)1 +)2
D(i l )XD+(i2)=  D(i 3 )+2
)3=0
-I
 D+(i 3 )
h--lJ-h-1
- )1-)2 - 2
+  D+(i 3 )
-)3=-)1+)2
(i 1  - i] ) (17.113)
for il,i2 both integers or half integers.
If il is an integer andi2 a half integer, or vice versa, we have
)1 +)2 -3/2
D(i.)XD+(i2)=  D(i3)+2  D+(i 3 )
)3 = 1/2 )3 = -)1 -)2 - 1
-)1-)2- 2
+  D+(i3)+D+( -i)
)3= -)1 +)2
(17.114)
It is apparent from the above that for i 1> - i2 the Kronecker products
are not simply reducible. However, if we restrict our interest to the range
i 3  - i. + i 2' we avoid the difficulties associated with the duplicated repre-
sentations and can obtain the coupling coefficients, to within an overall
phase, by analytic continuation of the coefficients found for SO(3) under
the transformation
J.Jl'
i2-i2'
. .
J3-J3
(17.115)

COUPLING COEFFICIENTS AND ANALYTIC CONTINUATION 205
yielding
C JIi2i3 =
m l m 2 m3
(_I)h-J3- m l( -2j3- 1 )(jl +j2-j3)!(jl-j2+j3)!
x( -jl-j2-j3-2)!
(jl-j2-j3- 1 )!
1/2
(jl + m 1 )! (jl - m 1 )! (j2 + m 2 )! (j3 + m 3 )!
(m 2 -j2- 1 )!(m 3 -j3- 1 )!
X L (j\-h+ m 2- r - 1 )!
r r! (jl - m 1 - r)! (j2 + m 2 - r)! (jl + j2 - j3 - r)! (j3 - j2 + m 1 + r)!
x
(17.116)
We note the appearance of a phase under the first square root. Unlike the
earlier coupling coefficients we have encountered, Eq. 17.116 can yield
either real or imaginary values.
These coupling coefficients do not satisfy the unitary condition, but
rather we find
 C JIi2i3 *Cjti2i3, ( _I ) J2-J3-ml= ,.,'
 mlm2 m 3 mlm2 m 3 m3 m 3 J3.h
m., m2
(17.117)
EXERCISES
17.8. Verify the above result for j3 = - m3' making use of the identity
r
 (-1) (n-r)! (n-m)!(n-/)!
£.J r!(m-r)!(/-r)! = n!m!/!(n-m-/)!
r
(17.118)
ifn >m >O,n >1 >0.
17.9. Use Eq. 17.116 to derive the following algebraic expressions for D(jl)
D+U2):
C I'.. I
2.1 212 + 2 =
I I
- 2 m2 + 2" m2
. + I
J2 - m2 1:
2j 2 + 1
C Ij2h+ 1 =
-I m2+ 1 m2
(j2-m2)(j2-m2+ 1)
2(j2+ 1)(2j2+ 1)
C 1J2h + 1 =
Om2 m 2
(j2+m2+ 1)(j2-m2+ 1)
(j2+ 1)(2j2+ 1)

206 REPRESENTATIONS OF SOME THREE-PARAMETER LIE GROUPS
C Ij2 =
- 1 m2 + 1 m2
(J2+ m 2+ 1)(J2- m 2)
2j2(J2+ I)
C 1 0. m 2
J2I2 _
Om2 m 2 - V
J2(J2+ I)
17.10 Show that the above algebraic forms are ( - I).it -j2+j3 times the correspond-
ing algebraic forms for 80(3).
The expression found in Eq. 17.116 for D(jl)XD+(j2)XD+(j3) of
80(2, 1) by analytic continuation of the corresponding result for SO(3) is
naturally closely related to that found in Eq. 17.101 for SO(3). Indeed, the
algebraic forms found from Eq. 17.116 are simply (- l)h -}2 +}3 times the
corresponding algebraic forms found from Eq. 17.101, though of course
the ranges of the quantum numbers j; and m; are quite different. The
extensive tables of the algebraic forms of coupling coefficients for SO(3)
can thus be used for algebraic forms for D(jl)XD+(j2)XD+(j3)'

18
Some su(l,l )-Type Spectrum-
Generating Algebras
18.1 INTRODUCTION
Let us now apply some of the foregoing material to the evaluation of the
spectra associated with a number of simple systems that are of consider-
able general interest in physics. Infeld and Hull I 49 have noted that most of
the analytically solvable second-order differential equations involving a
single variable that are of interest in electromagnetic and quantum theory
can be transformed into the standard form
d 2 y
- +f{y)Y=O
dy2
( 18.1 )
where Y = Y(y).
In this chapter we shall obtain a realization of the generators of the
noncompact Lie algebra su(I,I) in terms of a single variable, and then
show that many of the second-order differential equations can be ex-
pressed in terms of these generators. Since we know the spectral properties
of the su(l, 1) generators, it follows that we can immediately generate the
spectrum associated with the relevant second-order differential equation.
207

208 SOME su(l, I)-TYPE SPECTRUM-GENERATING ALGEBRAS
18.2 A REALIZATION OF su(l, 1)
The Lie algebra associated with the noncompact group 80(2, I) and
8U(I, 1) is characterized by the commutator relationships
[ f l' f 2] = - if 3;
[ f 2' f 3] = if I ;
[f 3' f I] = if 2
( 18.2)
We may obtain a realization of this Lie algebra in terms of a single
dimensionless variable y by writing
a 2
f l =2+ al (Y);
ay
f2=;[ k(y)  +a2(Y)];
a 2
f 3 = 2 +a 3 (y)
ay
( 18.3)
(Realizations in terms of two variables have been discussed by Miller. I 
We readily find that Eq. 18.2 will be satisfied if
2
a= a + (P-Y)
I ({3_y)2 16
a --J.
2- 4
2
a (f3-y)
a 3 = 2 - 16 +y;
(f3-y)
f3-y
k=
2
( 18.4)
where a, 13, and yare constants of integration.
The existence of the Casimir invariant of su(l, I),
r 2 = f - fi - f
requires that y = 0, and we find
r2= _ a _1-
4 16
( 18.5)
( 18.6)
If we choose 13 = 0, we obtain the standard form for the generators of
su(l, 1) in terms of a single variable y as l50
a 2 a y2
f I = oy2 + y2 + 16
-i ( a )
f =- y-+t
2 2 ay
( 18.7)
a 2 a y2
f =-+---
3 ay2 y2 16

DISCRETE EIGENVALUE SPECTRUM 209
Let us consider the particular case of those second-order differential
equations that may be transformed into the standard form of Eq. 18.1 and
that have
f(y) = a 2 +b y 2+ C
y
( 18.8)
We may write our second-order differential operator In terms of the
su(l, I) generators given in Eq. 18.1 as
a 2 a
2 + 2" + by2 + c = ( t + 8b ) r 1 + ( ! - 8b ) r 3 + c
ay y
( 18.9)
and make the identification
a=-4r 2 -i
(18.10)
Thus in terms of the su(l, 1) generators, we have for Eq. 18.1
[ ( ! + 8b ) r 1 + ( ! - 8b ) r 3 + c ] Y = 0
( 18.11 )
18.3 DISCRETE EIGENVALUE SPECTRUM
Equation 18.11 may be simplified by performing a rotation through an
arbitrary tilting angle (), in essentially the same manner as the familiar
Foldy-Wouthuysen- Tani transformation of relativistic quantum me-
chanics,I03 such that
e - iO r 2 r 1 e ior 2 = r 1 cosh () + r 3 sinh ()
and
e - ior 2 r 3 e iO r 2 = r 1 sinh () + r 3 cosh ()
(18.12)
to give
{ [(! + 8b) cosh(} + (! - 8b) sinh(}]r 1
+ [ ( ! + 8b ) sinh () + ( ! - 8b ) cosh () ] r 3 + c } Y = 0
(18.13)
where
Y = e- ior2 Y
(18.14)
The tilting angle () may be chosen to diagonalize either the compact
generator r 3 or the noncompact generator r l' In the latter case we obtain

210 SOME su(l, I)-TYPE SPECTRUM-GENERATING ALGEBRAS
the continuous part of the spectrum, and in the former the discrete part of
the spectrum.
F or the discrete spectrum we put
!+8b
tanh{} = - t - 8b
( 18.15)
to reduce Eq. 18.13 to just
- C
f 3 Y= Y
4 Y=b
(18.16)
where the eigenvector Y is a simultaneous eigenvector of r 2 and f 3' and
thus must span one of the dIscrete infinite-dimensional representations
D+() or D-() of SO(2, I) or SU(I, I). In the case of D+(<P) the
eigenvalues of f3 will have the lower bound «1>, which increases in steps of
unity with no upper bound; and conversely for D - (<P).
It should be apparent from the preceding remarks and our discussion in
Chapter 17 that we may now write the eigenvalue solution for Eq. 18.16 as
- - c-
f3Y:x=(-+m)Y:x= Y: x
V - 16b
(x=O, 1,2,...) (18.17)
with
r 2 f+ = ( <p+ 1 ) y+
x x
(<P < 0)
(18.18)
Thus the existence of a discrete eigenvalue spectrum associated with the
second-order differential equation
( d2 a )
_+_+b y 2+ C y=O
dy2 y2
(18.19)
requires that
c
4( -+x) =
Y=b
( 18.20)
This equation may be put into a more direct form by noting from Eq. 18.6
that
(<p+l)=_ a _1-
4 16
(18.21)
and hence
cI»=-t(l+ Y !-a)
(*-aO)
( 18.22)

THREE-DIMENSIONAL ISOTROPIC HARMONIC OSCILLATOR 211
where, since (f) < 0, we have kept only the negative root. Using this result in
Eq. 18.22 finally gives the key result,
4x+2+ VI-4a = c

x = 0, 1, 2. ..
(18.23)
18.4 CONTINUOUS EIGENVALUE SPECTRUM
The continuous eigenvalue spectrum may be found by putting
!-8b
tanh 0 = - t +8b
( 18.24 )
to reduce Eq. 18.13 to just
- -c
f 1 y= y
4 
( 18.25)
In this case e must diagonalize the noncompact generator f l' and the
eigenvectors Y will form a continuous basis. The eigenvalue spectrum is
characterized by a continuous spectrum A, where
-c
A=
4 
( 18.26)
We note that the continuous part of the spectrum will exist only where
tanh () exists. In what follows we shall largely restrict our attention to the
discrete spectrum.
18.5 THREE-DIMENSIONAL ISOTROPIC HARMONIC OSCILLATOR
The preceding results may be readily applied to a wide range of solvable
problems involving second-order differential equations. In each case the
relevant second-order differential equation is cast into the standard form
of Eq. 18.19, and then its eigenvalue spectrum is solved for, using Eq. 18.23
for the discrete part and Eq. 18.26 for the continuous part.
In the case of the three-dimensional isotropic harmonic oscillator, the
appropriate radial differential equation is 103
( d2 l( I + 1) )
- - -r 2 +2E R(r) =0
dr 2 r 2
( 18.27)

212 SOME su(l, I)-TYPE SPECTRUM-GENERATING ALGEBRAS
Comparison with Eq. 18.19 requires
a= -/(/+ 1), b= -1, and c=2E
Using these values in Eq. 18.23 gives
E=2x+/+ t
(x = 0, 1,2,.. . )
Putting n = 2x + / gives the well-known result (in atomic units)
En = (n + ! )
( n = 0, 1, 2,. . . )
( 18.28)
If a perturbing term £ / r 2 (£  0) is added to the Hamiltonian, we find
that Eq. 18.27 becomes
( d2 / ( / + 1 ) + £ )
- - -r 2 +2E R(r) =0
dr 2 r 2
and we are immediately led to the result I51
E=2x+ 1 + V(l +1)2+£
( 18.29)
We note that putting b = -1 in Eq. 18.24 gives a tilting angle outside the
allowed limits of tanhO, and thus in both of the above cases the spectrum
is entirely discrete.
18.6 THE GENERALIZED KEPLER PROBLEM
The second-order differential equation
( d2 2 d t u )
-+--+-+-+v R(r)=O
dr 2 r dr r r 2
( 18.30)
arises in the generalized Kepler problem for motion in three dimensions.
Equation 18.30 may be transformed into the standard form by putting
r=y2 and R(r) =y-3/ACR(y)
( 18.31 )
to give
( d2 4u -  )
dy2 + y2 4 + 4vy2 + 4t at (y ) = 0
( 18.32)

THE GENERALIZED KEPLER PROBLEM 213
Using Eqs. 18.10 and 18.23 gives for the discrete spectrum
2x + 1 + v' 1 - 4u = t
v=v
( x = 0, 1, 2, . . . )
( 18.33 )
In the case of the nonrelativistic hydrogenic atom, 103 we have t = - 2Z,
u= -1(1+ I), and v=2E, and hence from Eq. 18.33,
_Z2
En= 2
2n
(18.34)
where n=x+l+ 1.
If an inverse-cube potential is added, we then have f3 = -1(1 + 1) - € and
obtain for the discrete spectrum I 52, 153
_Z2
E = 2 (x = 0, 1,2,... ) (18.35)
2[ x+t+ V (l+t)2+ 2 £ ]
which lifts the degeneracy of the H-atom states in much the same manner
as the normal fine structure.
In the case of the Klein-Gordon equation we have for a hydrogenic
atom t= -2Za 2 E, u=Z 2 a 2 -1(1+ 1), and v=(a 4 E2-1)/a 2 , to give from
Eq. 18.33 the result I 54
( 2 2 ) -
a 2 £= 1 + Zn
( 18.36)
where n = x + t + V (l + 1) 2 - Z 2a 2 and a is the fine-structure constant.
Biedenharn 155 has shown that the second-order iterated Dirac equation
may be written as
( d2 + 2 !!... _ f(f-I) _ 2Za 2 £ + a 4 £2_1 ) rl,.=0
dr 2 r dr r 2 r a 2 'j" (18.37)
where in Biedenharn's notation
f = P3K + iaZPlo' r
( 18.38)
Putting t= -2Za 2 E, u= -f(f-l), and v=(a 4 E 2 -1)/a 2 In Eq. 18.33
gIves
x+f= -Za 3 E
Y l-a 4 E 2

214 SOME su(I, I)-TYPE SPECTRUM-GENERATING ALGEBRAS
F ollowing Bied enharn, we may take the eigenvalues of r as
.y U + t)2 - Z 2 a 2 to give
a 2 E= ( I+ I+Z2a2 )
(X+!+VU+t)2-Z 2 a 2 f -I
( 18.39)
18.7 THE 1WO-DIMENSIONAL KEPLER PROBLEM
The second-order differential equation
( d2 I d t u )
-+--+-+-+v R(r)=O
dr 2 r dr r r 2
(18.40)
arises in discussing the somewhat artificial problem of generalized
Keplerian motion in two dimensions. Putting r=y2 and R(r)=y-<R(y)
gives the standard form
( d2 4u + * )
dy2 + y2 +4vy2+4t <R(y)=O
(18.41)
In this case Eq. 18.27 yields for the discrete spectrum
2x+ 1 +2 Y=U = t
v:::v
( x = 0, I, 2, . . . )
( 18.42)
For the two-dimensional nonrelativistic hydrogenic atom, we have
t = 2Z, u = - m 2 , and v = 2E, with m = 0, + I, + 2,..., and thus obtain from
Eq. 18.42
_Z2
E=
2(n+!)2
( 18.43)
with n=lml+x;, in agreement with the result that Jauch and Hill 156
obtained by conventional methods. The addition of a perturbing term
- f./ r 2 (f. > 0) gives
_Z2
E=
2
2 ( x + ! + V m 2 + f. )
(18.44)

THE MORSE POTENTIAL 215
The solution of the energy eigenvalues of the Klein-Gordon hydro genic
atom where there are only two spatial and one timelike coordinate involves
the differential equation (in cylindrical polar coordinates)
( d2 +,!.f{ + 2Ea 2 Z + Z 2 a 2 -m 2 + a 4 E2-1 ) R(r) =0 (18.45)
dr 2 r dr r r 2 2
Putting t=2Ea 2 Z, u=Z 2 a 2 -m 2 , and v=(a 4 E 2 -1)/a 2 in Eq. 18.42 leads
to the result
.1
a 2 E= ( I+ Z2a2 ) 2
( X + t + vi m 2 - Z 2a2 ) 2
If the term -£/r 2 is added we obtain
( 18.46 )
.1
a 2 E= ( I+ Z2a2 ) -2
( x + t + vi m 2 + f. - Z 2a2 ) 2
This result is not without some interest, as the solution obtained even for
m = 0, irrespective of Z, is unphysical, whereas the customary three-
dimensional :Klein-Gordon equation yields unphysical solutions for 1=0
only when 2Z > 137.
(18.47)
18.8 THE MORSE POTENTIAL
The differential equation
( d2 +pe2TZ+qeTZ+r ) R(Z)=0
dz 2
( 18.48)
may be transformed into the standard form by putting
I
z=l ny 2 and R(Z)=y-2CR(y)
to yield
( d2 + 16r+7"2 + 4p y2+ 4 Q ) m(y)=0
dy2 4 1'y 2 1'2 1'2
Morse l5 ? has considered the energy eigenvalue spectrum associated with
( 18.49)

216 SOME su(l, I)-TYPE SPECTRUM-GENERATING ALGEBRAS
the differential equation
( d2 -2De-2Tr +4De-Tr +2E ) R(r) =0
dr 2
( 18.50)
Noting Eqs. 18.48 and 18.49, we obtain the standard form
( d2 + 32E+r 2 _ 8D y2+ 16D ) ffi.(y)=0
dy2 4 1'y 2 1'2 1'2
Use of Eq. 18.23 leads immediately to the result
( 18.51 )
E= 2 r2 ( V: D _<X+!)f
(x=0,1,2"...,x max ) (18.52)
where
1 V2n
Xmax+ 2 <
l'
( 18.53 )
Equation 18.50 is valid only for the bound S-state energy spectrum. To
take into account other states it is necessary to add a term 1(1 + 1)/ r 2 .
However, when this is done it is no longer possible to cast the differential
equation into the simple su(l, I) form.
We note that for the Morse potential, in contrast with our prevIous
examples, the number of bound states is finite. 1 58
18.9 LIMITATIONS OF su(l, I)
The preceding examples illustrate some of the applications of the
su(l, I)-type Lie algebras to the solution of certain problems in quantum
dynamics. These examples are by no means exhaustive. Barut 15 has dis-
cussed many examples relevant to the dynamics of magnetic charges, while
Solomon 159 has used su(l, I) to generate the energy spectrum of a su-
perfluid boson system.
Nevertheless, many (indeed, most) problems in physics cannot be ac-
commodated in the su(l, I) treatment. For example, if a term to include
the Stark effect is added to the hydrogenic atom Hamiltonian, we can no
longer express the Hamiltonian in terms of the generators of su(l, I). To
handle the Stark effect (and then we can do so only by a perturbative
method), we must enlarge our group algebra to include the possibility of
representing the variable z in terms of generators. A similar situation arose
in the Morse-potential problem, where we were unable to discuss the 1=1=0

LIMITATIONS OF su(l, I) 217
states in terms of the su(l, I) spectrum-generating algebra.
A number of authors160-162 have attempted to determine the Ham-
iltonians of quantum-mechanical systems that can be associated with a
given Lie spectrum -generating algebra. These studies have usually been
limited to rotationally and time-reversal-invariant Hamiltonians that are at
most quadratic in the momentum and have su(l, I) as their spectrum-
generating algebra.
There is clearly a need for much more study of spectrum-generating
algebras, and in particular for a way to decide directly on the spectrum-
generating algebra appropriate to a given Hamiltonian. For recent work in
this direction the reader is referred to the papers of Anderson, Kumei, and
Wulfman 163 and the works of Bluman 164 and Osvjannikov}65
EXERCISE
18.1 Show that the set of operators
f.=t(rp2+ r ), f 2 =r'p-i, f 3 =t(rp2_ r )
closes on 80(2, 1), and use this fact to solve for the eigenvalue spectrum of
the Hamiltonian
p2 a
H=---
2m r
(N.B. The quantity r(H - E) will be linear in the group generators.)

19
The Wigner-Eckart Theorem and Tensor
Operators
19.1 INTRODUcnON
Detailed quantum-mechanical calculations almost invariably reduce to
the evaluation of the matrix elements of interaction terms in some suitably
defined set of basis states. Group theory can be of great practical value in
the classification and construction of basis states that have well-defined
transformation properties with respect to the symmetry operations of a set
of nested groups. The interaction terms may be expanded into a set of
operators having well-defined transformation properties under the same set
of nested groups used to describe the symmetrized basis states. Having
classified the symmetry transformation properties of both the basis states
and the interaction terms, it is a comparatively simple matter to obtain
selection rules that allow us to immediately predict the vanishing of many
matrix elements and thus avoid much needless computation. 31 ,81, 166, 167
However, group-theoretical applications that do not go beyond the
simple symmetry classification of states and interactions and the prediction
of vanishing matrix elements are very tame affairs. Clearly we still have to
be able to calculate the nonvanishing matrix elements. The vital tool
needed to complete the group-theoretical calculation of the nonvanishing
matrix elements, and indeed also to predict the vanishing matrix elements,
is the celebrated Wigner-Eckart theorem. 127, 131 It is this theorem that lifts
218

SOME NOTATION 219
group theory from merely qualitative usefulness to its status as a powerful
tool for making quantitative predictions.
The earliest applications of the Wigner-Eckart theorem centered on the
calculation of the matrix elements of irreducible tensor operators as
developed by Wigner 5 and Racah l68 for the particular case of three-
dimensional symmetry as characterized by the group 80(3). The enuncia-
tion of the Wigner-Eckart theorem for the special case of 80(3) made
possible the calculation of the matrix elements of tensor operators between
angular-momentum states, while Wigner's introduction I 31 of generalized
angular-momentum vector coupling coefficients allowed the properties of
coupled products of angular-momentum states and of tensor operators to
be developed.
The general description of the irreducible tensorial algebra has been the
subject of many detailed works I 04, 105, 169, 170 and is now a standard part of
most quantum-mechanics courses. As a result we only sketch the main
features, assuming that the reader already has some familiarity with the
quantum theory of angular momentum.
The group 80(3) (and also 80(4)) has the great simplifying feature of
being simply reducible, and thus in the Kronecker product of two irreduc-
ible representations a given irreducible representation never occurs more
than once. For semisimple groups in general, this is quite exceptional
behavior. As a result the vector coupling coefficients for the groups 80(3)
and 80(4) are well established, 171, 172 while for most semisimple groups the
problem of determining the vector coupling coefficients remains a formid-
able problem, apart from a few special cases.
The generalization of the irreducible tensorial method to groups other
than 80(3) has proceeded slowly. The generalization of the Wigner-Eckart
theory to arbitrary compact groups has been frequently given, following
the work of Racah,9 Koster;73 Stone,86 and others}74, 175 The application to
finite symmetry groups is well established l76 and the relevant vector
coupling coefficients well known. In the case of noncompact groups much
less is known}77,178
19.2 SOME NOTATION
Consider a simple compact group 9 having elements denoted by g. Let
U g denote a unitary, not necessarily irreducible, representation of 9 on a
Hilbert space 3C. The various unitary representations are distinguished,
where necessary, by writing Ug(A). Where it is convenient and unam-
biguous we simply denote Ug(A) by (A). In the case of unitary irreducible
representations, A normally corresponds to the highest weight.

220 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
Let IAA> be the basis vectors of the representation (A), where A labels
the individual basis vectors; in the case of unitary irreducible representa-
tions, A normally designates a weight of the representation (A). Where
there are degeneracies in the weight space it may be necessary to supple-
ment A with some additional distinguishing labels, and this is taken for
granted except where explicitly noted.
We let the complete set of basis vectors IAA> span the infinite Hilbert
space 3C in which the linear operator Rg (or for brevity just R) correspond-
ing to the element g of 9 is represented by the block-diagonal matrix
I<AAIRIAA'>I. An indvidual matrix element corresponding to g in the
representation (A) of 9 will be designated as <AAIRIAA'>. The effect of the
linear operator R acting on a basis vector IAA> will be to produce a linear
combination of those basis vectors that span the representation (A), that is,
RIAA> =  <AA'IRIAA>IAA'>
A'
(19.1)
19.3 TENSOR OPERATORS
The set T(A) of [A] linearly independent operators T(AA) is said to form
a tensor operator under the group 9 belonging to the representation (A) of
9 if under the operations of the group it transforms according to the
representation (A}-that is, if
RT(AA)R- 1 =  <AA'IRIAA>T(AA')
A'
( 19.2 )
A tensor operator T(A) is said to be irreducible, reducible, or equivalent if
the group representation (A) is correspondingly irreducible, reducible, or
equivalent.
For an infinitesimal transformation in 9 we have
R=I+aaX
a
( 19.3 )
where aa are the infinitesimal parameters and Xa the corresponding
infinitesimal operators. Using this result in Eq. 19.2 and keeping only
terms to first order in the aa, we have
[X a , T(AA)] =  <AA'IXaIAA>T(AA')
A'
( 19.4)
while from Eq. 19.1,
XaIAA> =  <AA'IXaIAA>IAA'>
\'
( 19.5)

TENSOR OPERATORS IN SO(3) 221
showing that the same matrices I<AA'IXaIAA)1 can be used to transform
states and operators. As it stands, Eq. 19.4 suffices to identify any tensor
opera tor.
19.4 TENSOR OPERATORS IN 80(3)
For the group 80(3) having the infinitesimal operators Jz,J:t, we have,
in an angular-momentum basis that diagonalizes J2 and Jz,
JzIJM)=MIJM)
and
J :tIJM)= VJ (J + 1) - M(M + 1) IJM + I)
( 19.6)
which is the 80(3) equivalent of Eq. 19.5.
Suppose T(k) is an irreducible tensor operator in 80(3) transforming
according to the irreducible representation D(k) of 80(3). Then it follows
from Eq. 19.4 that the 2k+ 1 components T(kq) (q= -k, -k+ 1,...,k-
l,k) must satisfy the commutation relations
[Jz, T(kq)] = qT(kq)
[J:t, T(kq)] = Vk (k+ 1) -q(q + 1) T(k,q + 1)
( 19.7)
which indeed were taken by Racah 168 as the defining relations in develop-
ing his theory of irreducible tensor operators for 80(3). The tensor
operator T(k) will be said to be of rank k.
EXERCISES
19.1. Show that the angular momentum J of a system is a tensor operator of rank
one, that is, a vector operator.
19.2. Show that if
(k) _ .. ,----;;;- )
C q - V 2k+f Ykq«(J,cf>
where Y kq is a spherical harmonic, then C(k) is a tensor operator of rank k.

222 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
19.5 TENSOR OPERATORS FOR SEMI SIMPLE LIE GROUPS
We saw in Chapter 6 that it is possible to cast the infinitesimal operators
of a semisimple Lie group of rank 1 into the standard Cartan-Weyl basis,
namely,
[Hi,Hk] =0
(i,k= 1,...,/)
[Hi,Ea] =aiE
[ Ea' E {3] = N a{3 E a + {3
[Ea,E-a] =aiHi
( 19.8)
where the matrices of the self-commuting Weyl operators Hi may be
chosen to be diagonal and real, and thus Hi will be self-adjoint. Henc for
a compact semisimple group we may choose our infinitesimal operators so
that their adjoints are
H.t=H. and Et=E
I I a-a
( 19.9)
Since the Hi are associated with null roots, we may encompass Eq. 19.9 in
the single form
xt=x
(1 -(1
(19.10)
We note from Eq. 19.8 that the Hi are diagonal and the Ea acts as a shift
operator sending the weight A to A + a. Thus we may rewrite Eq. 19.4 for a
semisimple group as
[Hi' T(AA)] = \AAIHiIAA>T(AA) =Aj T(AA)
(19.11)
and
[Ea' T(AA)] =  <AA+anIEaIAA>T(AA+lX n )
n
(19.12)
19.6 COUPLING COEFFICIENTS
While the Kronecker product (AI) X (A 2 ) of two unitary irreducible
representations of a semisimple group is completely reducible, it is gener-
ally not simply reducible, and hence a given representation (A I2 ) may
appear more than once in the reduction of the Kronecker product. In these
cases we must distinguish the duplicated representations by an additional
label, say a.

COUPLING TO THE IDENTITY REPRESENTATION 223
If IA.A.> and IA2A2> are two basis vectors of (AI) and (A 2 ), respectively,
then the reduction of the Kronecker product is accomplished by the
coupling coefficients <AIAIA2A2IA.A2;aAI2AI2>' where
IA I A 2 ; aA 12 A 12 > =  <A.AIA2A2IAIA2; aA.2AI2>IAIAI>IA2A2> (19.13)
AI,A2
For the sake of brevity we normally write the coupling coefficient as
simply <AIA2IaAI2A.2>' In the case of 80(3) the coupling coefficients in
Eq. 19.13 are identical to the usual Clebsch-Gordan coefficients.
The inverse transformation can be written as
IAIAI>IA2A2> =  <aA.2AI2IAIA2>*IA.A2; aA. 2 A. 2 > (19.14)
a,A12,A12
Since the transformations are unitary, we have the usual orthogonality
relations
 <aAI2AI2IA.A2>*<AIA2Ia' A;2 A ;2> = 8 aa ,8 A 12A"128A 12A'12
AI' A2
(19.15)
and
 <AIA2IaA.2AI2>*<aAI2AI2IA;A;> = 8AIA'18A2A2
a,A 12 ,A12
(19.16)
19.7 COUPLING TO THE IDENTITY REPRESENTATION
The coupling of the basis states IA.AI> and IA2A2> of two representations
(A.) and (A 2 ) to form the identity representation (0) is of practical interest.
Since necessarily A I2 =A. 2 =0, we must have
Al = -A 2
(19.17)
Thus we can obtain the identity (0) only if the two representations (AI) and
(A 2 ) are of the same dimension and the weights of (A.) can be placed in a
one-to-one correspondence with those of (A 2 ) via Eq. 19.17. This will be
possible only if the highest weight of (A.) is equal to the negative of the
lowest weight of (A 2 ).
Two possibilities arise. 37,72-74, .79
1. The weight space Ll(A I ) of (AI) is completely symmetric with
+ Ai E Ll(A.) for all Ai in (AI)' In this case the representation (AI) is said to
be self-contragredient, and we have the condition that (A}) = (A 2 ). The
class of self-contragredient representations encompasses all the representa-

224 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
tions associated with the Lie algebras Bn' Cn' D 2k " G 2 , F 4 , E 7 , and E8
together with those representations of An (n> 1), D2k+ I (k> 1), and E6
whose weighted Dynkin diagrams are symmetrical, that is, of the form 37 ,74
An
D2k+ I (n = 2k + 1)
(19.18)
E6
a1
o
a2
o
a3 a3
o . . . 0
a2 a1
o 0
a1
o
a2
o
a3
o . .
< an
.a n - 2
an
a1
o
a2
o
a1
o
a 3 a 2
L 4 a
2. The weight space d(A I ) of (AI) is such that "A; Ed(A I ) but -"A;fld(A I ).
Only the algebras An (n> 1), D2k+ I (k> 1), and E6 have representations of
this type. In this case we obtain the identity only if (AI) and (A 2 ) are
conjugate representations. In these cases if A=(a l ,a 2 ,...,a n )Ed(A), then
- !(J(a l ,a2"" ,an) Ed(A) if and only if !o is one of the following permuta-
tions of the weights labeling the Dynkin diagram:
An (n> 1): !0(al,a2,...,an)=(an,...,a2,al)
D2k+1 (k> 1): !(J(al,a2,...,an-2,an-l,an) = (al,a2,...,an-2,an,an-l)
(19.19)
E6:
!0(al,a2,a3,a4,aS,a6) = (as,a4,a3,a2,al,a6)
The above results establish that the identity element will be obtained if
and only if
(AI) = (A 2 )*
( 19.20)
and
Al = - A2

THE WIGNER-ECKART THEOREM 225
In the case of self-contragredient representations, Eq. 19.20 reduces to just
(AI) - (A 2 )
We may evaluate the coefficient for coupling to the identity by using Eq.
19.20 in Eq. 19.15, with A I2 =A I2 =0, to obtain the important result that
apart from a phase factor
<AIAIAI*-AIIOO)= 1 (19.21)
vfAJ
EXERCISES
19.3. Verify that the Kronecker square of
2 1 2
o 0 0 of A3
contains the identity representation.
19.4. Verify that the Kronecker product
2 2
o 0 X 0 0 of A 2
contains the identity representation.
19.5. Show that Eq. 19.19 implies that in the Cartan notation of Section 12.6, two
representations of An will be conjugates if and only if
{II' 1 2 "" ,In} = {II' 11 -In' 11 -In-l,...,11 -/ 2 } *
( 19.22)
For example, the representations {432} and {421} of U 4 are conjugate.
19.6. Show that if, in the Cartan notation of Section 12.6, the irreducible unitary
representations of Dn (n odd) are labeled as [/ 1 ,/ 2 "" ,In], then the self-
contragredient representations all have In = 0, while for In =1= 0 the representa-
tions [/ 1 ,/ 2 "" ,In] and [/ 1 ,/ 2 "", -In] are conjugates.
19.7. Verify that for the coupling coefficients of 80(3)
.. ( _ 1 )J I - m I
<J.m.,J. - m.IOO) =
Y 2j. + 1
(19.23 )
19.8 THE WIGNER-ECKART THEOREM
We now sketch the derivation of the celebrated Wigner-Eckart theorem
as it applies to compact Lie groups. Here we essentially follow the
derivation given by Stone. 86
With the introduction of tensor operators comes the need to evaluate the
matrix elements
<AIAIIT(AA)IA2A2>

226 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
for the various components of the tensor operator T(A). Consider a
tensor-operator component T(M) acting on a basis state IA2A2>' We have,
upon noting Eqs. 19.4, 19.5, and 19.14,
T(M)IA 2 A 2 >=  <aAIAIIAAA2A2>*IT(A)A2;aAIAI> (19.24)
aAtAt
Thus the matrix elements of T(M) are given by
<AIAII T(AA)IA 2 A 2 > =  <aAIAIIAAA2A2>*<AJAII T(A)A 2 ; aAIAI>
a
( 19.25)
We now show that coefficients of the form <bAIAllaAIAI> are indepen-
dent of the component AI'
Consider the transformation
laAIAI> =  <bAIAllaAIAI>lbAJAI>
b
( 19.26)
Suppose that Xp. is an arbitrary infinitesimal operator of the group 9 and
that
la Al Al + IL> =  <bAIAI + ILlaAIAI + IL>lbAIAI + IL>
b
( 19.27)
Noting Eq. 19.12, we have for IL:FO
Xp.laAIA I >
la AI AI + IL> = <AI AI + ILIX,JAIA I >
=  <bAIAllaAIAI>lbAIAI + IL>
b
( 19.28)
Comparison of Eqs. 19.27 and 19.28 gives immediately that
<b Al Al + ILia Al Al + IL> = <bAIAllaAIAI>
( 19.29)
for all IL:FO, and hence the coefficients <bAJAJlaAJAI> must be indepen-
dent of the component AI'
Using the above result in Eq. 19.25 gives the Wigner-Eckart theorem as
<AIAII T(AA)IA 2 A 2 > =  <aA I A l IAA 2 >*<aA I II T(A) IIA2> (19.30)
a
where we have written <aAIII T(A)IIA 2 > in place of <AlAI T(A)A 2 ; aAA I >,
since the latter has been seen to be independent of AI' The double-barred

SELECTION RULES 227
matrix elements are independent of the weights of Ai of the representations
(Ai)' and are referred to as reduced matrix elements. The entire dependence
of the matrix element on the weights of the bra and ket representations
together with the component of the tensor operator T(A) is encased in the
coupling coefficients <aAIA.lAA2>*'
We note that Eq. 19.30 holds without modification for finite groups; in
this application it has been extensively reviewed by Griffith. 176 In the case
of noncompact groups we must proceed with great caution 177,178 and here
we confine our attention almost exclusively to compact groups.
The practical calculation of the matrix elements of the components of a
tensor operator is reducible, in terms of the Wigner-Eckart theorem, to the
calculation of the coupling coefficients of the group 9 used to establish the
basis states and the calculation of the reduced matrix elements of the
tensor operator. To this end we may readily invert Eq. 19.30, using Eq.
19.15, to give
<aAIIIT(A)IIA2>=  <AA2IaAIAI><AIAIIT(AA)IA2A2> (19.31)
A,A2
19.9 SELECTION RULES
The Wigner-Eckart theorem implies the existence of selection rules. The
coupling coefficients in Eq. 19.30 will vanish unless the weights of the bra,
ket, and tensor operator component satisfy the relation
A+A2=AI
( 19.32)
Any matrix elements that do not satisfy this relation are necessarily null.
Of course, the satisfaction of Eq. 19.32 alone does not assure us that the
matrix element is nonvanishing. As usual, selection rules tell us what is not
possible, but tell us nothing about what is possible.
A more stringent selection rule arises by noting that the coupling
coefficient will vanish unless the triple Kronecker product
(A I )* X (A) X (A 2 ):J (0)
(19.33)
Of, equivalently,
(A) X (A 2 ):J (AI)
(19.34 )
or
(A I )* X (A):J (A 2 )*
( 19.35)

228 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
For future use we write the number of times the identity representation
occurs in the reduction of the triple Kronecker product (A I )* X (A) X (A 2 )
as C(Al' A, A 2 ). A selection rule is obtained whenever c(A I , A, A 2 ) = O. The
number of terms in the summation on the right in Eq. 19.30 is just
c(AI' A, A 2 ).
19.10 APPLICATION TO 80(3)
The group 80(3) is simply reducible, and hence the Wigner-Eckart
theorem may in this instance be written as
<al}lmll T( kq) la 2 }2 m 2> = C!;,2 <al}111 T(k) Il a 02>
( 19.36)
where a l and a 2 stand for any additional labels that may be required to
uniquely specify the basis states, and where the coupling coefficients are
real. With a suitable redefinition of the reduced matrix element, we may
rewrite Eq. 19.36 using the 3-} symbol defined in Eq. 17.105 to give the
familiar expression
<adl m lI T (kq)l a :J2 m 2)= ( _1);,-m,(
JI
k
q
}2 ) <a l }IIIT(k)ll a :J2>
m 2
-m 1
(19.37)
The matrix elements of T(kq) vanish unless
m 2 +q=m l
( 19.38)
and the reduced matrix element vanishes, by Eqs. 19.31 and 19.33, unless
the triangular condition
}I +}2  k  I}I - }21
( 19.39)
is satisfied.
The reduced matrix elements of the angular-momentum tensor operator
J(I) can be readily determined by noting that J z =J(I,O) and hence from
Eq. 19.37
<a}mIJzlaJ'm'> = aa'Jj'mm,m
=(_I)J-m ( )
-m
1
o
 )<qj!lJ(1)!lqj)

APPLICATION TO SO(3) 229
But from Eqs. 17.101 and 17.105 we have
( _ 1 )J - m ( j
-m
1
o
) ) -
m Vj (j+ 1)(2j+ 1)
m
Therefore we have for the reduced matrix element
<ajIlJ(I) lIaJ') = yj (j+ 1 )(2j+ 1) aa'jj'
( 19.40)
With somewhat greater difficulty we may show that I70
< /11 C( k ) Ill') = ( - 1) 'vi (2/ + 1 )( 21' + 1) (
k
o
1 0 ' )
(19.41 )
From the triangular condition on the 3-j symbol we deduce the selection
rule
l+l'>k>I/-I'1
( 19.42 )
For k = 1 we obtain the selection rules
Llm = 0, + 1 and LlI = 0, + 1
(OO)
(19.43 )
The matrix elements of C (1) arise in the calculation of electric dipole
matrix elements. I80 At first sight the selection rule LlI = 0 seems to be
anomalous, as it is well known to spectroscopists that electric dipole
transitions with dl = 0 are forbidden. However, as long as we restrict our
attention to the rotation group there is no reason to expect thes transi-
tions not to be allowed, and yet specific calculation reveals that such
transitions are associated with vanishing matrix elements. To account for
the vanishing of these matrix elements we must go outside the group SO(3)
to the group of space reflections, where we find that parity conservation
requires that k and I-I' both be even or odd.
In general we can calculate the reduced matrix elements of a tensor
operator only by going outside of the group structure defining the coupling
coefficients. As we see later, it is frequently possible to construct a set of
basis states whose properties are described by a chain of nested groups and
to apply the Wigner- Eckart theorem systematically through the chain of
groups. In this way, at least formally, it becomes possible to factorize the
matrix element into products of coupling coefficients.

230 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
19.11 GENERALIZED RECOUPLING COEFFICIENTS
In practical applications it is often necessary to consider the coupling of
products of the basis states of three or more irreducible representations of
a group 9. The coupling of the products may be performed in various
possible sequences. The various resultant coupled states are related by a
unitary transformation. The elements of these unitary matrices are known
as generalized recoupling coefficients, or briefly as recoupling coefficients.
Consider the ket \AIAI,A2A2,A2A3) belonging to the space of the triple
Kronecker product (AI) X (A 2 ) X (A 3 ). We may use Eq. 19.14 to write either
IAIAI' A 2 A 2 , A3 A 3) =  <aI2AI2AI2IAIA2)*1 (AIA2)aI2AI2AI2' A 2 A 3 )
a12,A l2 ,Al2
  <aAAlaI2AI2A3)*<aI2AI2AI2IAIA2)*1 (AIA2)aI2AI2A3; aM)
a12, A l2 , A12 a, A, A
or
IAIAI,A2A2,A3A3)=   <a'AAIAla23A23)*
a23, A 23 , A23 a', A, A
(19.44 )
X <a23A23A23IA2A3)*IAI (A2A3)a23A23; a' AA) (19.45)
The two coupled triple-product states must be related by a unitary
transformation such that
I(AIA2)aI2AI2A3;aAA)=  <AI(A2A3)a23A23a'AI
a',a23,A 23
x (AIA2)aI2AI2A3; aA)IA I (A2A3)a23A23; a' AA)
( 19.46)
where we have employed the same argument used to derive Eq. 19.29 to
deduce that the coefficients effecting the unitary transformation are dia-
gonal in the representation (A) and independent of the weights A.
We may obtain an expression for the recoupling coefficient in terms of
the jm-coupling coefficients of Section 19.6 by first expanding both kets in
Eq. 19.46 using Eq. 19.13 and then equating coefficients of the resultant
uncoupled ket IAIAI' A 2 A 2 , A3A3)' The result is
<AIA21 a 12 A I2 A I2)< a 12Al2A31 aAA)
-  <A2A3Ia23A23A23)<Ala23A23Ia'AA)
a', a23, A 23
X <AI (A2A3)a23A23; a' AI (AIA2)aI2AI2A3; aA)
( 19.47)

GENERALIZED RECOUPLING COEFFICIENTS 231
Multiplying both sides by <a' 23A' 23A23IA2A3>* and summing over A 2 and A3
(using Eq. 19.15), we obtain
 <a 23 A 23 A 231 A 2 A 3> * <A I A 2 1 a 12 A 12 A 12> < a 12 A 12 A 31 aAA >
A2, A3
=  <Ala23A23Ia' AA><AI (A2A3)a23A23; a' AI (AIA2)aI2AI2A3; aA>
a '
( 19.48)
We then multiply both sides by <a" A'A'IAla23A23>* and sum over Al and
A 23 (using Eq. 19.15 again) to give the final result
<AI (A2A3)a23A23; a' AI (AIA2)aI2AI2A3; aA>
=  <a' AAIAla23A23>*<a23A23A23IA2A3>*<AIA2IaI2AI2AI2><aI2AI2A31aM>
AJ,A2! A 3"
A23,AI2
( 19.49)
Thus the triple recoupling coefficients may be expressed as a sum over a
quadruple product of coupling coefficients. Furthermore, they satisfy the
usual orthogonality condition
 «AIA2)aI2AI2A3; aAIAI (A2A3)a23A23; a' A>*
a23, A 23
X <AI (A2A3)a23A23; a' AI(AIA2)a;2A;2A3; aA>
=  I  A A '  I
a 12 a 12 12 12 aa
( 19.50)
In practical applications it is frequently necessary to couple more than
three basis states. For example, the unitary transformation from LS to jj
coupling in atomic spectroscopy requires a quadruple coupling of angular
momenta. All of these recoupling coefficients may be resolved stepwise
into sums over product of jm-coupling coefficients and triple recoupling
coefficients.
The fundamental problem in calculating recoupling coefficients for an
arbitrary group is essentially that of constructing explicit expressions for
the jm-coupling coefficients. In the special cases of SO(3) and SO(4), the
solution is completely known, and well-established diagrammatic methods
exist for performing summations over products of jm-coupling and re-
coupling coefficients. lOS, 181-187 A sophisticated machine program for im-

232 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
plementing the diagrammatic methods has been developed by Bor-
darier}88
The problem of constructing}m-coupling and recoupling coefficients for
finite groups has been considered by Griffith,176 while Wybourne 189 has
shown how the phase ambiguities for finite groups can be overcome for
finite groups by embedding them in an overlying continuous group.
The derivation of general explicit expressions for the }m-coupling coef-
ficients for arbitrary groups has, apart from a few special cases such as
SO(3) and SO (4), met with comparatively little success. The reasons for
such a situation are not difficult to find. The construction of explicit
expressions requires that the Clebsch-Gordan series for the arbitrary group
be known explicitly. While this is possible in principle for simply reducible
groups, it is not generally possible for non-simply-reducible groups. If the
group is not simply reducible, multiplicity labels must be assigned to
coupled representations. The situation is further complicated by the need
for great care in the assignment of phases and in the treatment of
complex-conjugate representations.
We do not explore here the various complexities associated with the
construction of generalized recoupling coefficients for arbitrary groups,
except to note that this is one of the great unsolved problems of group
theory and yet the most vital problem for the application of the Wigner-
Eckart theorem to non-simply-reducible groups. The reader interested in
pursuing this topic will find a diverse and interesting literature l31 , 19196 on
it.
19.12 RECOUPLING COEFFICIENTS FOR 80(3)
The problem of obtaining expressions for the recoupling coefficients for
SO(3) and its covering group SU(2) is much simpler. As the group is
simply reducible, there is no multiplicity problem, and the Clebsch-Gordan
series is just
)1 +)2
D(}I) X D(}2) =  D(}3) (}1  }2)
)3 =)1 -)2
( 19.51 )
Furthermore, since the explicit form of the}m-coefficient is known (see Eq.
17.101), every recoupling coefficient may be expressed as a sum over
products of }m-coefficients, which are capable of explicit evaluation.
In making practical calculations in SO(3) it is usually desirable to make
use of the highly symmetrical 3}-symbol rather than the unsymmetrical

RECOUPLING COEFFICIENTS FOR SO(3) 233
coupling coefficients. Thus we shall write
C!J('m3 = ( _ly,-h+ m 3V[jJ ( }I
m l
12
}3 )
-m3
( 19.52)
m 2
Remembering that the 3}-symbol is real, we have from Eq, 19.49 that the
triple recoupling coefficient for SO(3) is expressible as
<}I(}2J3)}23;}I(JI}2)}12i3;}>= (-1)JI+J2+J3+J V [}12,J23] ( ]
13
12
il2 )
1
123
(19.53)
where [a, b,. . .] = (2a + 1 )(2b + 1). . ., and the 6}-symbol is defined in terms
of the 3}-symbols via the relation
( i] 12 3 )= L (_l)k( 12 i3 )( i] 1 2 -3 )
II 1 2 m 2 m3 m l n2
3 all m, n I
x( II 12 13 ) ( I] 1 2 i3 ) (19.54)
-n l m 2 n 3 n l -n 2 m 3
Where k = II + 1 2 + 13 + n l + n 2 + n3'
The 6}-symbol is highly symmetrical: it is invariant with respect to any
interchange of its columns or interchange of the upper and lower argu-
ments of any pair of columns. By considering the triangular conditions for
the nonvanishing of the 3}-symbols in Eq. 19.54, we readily deduce that the
6}-symbol will vanish unless the triangular conditions are met by the
following arguments:
{}. r
o
J '{J}'{}
While it is possible to evaluate the 6}-symbol by using Eq. 19.54 with Eq.
17.101, Racah l68 has shown than an expression involving a single summa-

234 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
tion without reference to the projective quantum numbers IS possible,
namely
(: : ;) =!1( abc )!1( aej)!1( dbc )!1( dec)
[ (-I)x(x+l)!
X  (x-a-b-c)!(x-a-e- f)!(x-d-b- f)!(x-d-e-c)!
X (a+b+d+e-X)!(b+c+e+f-x)!(a+c+d+f-X)!] (19.55)
where
d( abc) = [ (a + b - c)! ( a - b + c)! (b + c - a)! / (a + b + c + I)!] 1/2
The 9j-symbol arises in considering the coupling of four representations
of 80(3) and may be defined in terms of the quadruple recoupling
coefficient
< (jlj2)j 12 (j j 4)j34;jl (j Ij3)j13 (j2j 4)j24;j>
JI J2 JI2
([' . . . ] ) 1/2 ( 19.56)
= JI2,J34,JI3,J24 J3 J4 J34
.113 J24 J
By repeated application of Eq. 19.46 we may readily show that the
9j-symbol may be expressed as a sum over triple products of 6j-symbols:
a b c
d e f =  (_1)2X[x]
h x
g I
/
( ; b :)(: e f)[g h )
( 19.57)
I x h l x a
The 9j-symbol finds considerable application in the evaluation of the
matrix elements of coupled products of tensor operators in 80(3) and in
the application of the Wigner-Eckart theorem to the group 80(4).
EXERCISES
19.8. Show that the 6j-symbols satisfy the orthogonality condition
 [c,f] (:  ;) (:  ;) =8 jg 8(a,e,f)8(d,b,f) (19.58)

RECOUPLING COEFFICIENTS FOR SO(3) 23S
where we write
8 ( a, b, c ) = 1 or 0
(19.59)
according to whether a, b, and c obey the triangular condition or not.
19.9. Show that the triple recoupling coefficients satisfy the associative property
 < (jlj2)jl213;jl (jlj3 )jI3j2;j)< (jlj3)jI3j2;jl (jlj2)jl213 ;j)
il3
= «jlj2)jIJJ3;jljlj23;j)
( 19.60)
and use this result to derive the Biedenham-Elliott l32 sum rule
 (_l)C+f+8[C) ( :
b
;)(:
b
d
;)=(:
e
) ( 19.61 )
e
d
19.10. Show that the 9j-symbol may be written as a sum over a sextuple product
of 3j-symbols:
jl j2 il2 = L ( jl j2 j12 )( j3 j4 h4 )
j3 j4 j34
ml m2 ml2 m3 m4 m23
jl3 j24 i all m' s
X ( jl3 j24 j)(h i4 h4 )( j12 j34 ) ( 19.62)
ml3 m24 m m 2 m4 m24 m 12 m34
19.11. Verify that the 9j - symbols satisfy the orthogonality condition
a b c a b c'
L [c,j,g,h) d e f d e f' = 8cc,8ff,8( a, b, c )8( d, e,f)8( c,f,k)
gh g h k g h k
( 19.63)
19.12. Remembering that
(
j
o ) = (- 1 )J - m
o VTJ]
( 19.64)
-m
show that
( :
b
) a+b+c
C = (- 1 ) 8 ae 8bd8 ( a, b, c )
o \/ [a,b]
(19.65)
e

236 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
and
a
b
c
= (-I )b+C+d+g8cgh ( a
Y [c,g] e
b
d
; )
( 19.66)
e
f
o
d
g
h
19.13 COUPLING COEFFICIENT FOR 80(4)
The group 80(4) is of special significance in the interpretation of the
degeneracies of the nonrelativistic hydrogen atom. 1 97-200 As we saw in
Section 5.11, the group 80( 4) is locally isomorphic to the direct-product
group 80(3) X SO(3). The group SO( 4) may be gnerated by six infinite-
simal rotation operators 32 ,53
JAI'=i(Xl' a: A -X A a :,.) (AoFp.=1,2,3,4) (19.67)
where JAil = - J p>..'
We may write the six infinitesimal operators as components of two
spherical rank-one tensor operators J(l) and N(I) by putting
J (1) = J
o 23
J (1)-+ 1 (J +' J ) }
:!: I - - V2 31 - I 12
N (1) 1 ( . )
z I = + V2 J 42 + iJ 43
( 19.68)
N(1) = J
o 41
to yield the commutation algebra
[J (1) J (1) ] = -c J (1) = [N (1) N (1) ]
q , q' qq'q" q" q' q'
[ J (1) N <,I) ] = - c N (1)
q , q qq'q"
( 19.69)
The six components of J(l) and N(I) may be divided into two sets. The
first set comprises the two Weyl self-commuting operators HI and H 2
constructed from suitable linear combinations of J6 1 ) and N6 1 ). The second
set comprises four operators Ea that are simultaneous eigenvectors of HI
and H 2 :
[HI,Ea] =alEa and [H2,Ea] =a2 E a
( 19.70)
and are constructed from linear combinations of J{ and N{. The
eigenvalues a l and a 2 define a two-dimensional weight space, which when

COUPLING COEFFICIENT FOR SO(4) 237
(-1, 1)
a 2
I
I
I
I
I
I
I
I
(1, 1)
(- 1 ,-1)
( 1 ,-1 )
Fig. 19.1. The root figure for 80(4) with HI =JJ I ) and H 2 = NJI).
plotted as a two-dimensional array will form the root figures of SO(4).
In defining the weight space we have two rather obvious choices:
I. We may choose
H -J(1) and H - N(1) (19.7Ia)
1- 0 2- 0
with
E+ =J(l)+N(1) and E- =J(1)-N(1) (19.71b)
I I I I I I
in which case we obtain from Eq. 19.70 the root figure shown in Fig. 19.1.
2. We may choose
J o (1) + N o (1) H _ J o (1) - N o (1)
H - - d
an 2-
122
(19.72a)
with
J1) + N1)
E+ = - 1 - 1
I 2
J(1) - N(1)
and K;;, I = :t I 2 :t I
(19.72b)
in which case we obtain the root figure shown in Fig. 19.2.
We may use either Eq. 19.71 or Eq. 19.72 to establish a systematic

238 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
a2
I
I
I
I (0, 1)
(-1, 0)
(1,0)
- - - a1
1(0,-1)
I
I
I
Fig. 19.2. The root figure for 80(4) with HI =(JJ I )+ NJI)/2 and H 2 =(JJI)- NJI»/2.
labeling of the irreducible representations of 80(4). Use of Eq. 19.71 leads
to the usual Cartan labels [pq], where p and q are both integers or half
integers with p  I ql, and although p is necessarily positive, q may be
positive or negative. Use of Eq. 19.72 leads to the labeling of the irreduc-
ible representations of 80(4) in terms of a pair of integers or half integers
J 1 and}2' which at the same time label the irreducible representations of the
direct product D(}I) X D(}2) of 80(3) X 80(3). We note that the represen-
tations D(}I)XD(}2) and D(}2)XD(}I) (}I=F}2) correspond to distinct
representations in 80(4).
We may readily establish that the two schemes are related by putting
P=}I+}2 and q=}1-}2 (19.73)
Furthermore, under the restriction 80(4)80(3) we must necessarily
have
[ pq ]  [ p ] + [p - I ] + . . . + [ I q I ] ( 19.74 )
As a result a given basis state of the representation [pq] of 80(4) may be
labeled as
I [ pq ]} m> ( 19.75)
where} labels the 80(3) representation arising from the restriction 80(4)
80(3) via Eq. 19.74, and m labels the 80(2) representations arising from

COUPLING COEFFICIENT FOR SO(4) 239
SO(3)SO(2). Thus the basis states are labeled under the group reduction
scheme 80(4)SO(3)SO(2).
The basis states specified by Eq. 19.75 can clearly be related to those of
the SO(3) X 80(3) scheme Ijlml,j2m2> by writing
I[pq ]jm> = Hjl + j2,jl-j2]jm>=  <m l m 2 Ijm>ljl m lj2 m 2> (19.76)
m.,m2
where the <m l m 2 Ijm> are the usualjm coefficients of SO(3).
The coupling coefficient for SO(4) arises in the reduction of the
Kronecker product [Plq.] X [P2q2]' By noting that every irreducible repre-
sentation of SO(4) is locally isomorphic to the simply reducible direct-
product group SO(3) X SO(3), we determine that SO(4) is also simply
reducible and that the Clebsch-Gordan series is just
t u
[ab]x[cd]=   [a+c-a-p,b+d-a+p] (19.77)
a=O {3=0
where t is the lesser of a + band c + d, while u is the lesser of a - band
c-d.
The SO(4) coupling coefficients may be defined as the coefficients
<[Plq.][P2q2]; [PI2qI2]JI2MI21[Plq.]jlml[P2q2]j2m2> that effect the unitary
transformation
I [Plql ]jl m l>1 [P2q2]j2 m 2>
 <[PIQl][P2q2]; [PI2QI2]JI2MI21[PIQl]jlml[P2Q2]J2m2>
[PI2Q12]J 12 M 12
X I[PIQl][P2Q2]; [PI2QI2]J I2 M I2 >
( 19.78)
If we make use of Eq. 19.76 to expand the left-hand side and then the
properties of the jm coefficients to perform the appropriate recouplings,
noting Eq. 19.62, we readily find that the SO(4) coupling coefficient can be
taken as real and written as
<[ PIQl ]jl m l [P2Q2 ]j2 m 21 [PIQl] [P2Q2]; [PI2QI2]J 12M12>
= [ (2jl + I) (2j2 + I) (2J I2 + I) (P12 + Q12 + I) (P12 - Q12 + I) ] 1/2
x <_l)J 12 -M 12 ( )\ J 12 )2 )
m l -M 12 m 2
!(Pl+Ql) !(P2+Q2) -1 (P12 + Q12)
X t(PI-Ql) t(P2-Q2) -!(PI2-QI2) ( 19.79)
Jl J2 J 12

240 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
-a result first obtained by Biedenharn. 172
Having constructed the 80(4) coefficients for a basis defined through
the group chain 80(4):) SO(3):) SO(2), we find it useful to construct
tensor operators Tpq]K that have well-defined transformation properties
not only with respect to 80(3) and 80(2) but also with respect to 80(4).
Using the Wigner-Eckart theorem we may write
<al [Plql ]JIMII r[pq] Q Kla 2 [ P2q2]J 2 M 2>
= ( - I ) J 1 - M 1 ( J I K J 2 ) < a I [ PI q I ]J III T[ pq] K II a 2 [ P 2q 2 ]J 2 >
-M I Q M 2
( 19.80)
where we have factored off the dependence of the matrix element on the
80(2) quantum numbers. The reduced matrix element on the right-hand
side may now be written as
<al [Plql]J III T[pq] Klla 2 [P2q2]J 2>
1
= { [K,J I ,J 2 ] } 2
-!(PI+ql)
t(PI-ql)
J 1
-! (P2 + q2)
t (P2 - q2)
J 2
-!(p+q)
-!(p-q)
K
x < a I [ PI q I ] II T [ pq] "a 2 [ P 2q 2 ] >
(19.81)
where we have absorbed a factor of [(PI + ql + 1)(Pl - ql + I)]! in our
definition of the reduced matrix element on the right-hand side. Thus the
80(4) coupling coefficient allows us to determine completely the depen-
dence of the matrix elements on the numbers used to label the 80(3) and
SO(2) representations required to specify the basis states in the 80(4)
:J 80(3):) 80(2) scheme.
19.14 RACAH'S FACTORIZATION LEMMA
In the preceding section we met for the first time the evaluation of a
coupling coefficient for a set of basis states labeled by the irreducible
representations of a chain of nested subgroups, in that case the chain
SO(4):) 80(3):) 80(2). We note .from Eq. 19.79 that the coupling
coefficient for SO(4) in the I[pq]jm> basis may be factorized into the

RACAH'S FACfORIZATION LEMMA 241
product of two coupling coefficients by writing
<[ Plql ]Jlm l [P2q2 ]J2 m 21 [Plql] [P2Q2]; [P12q12 ]J I2 M I2 >
= <} I m l}2 m 21}1}2; J I2 M I2 ><[ Plql ]}I [P2q2 ]}21 [Plql] [P2q2]; [P12q12 ]J I2 >
(19.82)
The first coupling coefficient involves only the irreducible representations
associated with the 80(3):) 80(2) part of the group chain and is simply a
}m-coefficient. The second coefficient involves only the irreducible repre-
sentations associated with the 80(4):) 80(3) part of the group chain and
may be written as
< [ PI q I ]} I [ P 2q 2 ]} 21 [ P I q I ] [ P 2q 2 ]; [p 12q 12 ] J 12 >
]
= ( (p 12 + q 12 + I ) (p 12 - q 12 + I) (2) I + I ) (2) 2 + I ) ) 2
JI
J2
-! (P12 + q12)
t(PI2-qI2)
J I2
(19.83)
-!(PI+ql)
X ! (PI - ql)
-!(P2+q2)
!(P2-q2)
It is natural to ask if this factorization process cannot be carried out for
arbitrary group chains. Let us consider a group 9 whose unitary irreduc-
ible representations are labeled f; and suppose 9 contains a subgroup3C
whose representations are labeled A; and have components 'A.;. Under the
group decomposition 9 3C we have
f;= ajAj
( 19.84 )
where a j is the number of times  occurs in the decomposition. Suppose we
construct a set of basis states If;a;A;A;>.
Using the coupling coefficients defined for the subgroup we can repre-
sent the coupling of two ket basis states as
IflaIAIAI>lf 2 a 2 A 2 A 2>
=  <aAAIAIAI,A2A2>lflaIAI,f 2a2A2; aAA>
aA'A
( 19.85)
The vectors on the right-hand side will form a ket basis for the irreducible
representation A of 3'C. We can form a ket basis for the irreducible repre-
sentation f of 9 by fOrmIng appropriate linear combinations and hence we

142 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
may write
IflalAIAI>lf 2 a 2 A 2 A 2>
=  <TfaAAlflaIAI,f2a2A2;aAA>I(AIA2)TfaAA> (19.86)
'Tra
But for the same reasons as given in Section 19.8, the coupling coefficient
must be independent of A, and hence, using Eq. 19.86 in Eq. 19.85, we
obtain, with an obvious notational change,
If laIAIAI>lf 2 a 2 A 2 A 2>
=  <aAAI A IAI,A2A2>< TfaAalf1alAI,f 2 a 2 A 2>1 (flf 2)TfaAA>
'T r aaA'A
( 19.87)
This result may be compared with the corresponding coupling
If lalAIAl>lf 2 a 2 A 2 A 2> =  < TfaAAlf lalAIA l , f 2 a 2 A 2 A 2>1 (f If 2)TfaAA>
'TraA'A
( 19.88)
to deduce Racah's celebrated factorization lemma 9 ,194
< TfaAAlf lalAIA I , f 2a2A2A2> =  <aAAIAIAI' A2A2>< TfaAalf lalA l , f 2a2A2>
a
( 19.89)
Racah's lemma can be used for chains of any number of nested groups
by simply applying it in succession to each group and its corresponding
subgroup. Used in conjunction with the Wigner-Eckart theorem, it pro-
vides a powerful tool in practical applications of group theory to physical
problems.
19.15 ISOSCALAR FACTORS
The factors <TfaAalflalAI,f2a2A2> ansIng In Racah's factorization
lemma are invariant under the group 3C and are here, following Ed-
monds,201 termed i'oscalar factors. The isoscalar factors are elements of a
unitary matrix satisfying the orthogonality conditions
 <f lalA I , f 2 a 2 A 2ITfaAa>*< TfaAalf laA, f 2 a ;A;> = 8alai8a2a28 AI A i 8 A 2 A 2
'Tra
( 19.90)

ADJOINT TENSOR OPERATORS 243
and
 < TfaAalflalA I , f 2a2A2>*<f lalA I , f 2a2A2IT'f' a' Aa> = TT'rr'aa'
aala2 A 1 A 2
(19.91)
Normally we can define our phases so as to ensure that the isoscalar
f actors are real.
The calculation of isoscalar factors is central to the application of the
Wigner-Eckart theorem in evaluating the matrix elements of tensor opera-
tors acting on basis states constructed for a chain of groups. Typically we
have
<€I f lalAIAII T( €faAA) 1€2 f 2a2A2A2>
=  <€Iflll T( €f) 1I€2f iT>< Tf laIAIAllfaAA, f 2a2A2A2>
T
=  < €I r III T( €f) 1I€2 f 2T><aAIAIIM 2 >< Tf lalAlalf aA, f 2a2A2> (19.92)
T,a
and the calculation reduces to the evaluation of three distinct parts: (1) the
reduced matrix elements, (2) the coupling coefficients, and (3) the isoscalar
factors.
19.16 ADJOINT TENSOR OPERATORS
The matrix elements of the adjoint or hermitian conjugate Tt of an
operator T are related to those of the operator T via the relationshi p 202
<AIAII T( AA) t1A2A2> = <A2A21 T( AA) IAIAI>*
(19.93)
We recall that a tensor operator T(A) may be defined for a semisimple Lie
group through the commutation relationship
[X a , T(AA)] =  <AA+ naIXaIAA>T(AA+ na)
n
( 19.94 )
It follows, by taking adjoints, that
[Xa' T(A;\) t] = - [X -a' T(AA)]t
= - <AA-naIX_aIAA>*T(AA-na)t
n
= - <AAIXaIAA-na>T(AA-na)t
n
( 19.95)

244 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
Thus T(A)t transforms contragrediently to T(A).
The components of T(A)t do not transform as a tensor operator. We
may, however, define an operator Tt(A) such that
Tt(AA) = ( _I)!(AA)T(A* -A) t
( 19.96)
where (- IY(AA) is a phase factor chosen to ensure that Tt(M) transforms
under the operations of the defining group in the same manner of T(M).
A given tensor operator T is self-adjoint if and only if Tt =T. It follows
from Eq. 19.96 that a tensor operator T(A) can be self-adjoint if and only
if (_1)2!(A,A)= I, and that the components of a self-adjoint tensor operator
will satisfy the condition
T(AA)t = (-I)!(A,A)T(A*-A)
( 19.97)
It follows that for a self-adjoint tensor operator T(A) we have
<AIAII T(AA) IA2A2> = ( - I )!(AA) <AIAII T( A * - A) t1A2A2>
= (-I)!(AA)<A2A2IT(A*-A)IAIAI>* (19.98)
If the Kronecker product (A) X (AI) is simply reducible, then we may
square Eq. 19.98, apply the Wigner-Eckart theorem, and then over A I ,A 2 ,
and A3 using the orthogonality of the coupling coefficients, to obtain
I<Alll T( A) IIA2>1 = V  I<A211 T( A *) IIA,>*I
( 19.99)
The phase relating these reduced matrix elements must be fixed in relation
to the phase convention adopted in defining the coupling coefficients as
well as to the phase required in Eq. 19.97.
If the Kronecker product is not simply reducible, then we have the
weaker result
 l<aAlllT(A)IIA 2 >1 2 = A2  l<a'A2I1T(A*)IIAI>12 (19.100)
a A I a'

SYMMETRY PROPERTIES OF COUPLING COEFFICIENTS 245
EXERCISES
19.13. Show that for SO(3) the components of a self-adjoint tensor operator
sa tisfy
t k-q
T(kq) = (-1) T(k-q)
(19.101)
and that k and q are necessarily integers.
19.14. Show that the redefined SO(3) reduced matrix elements of Eq. 19.37 satisfy
the relation
(ill T( k) Iii') = ( - I )k+ j - l (i'll T( k) lIi)*
( 19.102)
19.17 SYMMETRY PROPERTIES OF COUPLING COEFFICIENTS
The labor involved in the practical evaluation of coupling coefficients
can be greatly diminished by first investigating their symmetry properties
with respect to the permutation of their arguments and their relation to
complex-conjugate representations.
Since the Clebsch-Gordan series for (AI) X (A 2 ) and (A 2 ) X (AI) are
identical, we must have
<AIAIA2A2IaAA> = 111 <A2A2AIAllaAA>
( 19.103 )
where 111 = 111(A I A 2 A) is a phase factor that may be determined by the
phase convention adopted for the state of highest weight in the representa-
tion a A.
The eigenstates associated with a representation (A) and its complex
conjugate (A *) are related by a phase factor, and hence we must also have
<AIAIA2A2IaAA> = 1I2<AI * - AIA2 * - A 2 1aA * - A>
(19.104)
where 112 = 1I2(A I A 2 A) is again a phase factor.
If we apply the Wigner-Eckart theorem to Eq. 19.98 and use Eq. 19.99,
we find that if the Kronecker product (AI) X (A 2 ) is simply reducible, then
<A I A I A)\2IAA) =1/) V[[:I]] <A * -AA 2 A 2 I A * I -11. 1 )* (19.105)
with 113 a phase factor.
The symmetry relationships gIven by Eqs. 19.103-19.105 suggest the

246 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
possibility of defining for an arbitrary compact group a 3j-symbol of
greater symmetry than the coupling coefficients 190-194,203 analogous to that
defined for SO(3). .
Consider the Kronecker product of the representations (AI)(R) and
(A2)(R). Let us define a unitary matrix U whose matrix elements
u - [ A ] ! ( Al
03A3A3AIA2 - 3
Al
A 2
A 2
A3 )
A3 03
( 19.106)
reduce the Kronecker product. F.xplicitly,
 [A 3 J ( Al A 2 A3 ) :D Xl)(R)i?(R) ( Al
AIA'I Al A 2 A3 03 Al
2A2'
The symbol
( AI
Al
A 2
A 2
A 2
A 2
A' )
A: a;
-8 8 *(oJA3) ( R )
- 030) A3A) Ay')
( 19. 107)
A3 )
A3 03
( 19. 108 )
denotes a 3j-symbol, with a 3 supplying a multiplicity label for distinguish-
ing the c(A I A 2 A 3 ) representations (A 3 ) that occur in the reduction of the
Kronecker product (AI) X (A 2 ). The 3j-symbols will satisfy the usual unitar-
ity conditions
 [A 3 ]( I
03 A 3 I
A3 ) ( AI
A3 03 A;
A 2
A 2
and
 [A 3 ]( l
AIA2 1
A3 ) ( AI
A3 03 Al
A 2
A 2
A 2
A'
2
A3 ) = 8" A,8 A A' (19.109)
A "I I 2 2
3 03
A 2
A 2
A' )
3 = 8 ,8 ,8 ,
0303 A3 A 3 A3 A 3
A; 0;
(19.110)
It is not difficult to see that for a simply reducible group the absolute
magnitude of th 3j-symbols is invariant with respect to any permutation
of its columns. 131,204 The definition of the 3j-symbol in Eq. 19.106 is
sufficiently loose to permit multiplication by a phase 17 = 17(A I A 2 A 3 ) and to

SYMMETRY PROPERTIES OF COUPLING COEFFICIENTS 247
still preserve its unitarity. Taking advantage of this degree of arbitrariness,
we can define our phases so that the 3j-symbol remains invariant under an
even permutation of its columns and is multiplied by a phase ( - 1 )Al + A 2 + A3
for an odd permutation, where (- I)A, is a phase permanently associated
with each representation (Ai)'
The situation for non-simply-reducible groups is complicated by the
weaker permutational-invariace statement that 190
2 2
 ( AI A 2 A3 ) = ( Al A3 A 2 )
a3 Al A 2 A3 a3 a2 Aa A3 A 2 a2
2
= ( A 2 A3 AI) (19.111)
a} A 2 A3 Al al
which parallels the statement of Eq. 19.100. Now it is by no means clear
that the degree of arbitrariness in phase definition is sufficient to produce a
3j-symbol whose absolute magnitude is invariant under all permutations of
its columns. Indeed, while the absolute magnitude of the 3j-symbols for
SU(3) can be defined so as to be invariant under all permutations of their
columns,192 it is certainly not possible for all non-simply-reducible
groups. 191,205
Derome l91 has shown that if the three representations (AI)' (A 2 ), (A 3 ) are
inequivalent, then it is possible to define a suitable set of phases to yield a
3j-symbol whose magnitude is invariant under all permutations. If two of
the three representations are equivalent, say (AI) = (A2)(A3)' then it is
still possible to choose the phases appropriately to produce an invariant
3j-symbol. In this case it is necessary "to consider whether aA * 3 occurs in
the symmetric or the antisymmetric part of the Kronecker square (AI) X
(AI)' However, if all three representations are equivalent, then it may be
impossible to define a set of phases to produce a 3j-symbol that is
invariant under permutation of its columns. It is then necessary to analyze
the Kronecker cube of the representation (AI) into the terms that occur in
the symmetric part [(A I )@{3}], the anti symmetric part [(A I )@{1 3 }], and
the mixed-symmetry part [(A I )@{21}]. If the identity representation does
not appear in the mixed-symmetry part, then it is possible to express the
permutational properties of the 3j-symbol via a diagonal permutational
matrix, and thus to choose a set of phases to yield a 3j-symbol whose
magnitude is permutationally invariant. However, if the identity represen-
tation occurs in the mixed-symmetry part, the permutational matrix cannot

248 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
be chosen to be diagonal, and hence it is no longer possible to define a
3j-symbol whose magnitude is permutationally invariant. A careful exposi-
tion of this problem and its solution has recently been given by Butler.}94
The analysis of the Kronecker cube can be most readily made via the
method of Schur-function plethysms, as has been outlined elsewhere,3} and
using tables of plethysms. 206 , 207
The 3j-symbol introduced in Eq. 19.108 differs from the ordinary
coupling coefficient only by a phase and by the factor  introduced
to make it symmetrical. Since the 3j-symbol's magnitude is permutation-
ally invariant (apart from the special case of three equivalent representa-
tions just alluded to), we would expect the coupling coefficients to satisfy
the reciprocity relationship
../[A]
<A,A,A2A2IaAA)='I1J V1AJ <A* -AA 2 A 2 IaA,* -A,) (19.112)
even when c(A}A 2 A) > 1.
19.18 RECIPROCITY AND ISOSCALAR FACTORS
The amount of computation required to construct tables of isoscalar
factors can be greatly reduced by a judicious use of their symmetry and
reciprocity properties. 9 ,208-2}4 As with the coupling coefficients just consid-
ered, complexities arise if the groups are non-simply-reducible.
In the notation of Section 19.11 a typical generalized recoupling
coefficient is designated as
<f }a}A}A}f 2a2A2A2ITfaAA>
(19.113)
If the subgroup jC is simply reducible, or at least the Kronecker product
A] X A 2 is simply reducible, then Racah's factorization lemma involves just
one term, and the generalized recoupling coefficient becomes just the
product of a coupling coefficient and an isoscalar factor. In this case the
isoscalar factor satisfies the symmetry relationships
and
<f ]a]A]f 2 a 2 A 21TfaA> = 114<f 2a2A2f}a]A]ITfaA>
,
( 19.11
<f ]Q]A]f 2a4A21TfaA> = 115<f} *a]A] *f 2 *a 2 A 2 *ITf*aA *> (19.1 
where 114 and 115 are phase factors depending only on the f; and A;.
If the group '9 is simply reducible, or at least if f} X f 2 is simply
reducible, we have, after the manner of Eq. 19.105, the reciprocity re-
lationship

RECIPROCITY AND ISOSCALAR FACTORS 249
<f1a1A1f 2 a 2 A 21faA> = 116
[f][A 1 ]
<f*aA *r 2a2A21rTa1AT>
[r1][A]
(19.116)
where 116 is a phase factor dependent upon r i and Ai'
Caution must be exercised in applying the above symmetry and reciproc-
ity relationships to groups that are not simply reducible. In these cases,
Racah's factorization lemma involves more than one term, and auxiliary
conditions must be found if the above relations are to remain valid.
19.19 PHASE CONVENTIONS
Initially there is usually a high degree of arbitrariness in the assignment
of phases to coupling coefficients and isoscalar factors. This degree of
arbitrariness is frequently used to introduce phase conventions that
simplify the form of the coefficients. A minimal requirement is that the
unitarity conditions be satisfied. It is normally convenient to define the
phases to ensure that the coupling coefficients and isoscalar factors are
real. As we saw in the previous section, for simply reducible groups it is
possible to choose a phase convention that results in the construction of
3nj-symbols that are highly symmetrical with respect to permutations.
In practice the degree of arbitrariness of phase choice is progressively
restricted as we move through a chain of nested groups, primarily as a
consequence of the requirements produced by Racah's factorization
lemma. In some cases the choice of phases may become so restricted as to
involve the introduction of imaginary phases.2 15 ,216
A wide choice of phase conventions exists in the literature; this matters
little in practice, provided that the given phase convention is followed
consistently and different conventions are not mixed. In many cases,
though by no means all, the phases are fixed by an extension of the phase
convention adopted by Condon and Shortl ey 39 for the Clebsch-Gordan
coefficients of 80(3).
Let us first consider the phase of the so-called fully stretched isoscalar
factors. Suppose AT and A'; are the representations of a subgroup 3C( 9
:JJC) of highest weight contained in the reduction 9 3C of the representa-
tions r 1 and f 2 of g, respectively, and let r m be the representatIon
contained in r 1 X f 2 that contains the representation of highest weight
Am = A;n+A';. It follows from the unitary property of isoscalar factors that
we must have
<r lATf 2A'; Ir m Am> = e iw
(19.117)

250 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
where w is a phase angle. In order to fix our phase convention, we choose
w = 0 for all fully stretched isoscalar factors. If the representations of the
groups 9 and 3C are irreducible, then the usual multiplicity labels will not
be required (cf. Section 10.4). In the particular case of SO(3) this choice
amounts to putting
C/.l1./. +/2 = + I
lIlv. + l2
(19.118)
Extending the phase convention of Condon and Shortley,39 we choose
the generalized coupling coefficients to be real, and the isoscalar factors
<r lA;nr 2 A 21r Am»o
(19.119)
where now Am is the highest-weight representation of JC contained in r.
For simply reducible groups this choice creates no problems. However, for
non-simply-reducible groups, multiplicity problems may arise of the kind
encountered in Section 19.18, and great caution must be exercised. While
in these cases it is frequently possible to make an arbitrary phase choice
for the multiply occurring r 2 representations, it is not necessarily always
possible. 194
19.20 SIMPLE ISOSCALAR FACTORS
At the present time explicit formulas for the evaluation of isoscalar
factors (and coupling coefficients) have been found for only a few cases.
The most notable examples where success has been achieved are associated
with the simply reducible groups SO(4), SO(3), and SU(2), together with
results obtained by Gel'fand and Cetlin 217 ,218 and others219-222 for canoni-
cal chains of groups. We have already discussed the derivation of the
coupling coefficients for SO(3) and SO(2, 1) using spin or techniques which
can be generalized to arbitrary groups only with difficulty, especially in the
case of non-simply-reducible groups. While the methods of Gel'fand and
Cetlin are capable of a complete treatment in the case of the canonical
group chains 221 ,222
SU(n):JSU(n-l)... :JSU(2):JSU(I)
SO( n) :J SO( n -I) · · · :J SO(2)
these particular group chains are seldom of physical significance, though it
can sometimes be useful to calculate first in the Gel'fand-Cetlin basis and
then make a transformation to a physical basis. 223

THE BUILDING-UP PRINCIPLE 251
In many practical cases it is necessary to be able to evaluate particular
isoscalar factors in the absence of any explicit formula. Certain simple
isoscalar factors present no difficulties. We have already cited the fully
stretched isoscalar factors (cf. Eq. 19.117). If the summations arising in the
unitarity of the isoscalar factors (cf. Eqs. 19.90 and 19.91) contain just one
term, then the isoscalar factor may be assigned, to within a phase factor,
the value of unity. If the triple Kronecker product of the coupled repre-
sentations does not contain the identity representation, then the isoscalar
factor is null. More elaborate selection rules for null: isoscalar I factors have
been reviewed by Judd: 67
With these few simple isoscalar factors evaluated, it becomes possible to
build up systematically the more complicated isoscalar factors in terms of
the previously known factors. At each step the phases may be chosen to be
consistent with the adopted phase convention and to ensure that factors
are real and satisfy the unitarity conditions.
19.21 THE BUILDING-UP PRINCIPLE
The practical calculation of isoscalar factors can be divided up into five
steps.
A. Branching Rules
Initially it is necessary to establish the branching rules for the decompo-
sition of the irreducible representations r of 9 under the group restriction
9 3C.. This operation may be carried out either by the use of Littlewood's
S-function methods 26 ,224,225 or by standard group-character theory. Ex-
tensive tables appear in the literature. 31 The procedure using S-function
methods is briefly summarized in Appendix I and has been discussed in
detail elsewhere. 31 ,226,227 A typical table for Sp(4)SO(3) is shown in
Table 19.1.
B. Kronecker Products
Having established the branching rules, it is then necessary to determine
the Clebsch-Gordan series for the Kronecker products of the representa-
tions of both the group 9 and its subgroup 3C. This operation may be
performed either by use of the methods of weights developed in Chapter 11
or by use of the S-function method summarized in Appendix I. Again,
extensive tables exist in the literature. 31 A typical table for Sp (4) is shown
in Table 19.2.

Table 19.1. Branching Rulesfor the Reduction Sp(4)SO(3)
Sp( 4) SO(3)
<00) [0]
<10) [f]
<II) [2]
<20) [1]+[5]
<21) [!]+[]+[]
<30) []+ [] +]
<22) [2] + [4]
<31) [1] + [2] + [3] + [4] + [5]
<40) [0] + [2] + [3] + [4] + [6]
Table 19.2. Reduction of the Kronecker Products for Sp(4)
<(0) x <(0) = <(0)
<10) x <(0) = <10)
<10) x <10) = <(0)+ <11)+ <20)
<II) x <(0) <II)
<II) x <10) = <10) + <21)
<II) x 1.11)= <(0)+ <20)+ <22)
<20> x <(0) = <20)
<20) X<IO)= <10)+ <21)+ <30)
<20) x <11)= <11)+ <20)+ <31)
<20) x (20)= <(0)+ <11)+<20)+<22)+<31)+<40)
(21) X (00) = <21 )
<21) x <10) = <II) + <20) + <22) + (31)
<21) x <II) = <10)+ <21)+ <30)+ <32)
(21) x <20) = < 10) + 2<21) + <30) + <32) + <41)
<21) x <21)= <(0)+ <11)+ 2<20)+ <22)+2<31)+ (33)+ (40)+ <42)
<22) x <(0) = <22)
<22) x <10)= (21)+ <32)
<22) x <11)= <11)+ <31)+ <33)
<22) x <20) = <20) + (22) + <31) + (42)
<22) x <21)=<10)+ <21)+ <30)+ <32)+ <41)+<43)
< 22) x <22) = <(0) + (20) + <22) + (40) + <42) + <44)
I
252

THE BUILDING-UP PRINCIPLE 253
c. Simple Isoscalar Factors
With the relevant Kronecker products evaluated, it is now possible to
determine which isoscalar factors are necessarily null and to assign the
value of unity to some others. In the case of the isoscalar factors
< <A 1 ILI>J 1 <10>t II <AIL>J> for Sp(4)::) SO(3), we readily establish the results
shown in Table 19.3, where the null entries are indicated by a dash and the
entries remaining to be determined by an asterisk.
D. Symmetry and Reciprocity
The symmetry and, more particularly, the reciprocity relations of Sec-
tion 19.18 may now be used to establish further isoscalar factors. The
phases may either be chosen arbitrarily (though consistently), or by a
Table 19.3. Simple Isoscalar Factors < <A 1 ILI>J 1 <10>tll<AJL>J> for
Sp(4)::) SO(3)
<AI ILl) <(0) (1O) <II) <2O) <21) <30)
J I ° 3 2 I 3 1  7 3  9
2 2 2 2 2 2 2
<AIL) J
<(0) ° I
<1O) 3 I I . .
2
<II) 2 I . . .
<2O) I I . . . .
3 I . . . . .
<21) 1 I .
2
5 I . .
2
7 I I
2
<22) 2 . . .
4 . .
<31) 1 . . . .
2 . . . . .
3 . . . . .
4 . . . .
5 1 1

254 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
particular convention. In the particular case G::) SO(3) we can use the
reciprocity property
J 2 +M-J,"" 
<J,M,J 2 M 2 IJM)=(-1) V1LJ <J-MJ 2 M 2 IJ 1 -M 1 )
( 19.2(0)
of the coupling coefficients to give
<flaIJlf2a2IfaJ>= (_I)J+J 1 -J 2 +x
[f][J 1 ]
<f*aJf 2 a 'li 211 f 1 *a I J 1 >
[fl][J]
(19.201 )
where x = x (f 1 f 2f). Normally we fix x to preserve the generalized Con-
don-Shortley phase convention. We note that x has not always been
chosen to satisfy this phase convention. Thus J ahn et al. 228,229 chose their
phases to make x = 0, while Racah 9 chose x = J 2 for the group SO(7)
::) SO (3). In the general case such a simple choice is not possible. We also
note that, strictly, Eq. 19.201 as written applies only to the multiplicity-free
case.
Use of the reciprocity relationship and the simple isoscalar factors given
in Table 19.3 immediately establishes the results for < <20>J 1 <10>111<10>1 >
and «21>Jl<10>II<II>2>. The isoscalar factors «20>J2<10>11<30>> are
then determined by their orthogonality with the isoscalar factors
«20>J 1 <10>111<10>1> (cf. Eq. 19.91).
E. 1be Building-Up Principle
To establish further isoscalar factors requires the use of the so-called
building-up principle209-212 whereby we construct the more complex
isoscalar factors from known less complex isoscalar factors.
As usual let f i label representations of 9 and Ai those of the subgroup
:IC. By standard recoupling theory we write
I (f If 2)a 12 f 12 f 3; a 12 , 3 f lXAp'A> =  <f 12f.12f 3f.31a 12, 3 f f.>1 (f2) a 122f.12f3f.3>
£12,£3
=   <f 12f.12f 3f. 3 I a I2, 3 fE ><f If. l f 2f. 2 1a 12 f 12f.12> If 1f. 1 , f 2f. 2 , f 3f.3>
£12'£3 £h£2
( 19.202)
 <f 1 (f 2 f 3)a 23 f 23; ai, 23 f l (f If 2)a 12 f 12 f 3,a I2 , 3 f >
a23, r 23, aI, 23
x If I (f 2 f 3)a 23 f 23; a l ,23 f f.> (19.203)
where we have put f. i = lXjAipiA.;.

THE BUILDING-UP PRINCIPLE 255
The generalized recoupling coefficient appearing in Eq. 19.203 may be
expressed in terms of sums of products of generalized coupling coefficients
by tediously expanding the ket in Eq. 19.203 as for Eq. 19.202 and
equating coefficients of the uncoupled kets:
 <f 12f.12f 2f.2IaI2, 3 f f. ><f 1f. 1 f 2f. 2 1a 12 f 12f.12>
£12
  <fl (f 2 f 3 )a23 f 23; aI, 23 f l (f If 2)a 1 2 f 12 f 3; a 12 , 3 f >
a23,r 23 ,al,23 £23
x <f 2f.2 f 3f. 3 1a 23 f 23f.23><f If. l f 23f.23Ial,23ff.>
(19.204 )
The unitarity properties of the generalized coupling coefficients may then
be exploited to give
<fl (f 2. f 3)a 23 f 23; a l ,23 f l(f l f 2)a 12 f 12 f 3; a I2 ,3 f >
=  <flf.lf 2f.2IaI2fI2f.12><flf.lf 23f.23Ial,23ff.>
X <f 12f.12f 3f. 3 1a 12 3 f f.><f 2f.2 f 3f.31a 23 f 23f.23>
,
( 19.205)
where the summation is over f. 1 , f. 2 , f. 3 , f. 12 , and f. 23 .
If the representations Ai have simply reducible Kronecker products,
then we may use Racah's factorization lemma together with the unitarity
properties of the coupling coefficients to carry out the summations over
the labels PiAi' giving
<f I (r 2 f 3 )a 23 f 23; aI, 23 f l (f If 2)a 12 f 12 f 3; a 12 , 3 f >
=  <flalAlf 2a2A2I1aI2fI2aI2AI2><fI2aI2A3a3A31IaI2,3faA>
X <f lalAlf 23a23A231Ial,23faA><AI (A 2 A 3 )A 23 ; AI (AIA2)AI2A3; A>
( 19.206)
where the summations are over al' a 2 A 2 , a 3 A 3 , a 12 A 12 , a 23 A 23 , and aI2,3A.
The last factor in Eq. 19.206 is the triple recoupling coefficient defined in
Section 19.11.

256 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
If we multiply Eq. 19.206 on both sides by <f'laA;f;3a3A;31
al,23raA> and sum over al,23faA, we obtain the key result
 <f I (f 2 f 3 )a23 f 23; aI, 23 f l (f If 2)a 12 f 12 f 3; a 12 , 3 f >
°1,23
x <f lalAlf 23a23A2311al, 23faA>
=  <f lalAlf 2 a 2 A 211a l2 f 12 a I2 A I2><f 12al2AI2f 3a3A311a12, 3faA>
x <f 2 a 2 A 2 f 3 a 3 A 311 a 23 f 23a23A23><AI (A 2 A 3 )A 23 ; AI (AIA2)AI2A3; A>
( 19.207)
where the summation on the right is over a 2 A 2 , a 3 A 3 , and a 12 A 12 . This
result makes possible the building up of successively more complex isosca-
lar factors from less complex ones. These new isoscalar factors may in turn
be used in Eq. 19.206 to build up tables of the generalized recoupling
coefficients.
Let us continue with the calculation of the isocalar factors for Sp (4)
:J 80(3). To use the building-up principle we start by constructing tables
for f 23 = < 10> and < 11 >, which correspond to the two basic representations
of 8p(4). From these results we can systematically construct all other
isoscalar factors.
The simple isoscalar factors of Table 19.3 are used, together with the
reciprocity relationship of Eq. 19.201, to establish simple isoscalar factors
< <AI ILI>J 1<11>211<AIL>J>, noting that the phases chosen in Table 19.3
dictate the phase choices in these later calculations. Thus we have from Eq.
19.207
a< <11>2<11>211<20>J>
= < <11>2<10>111<10>! >< <10>!<10>! II <20>J >
x «10>!<10>!II<II>2><2( 1t )2;JI(21 )t!;J>
= ( -1)J+1 2V5 ( 
1
2
 )
( 19.208 )
J

THE BUILDING-UP PRINCIPLE 257
where we have put
a = < <11>( <10><10> )<11>; <20>1 «11><10> )<10><10>; <20> >
and used Eq. 19.53 to express the triple recoupling coefficient as a
6j-symbol. The isoscalar factors on the right-hand side may, as a con-
sequence of the choices made in Table 19.3, all be put equal to unity.
Using the tables of 6j-symbols,134 we obtain
0«11>2<11>211<20>3>= - '1
and
0«11>2<11>211<20>1>= '1
Remembering that our phase convention requires «11>2<11>211<20>3>
= + 1, we have a= - V5 /5, from which we conclude that
< <11>2<11>211<20>3> = 1 and «11>2<11>211<20> 1> = -1
The isoscalar factors < <20>J 1 <11>211<20>J> follow directly by considera-
tion of the equation
a< <20>J r 1 <11>211<20>J> = < <20>J 1 <10>1 11<10>1 >< <10>!<10>1 11<20>J>
x «10>1<10>111<11>2> (J 1 ( 1t )2;JI(J 1 1 )tt;J>
with
a = < <20>( <10><10> )<11>; <20>1 «20><10> )<10><10>; <20> >
Again the phase of a is fixed by requiring «20>3<11>211<20>3> be positive,
giving a = - Y30 /10.
The isoscalar factors < <10>1<11>211<21>J> may be determined from Eq.
19.207 by using the known isoscalar factors «11>2<10>!1I<21>J>, and
these in turn may be used to construct the isoscalar factors
«20)J 1 <10>tll<21>J>. The isoscalar factors «20>Jl<10>11<30>J> may
then be deduced by use of the unitary property of isoscalar factors.
Continuing in this manner, it is a comparatively simple task to establish
the results of Tables 19.4 and 19.5.

258
0\11"1 I
8'\ 10 V) IV)
M
M onlN > -
'v/ N
 V)  V)
f'OIlN M
-
,.-....

---
C
V) IN I
n -IN
,.-....

--- I\O
 /' I I
- nlN
, N
 v I I
/'..:
"-'") L>I r;I\O
./"-...
:t -IN I
,<
'-./ I
MIN
./"-...
0 18 
....-4 V)
'-./ - -
- rr')
"-'") N
./"-... /'. I
- 0
:t N
'V
- 18 V)
,< - 0 V)
'-./ - I - -
'-./ >

 /'-.
.....
<:.J -
- N - - - -
 'v/
, /'...
 0 -
- I"IIN - - -
 'v/
<:.J
 /'-.
 8 0 -
'v/
 /'-.
0\ - -  0 f'OIlN N - M
 ::s.. -IN onlN !"-IN
-
 -<
'v/
...t:) /'-. /'-. /'-. /'-. /'-.
 8 0 - 0 -
,-< - - N N
"-/ "-/ "-/ "-/ 'v/ "-/

I I :;Iv -
I I
 I I Iv I
I
!  'I  
I
I   Ir- I  g Ir- Ir- -
:! !r- I I !r- Ir- I
18 I I 
18   _
18  I
I N
o
N  - N M  V)
"'1M VlIM o-IM
/"- /". /".
0 N -
M N M
"-/ 'v' V
259

260 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
Table 19.5. Isoscalar Factors < <AI JLI)J I <II)211<AJL)J)for Sp(4)=> SO(3)
<AI J.l.1) <(X» (10) <II) (20)
J I 0 1 2 1 3
2
<AJ.I.) J
<(X» 0 1
(10) 3 1
2"
<II) 2 1 Y30 v'7O
--
10 10
(20) 1 -1 2V30 YI05
15 15
3 1 Y5 2Y5
-- -
5 5
(21) I -1
2"
5 -1
2"
7 1
2"
(22) 2 1
4 1
(31) 1 YI05 2Y30
15 15
2 V70 v'3O
-- --
10 10
3 2Y5 Y5
-- --
5 5
4 -1
5 1
The isoscalar factors calculated for Sp(4)=> SO (3) may be used in Eq.
19.206 to construct tables of the Sp(4) triple recoupling coefficients. These
in turn can be used in the right-hand side of Eq. 19.207 to calculate
isoscalar factors for SU(4)=> Sp(4). The group chain SU(4)=> Sp(4)=> SO(3)
finds practical application in constructing group symmetrized wave func-
tions for the (!)n jj-coupled nucleon configurations. 230
The results obtained for Sp(4)=> SO(3) may be taken over to the case
SO(5)=> SO (3) if we recall from Section 14.2 that SO(5)Sp(4) with
[/112]</1 + 1 2 / 1 -1 2 ),

ALTERNATIVE CALCULATION OF ISOSCALAR FACTORS 261
EXERCISES
19.15. Racah's calculation 9 of the coefficients of fractional parentage for the jn
atomic shell required the construction of tables of the isoscalar factors
(Ua 1 L 1 (10)FII UaL) for G 2 ::J 80(3). Show that these isoscalar factors may
be readily computed using the building-up principle. (Racah did not use
this method in his calculation and made a phase choice different from the
generalized Condon-Shortley convention we have adopted.)
19.16. Obtain the equations analogous to Eqs. 19.206 and 19.207 for a group
g:J JC 1 X Je1 2 , where JC 1 and X 2 are simply reducible, and investigate the
reciprocity properties of the isoscalar factors. 212
19.22 ALTERNATIVE CALCULATION OF ISOSCALAR FACTORS
In the absence of explicit formulas, the building-up principle gives a
relatively simple and systematic method of calculating isoscalar factors,
especially when the relevant Kronecker products are multiplicity free. This
method has the added merit of not requiring a detailed construction of
explici t bases.
In common with all methods for calculating isoscalar factors, the appli-
cation to non-simply-reducible groups is complicated by the multiplicity
problem. Fortunately, in many cases of practical interest the subgroup U is
simply reducible, Eqs. 19.206 and 19.207 remain valid, and the multiplicity
problem is restricted to that of the group 9 :)3C. Multiplicity problems of
two types arise, those associated with the reduction 9 3C and those
associated with the reduction of the Kronecker products f; X fj of repre-
sentations of g. Both multiplicity problems require the introduction of
some means of distinguishing the repeated representations. Two
approaches are usually considered. Either an arbitrary separation is made
(subject to the usual unitary restrictions), or operators from outside the
group are introduced and used to supply the additional distinguishing
labels. 211,221,231. 232
The building-up principle represents one of several methods for calculat-
ing the isoscalar factors associated with an arbitrary group chain. Here we
only briefly enumerate some of the alternative methods.
Racah's method S ,9 of calculating isoscalar factors for use in his calcula-
tion of coefficients of fractional parentage is essentially a chain calculation
based on the action of infinitesimal operators Ea=Ea(I)+Ea(2) on the
state of a coupled system constructed from systems I and 2. The infinites-
imal operators are chosen from those belonging to 9 but n0 1 , :JC so that

162 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
<f 1€l f 2€2IEalf;€f;€;>
=  <f 1€l f 2€21a 12 f 12€12><a I2 f 12€12IEala2f2€2><a2f2€2If€f;€;>
( 19.209)
== <f 1€IIEa (1) If I€>rlrir 2r2E2E2 + <f 2€21 E a (2) If 2€;>rlrir 2r2EIEi
( 19.210)
where the summation is over a l2 f 12€12 and a' 12 f ' 12€' 12 with €; = a;A;p;A;.
For the sake of simplicity let us assume that3Cis simply reducible, or at
least that the relevant Kronecker products involved in 3C are multiplicity
free. In this case we may make use of Racah's factorization lemma on the
right-hand side of Eq. 19.209, together with the properties of coupling
coefficients, to deduce that
<flwlf 2 W 2; AI21IEallf€f;w;; A2>
=  <flwlf 2w21IaI2fI2wI2><aI2f12W12I1Ealla12fI2w;2>
x <aI2fI2w;2I1f;w;f;w;>
( 19.211 )
where the summation is over a l2 f 12a12 with Wi - aiA i and we have used the
fact that the matrix elements of the infinitesimal operators of 9 are
diagonal in aif;. Upon multiplying both sides by <a" 12 f " 12 W " 1211 f IWl f 2w2>
and summing over WI and W 2 , we find
 <a l2 f 12 W I211 f IWl f 2 W 2><f IWl f 2 W 2; AI211 E a Ilfwf;w;; A12>
"'1,"'2
=  <a l2 f 12WI21IEallaI2fI2W;2><aI2f12W'12I1fw;f;w;> (19.212)
al2
This result can be used to initiate a chain calculation in much the same
manner as in the application of the building-up principle. The method
proceeds by calculating the simplest possible reduced matrix elements and
isoscalar factors, and then using these in Eq. 19.212 to commence a chain
calculation of reduced matrix elements and isoscalar factors of increasing
complexity. A detailed account of the calculation for the d N shell has been
given by Racah, 8 and for the jN shell by Judd. 170
Methods also exist for the calculation of isoscalar factors and coupling
coefficients by the explicit construction of basis states, usually in terms of
polynomials of boson 231 or fermion 233 annihilation and creation operators;

COUPLED TENSOR OPERATORS 163
in that respect they are similar to the spin or technique discussed in
Chapter 17. These methods are usually afflicted by severe combinatorial
problems in all but the simplest cases and, along with most methods,
encounter the usual multiplicity problems.
Elegant methods using the Young and Yamanouchi representations of
the symmetric group28 have been developed by Hassitt 234 and Jahn. 235
The application of these methods to the symmetric group has been dis-
cussed by Hamermesh 204 and is not pursued here.
19.23 COUPLED TENSOR OPERATORS
The isoscalar factors and coupling coefficients may be used to construct
coupled products of tensor operators symmetrized according to the repre-
sentations of a whole chain of nested subgroups.
Using the notation of Section 19.3, we introduce the coupled tensor
operator X(aA) via the relationship
X(aM) = [T(A')U(A")] (aAA)
=  T(A'A') U(A"A")<A'A"laAA>
A'A"
( 19.213 )
It may be readily verified, using Eq. 19.4, that X(aA) is indeed a tensor
opera tor.
The matrix elements of X( aA) may be evaluated in terms of the reduced
matrix elements of T(A') and U(A") by use of the standard recoupling
techniques of Section 19.11. Briefly, we have
<aI2AI2AI2IX( aAA) la2A;2A'12>
=  <a 1 2 A 12 A 121 T( A'A') U( A"A") la2A;2A;2><A'A" laM>
A'A"
=   <aI2AI2AI2IT(A'A')la2A;'2A2>
A'A" (X./2Al'2A12
x < "A" A" I U ( A"A" ) l a' A' '\' >< A'A" l aM >
a 12 12 12 12 12/\ 12
( 19.214 )
where the tensor operators have been separated by the insertion of a
complete set of states la'2A2A2><a;'2A2A21.
The Wigner-Eckart theorem (Eq. 19.30) may now be applied to both
sides, and then both sides multiplied by <AA'I€12AI2AI2> and summed over
A, A' 12' The left-hand side simplifies to the reduced matrix element of

264 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
X(aA), and the right-hand-side sum over the four coupling coefficients is
expressible as a triple recoupling coefficient via Eq. 19.49, to give finally
<€12; al2 A l211 X ( aA) lIa2A2>
=   <a I2 ; a l2 A l2 11 T(A') lIa'2A;'2> <at2A211 U(A") Ila2A7>
a12 a .'2 £Xl2 A l2
x <A' (A" A;2)a'2A2; a 12 A 12 / (A' A" )aAA2; €12AI2>
( 19.215 )
This result is valid independently of whether T(A') and U(A;) act on the
same or different parts of a given system.
In the case of a two-part coupled system where T(A') acts on the first
part and U(A") on the second, we find by a similar recoupling procedure
that
<€12; T A I A 2 ; al2Al211X( aA) liT' A;A;; aI2 A ;2>
=  <€I;TAIIIT(A')IIT"A><€2;T"A21IU(A")IIT'A;>
EIE2'T"
x < (A' A; )€IAI (A" A;)€2 A 2; a l2 A l2 1 (A' A")aA(A;A;)a;2 A ;2; €12 A I2>
( 19.216)
where the recoupling coefficient involves a quadruple coupling of repre-
sentations.
The above results hold for an arbitrary group g, but there is no
difficulty in generalizing the treatment to cover tensor operators
symmetrized along a whole chain of nested groups. For example, for the
group-subgroup combination 9 :)3C we have, as the analog of Eq. 19.213,
X( afaAA) = 1 T(f')U(f") 1 (afaAA)
=  T(f' a' A'A') U(f" a" A"A")<f' a' A'A'f" a" A"A"/afaAA> (19.217)
where the summation is over a' A'A' and a" A"A". Racah's factorization
lemma may be applied to the generalized recoupling coefficient, giving
<f' a' A'A'f" a" A"A"/afaAA> =  <f' a' A'f" a" A"lafaA€><A'A"/€AA>
E
( 19.218 )

COUPLED TENSOR OPERATORS FOR SO(3) 265
EXERCISE
19.17. The group chain SU(4)::J SU(2) X SU(2) is used in the Wigner supermul-
tiplet scheme for nuclei. 212 Discuss the formation of coupled products of
tensor operators symmetrized according to this group chain.
19.24 COUPLED TENSOR OPERATORS FOR SO(3)
The results given above for arbitrary groups undergo considerable
simplification in the case of the simply reducible group 80(3). In terms of
the customary notation t04 for SO(3), we have for Eq. 19.213
X(KQ) = [T(k t )U(k 2 ) 1 (KQ)
=  T(ktql) U(k 2 Q2)<Qlq2IKQ)
qlq2
( 19.219)
and Eq. 19.215 reduces to
<aJIIX(K)lla'J')= ([K])!( _1)J+J'+K  <aJIIT(kt)lla"J")
a"J"
x<a"J"IU(k 2 )lla'J') ( J
k 2
K
J' )
k l
( 19.220)
J"
In the case of T(kt) and U(k 2 ) acting on separate parts of a coupled
system, we have from Eq. 19.216
<ai t i2 J IIX( K) II a'ii;J')
I
=([J,J',K])"2 i2
J
it
.,
it
.,
i2
J'
kt
k 2
K
 <ailll T( k l ) Ila"i; >
a"
x <a"i211 U(k 2 ) IlaJ;
(19.221 )
where we have made use of the 9j-symbol defined by Eq. 19.56.

266 THE WIGNER-ECKART THEOREM AND TENSOR OPERATORS
Setting k 2 =0 in Eq. 19.221 and using Eq. 19.66, we obtain
<a}I} II T( k l ) Ilaj;};.!'>
= (-li,+h+J'+k'8h.i,([J,J,])tf :
t 11
J'
I ) <a}IIIT(kl)!la j ;>
12
11
( 19.222)
while with k l =0 in Eq. 19.221 we have
<a}I}2 J II U(k 2 ) lIaj}2J'>
= (_1)Ji+J2+J+k2. ,,([J,J'])! I J J'
1111 .,.
12 12
; ) < a}211 U( k 2 ) II aj> (19.223)
11
The scalar product of two tensor operators is of considerable practical
importance. Following Racah,168 we define the scalar product by the
equation
T(k). U(k) =  ( -1 )qT(kq) U(k - q)
q
(19.224 )
Putting K=Q=O in Eq. 19.219 and noting Eq. 19.23 we have
{ T ( k ) U ( k )] (00) = [k ] - 1/2 ( - 1 ) k T ( k ). U ( k )
( 19.225)
If we have K=O in Eq. 19.221, we find via Eq. 19.66 that
<a}I}M IT(k). U(k )laj;}Y' M'>
_   ( _ 1 ) Ji + J2 + J I ) I
- UJJ'UMM'
.,
12
12
 )
.,
11
x  <a}111 T( k) Ila"}><a"}211 U( k) !laj;>
( 19.226)
"
a
EXERCISES
19.18. Given that
I 
b
C ) (_I)a+b+c a(a+ 1) -b(b+ I) -c(c+ 1)
- (19.227)
b - 2 [b(b+l)(2b+l)c(c+l)(2c+l)]
c

COUPLED TENSOR OPERATORS FOR 80(3) 267
show that
<s(jmls'lls(jm) = -!-[j(j+ I) -/(/+ I) -s(s+ I)]
19.19. Demonstrate that
<SUMIL z +2S z ISUM) = Mg
where
J(J + I) + S(S+ I) -L(L+ I)
g=l+ 2J(J+I)
the so-called Lande g factor. 39
19.20. Calculations of central two-body interactions in atoms and nuclei usually
involve the calculation of the matrix elements of the Legendre polynomials
Pk(coswij), where in terms of the spherical-harmonic addition theorem 39
4'17'
P k (cos wij) = 2k + I  Yk (OJ, <pj) Y kq (01' <p})
q
= e/). elk)
Calculate the matrix elements of ek). eyk) for the SL terms of the d 2
configura tion.
19.21. Show that
<aJMIT(k). U(k )la'J' M')
J-J"
=/jJJ,/jMM'  \-;,; I) <aJIIT(k)lIa"J"><a"J"IIU(k)lIa'J'),
( 19.228)
a")"

20
Case Study I: The Isotropic Harmonic
Oscillator
20.1 INTRODUcnON
The time has now come to apply much of our previous discussion to
some practical case studies in physics. In the concluding three chapters we
consider the isotropic harmonic oscillator, the hydrogen atom, and many-
particle systems.
The isotropic three-dimensional harmonic oscillator (or for brevity, the
harmonic oscillator) has long been of interest to physicists. This interest
was heightened with the introduction of the nuclear shell model indepen-
dently by Mayer and by Haxel, Jensen, and Suess, who considered the
properties of nucleons moving in an isotropic harmonic-oscillator poten-
tial. Their ambition was to account for the so-called nucleon "magic
numbers." They found that while the degeneracies of the harmonic os-
cillator failed to reproduce the magic numbers for shell closure, a satis-
factory accounting could be made by introducing a spin-orbit-type interac-
tion that partially lifted the degeneracy. Later Elliott 13 ,236,237 was able to
bring about a reconciliation of the shell model and the collective model of
the nucleus by using the properties of the group SU(3), which (as we see
shortly) is the degeneracy group of the isotropic three-dimensional har-
monic oscillator.
268

SECOND QUANTIZATION AND THE HARMONIC OSCILLATOR 269
20.2 SECOND QUANTIZATION AND THE HARMONIC OSCILLATOR
The Hamiltonian H of a normalized isotropic harmonic oscillator (i.e.,
with m=h=w= I) in three dimensions may be written as 238
H = t (p2 +r2)
( 20.1 )
From Heisenberg's quantization postulate, the coordinates qi and momenta
Pi (i = 1,2,3) satisfy the commutation reiations 202
[qi' qj] = [Pi,Pj] = 0, [qi,Pj] = iij (20.2)
It is convenient to introduce the boson annihilation and creation opera-
tors (a and at, respectively)238,239
a=  (r+ip),
at = -L (r- ip)
Vi
( 20.3 )
which, from Eq. 20.2, satisfy the commutation relation
[ a" a:t ] =..
I J l}
(20.4 )
The Hamiltonian can now be written as
H=a t .a+!
(20.5)
Making use of Eq. 20.4 gives
[H,a}] = a},
[H,aj]=-a j
(20.6)
from which we readily deduce that a} creates and a j annihilates a quantum
in the j direction. Using this result, or those of Section 18.5, we find the
energy eigenvalues of H as
En = n + !
( n = 0, 1, 2,. . . )
(20.7)
where the normalized state vectors are
3 a:tllt
In 1 n 2 n 3 ) = II I 1(00)
;=1 
(20.8 )
with
n=n 1 +n 2 +n 3
(20.9)
and 1(00) the vacuum state with 202
ajlOOO) = 0
(20.10)

270 CASE STUDY I: THE ISOTROPIC HARMONIC OSCILLATOR
Noting that at =a*, we have
3 a,n;
(n\n 2 n 3 1 = (0001 II '
i=l 
(20.11 )
with
<0001 a] = 0
(20.12)
20.3 THE GROUPS U(3) AND SU(3)
Let us consider the nine operators
Tij = t { aJ, a j }
( i,j = 1, 2, 3 )
(20.13 )
where {a, b} = ab + ba. Using the basic boson commutation relations of Eq.
20.4, we find that
[ T;j' Trs ] = jr T;s - ;s T 1j
(20.14 )
and hence the T;j are closed under commutation and must describe a Lie
algebra. The operators H; = T;; form a self-commuting set, and since
[ H;, 1}r ] = ( ij - ;r ) 1}r
(20.15)
the roots are all of the type e; - e j . However, the algebra is not semisimple,
since the operator
H=Hl+H2+H3
(20.16)
which corresponds to the Hamiltonian in our case, commutes with all Tij'
making it possible to find eight linearly independent combinations of the
T;j' other than H, which describe the Lie algebra A 2 associated with the
group SU(3).
The three operators
'-1
(20.17)
H;'=H; - 3
taken with the Tij (i=l=j), can be taken as the generators of SU(3), if we
remember that since ;H; =0, the H/ are not linearly independent.
The set of nine operators T;j may be identified with the generators of the
unitary group U(3) rather than GL(3), since we are interested in unitary
transformations that will preserve the orthonormality of our state vectors.

ROTATIONAL SYMMETRY 271
As we saw in Chapter 12, the irreducible representations of U(3) may be
labeled by the maximal weights of the Hi' The transformation properties of
a and at under U(3) follow if we note that
[ Hi' a]] = ija] and [ Hi' a j ] = - ijaj
( 20.18 )
Thus the components of at give rise to the set of weights (100),(010),(001)
of the representation {I oo} of U (3), while those of a give rise to the
weights (-100), (0 - 10), (00 -1) of the representation {OO -I} of U(3).
The representations {AIA2A3} and {- A3 - A 2 - AI} are contragredient to
one another. 84
Under restriction from U(3)SU(3) the representations
{AIA2 A 3} = {AI + aA 2 + aA 3 + a}
(20.19)
become equivalent for a a positive or negative integer. Thus for SU(3) it is
always possible to choose a to give A3 = 0, and hence the representations of
SU(3) may be labeled by just two integers (AJ1.), where following Elliott l3
we put
A-A -A
- I 2
J1. = A 2
(20.20)
Thus in Elliott's notation at transforms as the (10) and a as the (01)
representation of SU(3).
20.4 ROTATIONAL SYMMETRY
The harmonic-oscillator Hamiltonian (Eq. 20.5) commutes with all the
components of the angular momentum operator
L=rxp=iaxa t
(20.21 )
and hence H is rotationally invariant. The components of L form under
commutation the Lie algebra associated with the group SO(3). Noting Eqs.
20.13 and 20.21, we have
LI=-i(T23-T32)' L2=-i(T31-TI3)' L3=-i(TI2-T21)
(20.22 )
We may choose L3 as the generator of the group SO (2), and hence we
have established the group structure
U(3):J SU(3):J SO(3):J SO(2)
for the harmonic oscillator.
(20.23)

272 CASE STUDY I: THE ISOTROPIC HARMONIC OSCILLATOR
It is convenient to exploit the rotational symmetry of the isotropic
oscillator by working in an angular-momentum basis Inlm). It follows from
Section 18.5 that since n =0, 1, ... and n = 2x + I with x = 0, 1,2,..., the
values of I associated with a given value of n are
n odd:
1= 1,3,5, . .. ,n
n even:
1= 0, 2,4, . . .,n
( 20.24 )
and thus for a given n there is a set of (n 2 +3n+2)j2-fold degenerate
states Inlm).
The branching rules for SU(3)SO(3) may be deduced either by use of
the S-function methods summarized in the Appendix, or by a chain
calculation starting from the observation that (00) contains a single S state
and (10) a single P state with (AJL) and (}LA) having the same L-values. The
branching rules for other SU(3) representations are then established by
considering Kronecker products of both the SU(3) and SO(3) representa-
tions. Thus since by Chapter 11 we have
( 10) X ( 10) = (20) + (01 )
and know that under SO(3) the left-hand side contains SPD and on the
right-hand side (01)::) P, we deduce that (20)-:) SD. Continuing in this
manner, Elliott 13 has deduced the general rule that states that for a given
representation (AJL) of SU(3) the values of L associated with the decompo-
sition under SU(3)SO(3) are
L = K, K + 1, K + 2, . . . ,K + max {A, JL } (20.25)
where the integer K is min {A, JL}, min {A, It} - 2,..., 1, or 0, with the
exception that if K = 0,
L = max { A, It }, max {A, It } - 2, . .. , 1, or O.
(20.26)
Noting this result and Eq. 20.24, we see that the states Inlm) of the
harmonic oscillator associated with a given value of n span the representa-
tion (nO) of SU(3).
20.5 SOME SU(3) TENSOR OPERATORS
Having introduced the rotational symmetry of the harmonic-oscillator
states, it is now convenient to abandon the Cartesian annihilation and
creation operators and use the spherical operators
---L ( +' )
a:tl-+ Y2 a 1 _la 2 , a O =a 3

SOME SU(3) TENSOR OPERATORS 273
at = + .-L ( at + ia t )
:!:1 V2 1- 2'
Under commutation,
[ a q , aJ, ] = ( - I) q q _ q'
a t - a t
0- 3
(20.27)
(q, q' = 0, + I)
(20.28 )
The spherical annihilation and creation operators will transform under
SU(3) as the tensor operators
T(IO)I + I=al'
T(OI)1 + l=a:!:I'
T(10)10=a6
T(OI)IO=a o
(20.29)
These basic tensor operators can be used to build up other SU(3) tensor
operators using the methods of the previous chapter-in particular, the
results of Section 19.23. The tensor operators X(ll) are of particular
importance in what follows, and may be built up by use of Eq. 19.217 to
give
X(II)KQ= - V2  T(IO)lqT(OI)lq'«IO)I(OI)lll(ll)K><qq'IKQ>
qq'
.t (20.30)
where a factor of - V2 has been inserted for later convenience. Choosing
the SU(3) isoscalar factor to be + I, we have
X(II)KQ= (-I)Q+I V2 (2K+ I)  T(IO)lqT(OI)lQ-q
q
X ( q l I K ) (20.31 )
Q-q -Q
from which we readily deduce the symmetrized operators of Table 20.1.
The set of eight components of the SU(3) tensor X(II) are closed under
commutation, as may be readily seen by evaluating the commutator
[X(II)K 1 Ql,X(II)K 2 Q2]' The evaluation is made by first expanding the
components of X(II) in terms of those of T(IO) and T(OI), using Eq. 20.30,
and then using Eq. 20.29 followed by Eq. 20.28 to give
[X( II )K 1 Ql'X( II )K 2 Q2]
 [ ( _I ) q2 at a _ ( _I ) qI at a ]
 q2-q3 qI q4 ql-q4 q3 q2
qlq2q2q3q4
=2
X < (10) 1(01) III (II )K 1 >< (10) I (01) III (II )K 2 ><qlq2I K l Ql><q3q4IK 2 Q2>
(20.32)

274 CASE STUDY I: THE ISOTROPIC HARMONIC OSCILLATOR
Table 20.1. The SU (3)-Symmetrized Operators X(II)
Lo=X(II)10=alal-ara_1
L:t I = X ( 11) 1  1 = =+= (a 1: lao - a6a:t I)
Q 0 = X ( 11 ) 20 c::: -  ( 2a 6ao + a!. I a I + a r a - I )
v'3
Q:t I =X(II)2  1 = - (a1: lao+aba:t I)
Q:t2=X(II)22== -Y2 a1:la:t1
The summation over q 2 in the first term and qll
performed using the identi ty l69
L ( - 1 /. +/2 + /1. + /12 ( J I J 2 J 3 ) ( II
m) m l m 2 m 3 ILl
in the second can be
1 2
J 3 )
m 3
- IL2
= L ( -1)/ 3 +/13(2/ 3 + 1) ( j] J2
II 1 2
1),J.L)
X ( I] J2 13 ) ( j] 1 2
- ILl m 2 IL3 m l IL2
J 3 )
13
13 )
- IL3
(20.33 )
After collecting terms, we find that
...  "" Q ( ) 1/2
[X(II)K 1 Q.,X(II)K 2 Q2] = v2 £.J (-I) [KI,K2,K]
KQ
x ( ]
K 2
1
K ) ( KI
1 QI
K 2
Q2
K ) {(_I)KI-K2+K_1}X(II)KQ
-Q
( 20.34 )
which establishes that the eight components of X(II) are closed under
commutation and indeed form the generators of the group SU(3). Using
Eq. 20.34, we readily obtain the commutation relations
[ Lq, Lq' ] = ( - 1 ) q + q v'6 ( 
[ Lq, Qq' ] = ( - I) q + q' Y30 ( 2
q'
1
1 I ) L q + q ,
-q-q
2 I q l ) Qq + q'
-q-q
(20.35)
q'
(20.36)

REDUCED MATRIX ELEMENTS 275
[ Qq, Qq' ] = ( - 1 ) q + q' v'3O ( 
2
1 I ) L q + q , (20.37)
-q-q
q'
The generators of SU(3) can be divided into the three components of the
angular-momentum vector L, which fOlm the generators of the SO(3)
subgroup of SU(3), and the five components of a quadrupole tensor Q. In
addition to these eight operators, we can construct a scalar-tensor operator
T{OO)OO=  (a6ao-ara-l-alal)
(20.38 )
that commutes with all the components of Land Q. Taken with Land Q,
the operator T(OO)OO enlarges SU(3) to U(3). Since
H="'"' ( -l ) qata +1
 q -q 2
q
(20.39)
we may write
T{OO)OO= (B-1)
( 20.40 )
20.6 REDUCED MATRIX ELEMENTS
The matrix elements of the SU(3) generators for harmonic-oscillator
states may be found from those of at and a. The reduced matrix elements
of at are determined by noting from Eqs. 20.7 and 20.40 that
<nlmIT{OO)OOlnlm) = VI n
(20.41 )
Use of Eq. 20.38 then gives
<nlml T{OO)OOlnlm) = V t  )nlmlaJ ( - ]) q a _qlnlm)
q
= VI  <nlmlaJln -11'm,)2
[',m',q
= VI <nlla t lln-l)2
(20.42 )
where aJ and ( - 1 )qa _ q have been separated by the insertion of a complete
set of states and the Wigner- Eckart theorem has been applied. The

276 CASE STUDY I: THE ISOTROPIC HARMONIC OSCILLATOR
summation over I',m',q is made via the orthogonality property of the
coupling coefficients and isoscalar factors. Comparison of Eq. 20.41 with
20.42 gives the. doubly reduced matrix elements of at and a as
<nlla t lln-l)=V1l =<n-1Ilalln)
(20.43 )
The m-dependence of the matrix elements of an arbitrary SU(3) tensor
operator T(Ap.)kq is given by
I-m ( I k
(nlmIT(AJL)kqln'l'm')= (-1) -m q
where
I' ) <n/ll T(Ap. )kll n' I')
m'
( 20.44 )
< n/ll T (AIL) k II n' I') = ( - 1) k - J - I' ( [ I ] ) 1/2 < (Ap. ) k ( n' 0) I'll ( nO) I)
x < ( nO) II T (AIL) II ( n'O ) \
(20.45 )
Thus if the doubly reduced matrix element is known, the singly reduced
matrix element may be found using the SU(3) isoscalar factors. Following
the methods of Chapter 19, we readily find that
1/2
< n + II + 111 a t II nl) = - [ ( I + 1 ) ( n + I + 3) ]
1/2
< n + 1/- 111 at II nl) = + [ I ( n - I + 2 ) ]
(20.46)
The corresponding reduced matrix elements for a follow by noting that
<n/IIX (AIL) Iln' I') = ( - 1 )/-1' <n' I'IIX (Ap.) t l1nl )
(20.47)
and hence
1/2
< n - 1/- 111 a II nl) = [I ( n + I + 1 ) ]
1/2
< n - II + 111 a II nl) = - [ ( I + 1 ) ( n - I ) ]
( 20.48 )
Thus at and a act as raising and lowering operators for n and I.
The reduced matrix elements of the SU(3) generators X(II) may be
calculated by noting that
(nlllX (11 )Klln'l') = ( - 1) K [2(2K + 1) f/2 L (nliia t II n"l")(n'l'lIa t IIn"l")
n"l"
x ( 
1
:J
(20.49 )
I'

THE QUADRATIC CASIMIR OPERATOR 277
Table 20.2. Reduced Matrix Elements for Some SU(3)-Symmetrized Tensor
Operators with Harmonic-Oscillator States
<nlll T(OO)Ollnl)
+ Vj (21 + 1 )1/2 n
<n + 11 + 111 T(10)lllnl)
<n + 11-111 T(10)lllnl)
<nlIIX(11)lll nl )
- [(I + 1)(n + 1+3)]1/2
+ [l(n -I + 2)]1/2
+[1(1+ 1)(21+ 1)]1/2
1/2
[ l( I + 1) (21 + 1) 1
+ (2n + 3 ) 3 (2/- I )( 2/ + 3) J
1/2
[ 2(1+ 1)(1+2)(n-l)(n+l+3) ]
(21+3)
<nlIIX(II)21Inl)
<n 1+ 2I1X(II)2I1nl)
This result is obtained by first using Eq. 20.30 to expand the X(II)KQ in
terms of matrix elements of at and a, and then applying the Wigner-Eckart
theorem. The resultant summations are then performed, first using the
identity of Eq. 20.33 and the orthogonality property of 3j-symbols, then
applying Eq. 20.47. Thus the reduced matrix elements of the X(ll) may be
calculated by a building-up principle starting with the known reduced
matrix elements of at and a. In this way we readily establish the results of
Table 20.2.
EXERCISE
20.1. Give an alternative derivation of Eq. 20.49 making use of Eq. 19.220.
20.7 THE QUADRATIC CASIMIR OPERATOR
The operators Q2 and L 2 are both scalars under SU(3); however, Q2
does not commute with all the generators of SU(3). A quadratic Casimir
operator C that commutes with all the generators of SU(3) may be formed
by writing
C=aQ2+L 2
where a is a constant, which is fixed by demanding that
[C,X(ll)KQ]=O
(20.50)

278 CASE STUDY I: THE ISOTROPIC HARMONIC OSCILLATOR
Detailed calculation using Eq. 20.34 gives a = 1, and hence
C=Q2+L 2
(20.51)
The eigenvalues of L 2 acting on a harmonic-oscillator state Inlm> are
simply
L 2 lnlm> = l( I + I) Inlm>
(20.52)
The eigenvalues of Q2 are determined by noting from Eq. 19.220 (with
K=O) that
1
<nlmIQ-Qlnlm)= 21+ 1 :t <n I IlQllnl')2
Use of the results of Table 20.2 gives
(20.53 )
Q2lnlm) = [t(2n + 3)2 - (12 + 1+ 3) ] Inlm)
= (jH 2 - L 2 -3)lnlm>
(20.54)
Combination of Eqs. 20.52 and 20.54 yields the desired result,
Clnlm) = [t(2n + 3)2 - 3 ]Inlm)
(20.55)
which can also be derived using the methods of Chapter 15.
EXERCISE
20.2. Demonstrate by explicit construction that
Q2=H2_L2_3
and hence that the eigenvalues of H are just n + .
20.8 LADDER OPERATORS IN SU(3)
The operators Land Q supply a complete set of ladder operators for
moving throughout the -weight space of representations of SU(3). The
operators Lz} raise or lower m in unit steps, permitting us to move
throughout the weight space of representations of the SO(3) subgroup, but
not between different representations of SO(3). The operators Q can raise
or lower I in steps of two units, and thus can couple the different SO(3)
representations covered by a particular SU(3) representation. However, the

COMMUTATION RELATIONS 279
operators Q cannot permit us to move between different representations of
SU(3). To move between different SU(3) representations requires the
introduction of operators that lie outside of the SU(3) grouo generators.
20.9 SOME FURTHER SU(3) TENSOR OPERATORS
Before attempting to construct ladder operators to permit us to move
between SU(3) representations, we shall construct some additional SU(3)
tensor operators. The operators X(20) and X(02) are of particular impor-
tance, and may be built up by the use of Eq. 19.217 to give
X(20)KQ= -Vi  T(10)lqT(10)lq'«10)1(10)111(20)K><qq'IKQ>
qq'
(20.56)
and
X(02)KQ = - Vi  T(Ol) 1 qT(Ol) 1 q' < (01) 1 (01) 111 (02) K><qq'IKQ>
qq'
(20.57)
where K is limited to the values 0 and 2. Following the same procedure
used to construct X(II), we obtain the results given in Table 20.3.
The reduced matrix elements of X(20) may be readily calculated using
Eqs. 19.220 and 20.46 to give the results shown in Table 20.4. The
corresponding reduced matrix elements of X(02) follow by application of
Eq. 20.47. We note that the components of X(20) and X(02) raise or lower
n and I by 0 or + 2 units, and thus can couple different SU(3) and SO(3),
but cannot link harmonic-oscillator states of different parity.
20.10 COMMUTATION RELATIONS
The components of X(20) and X(02) separately commute among them-
selves. The commutation relations for the components of X(20) with those
of X(02) follow in the same manner as for Eq. 20.34, and in particular
[X(20)K 1 QI,X(02)K 2 Q2 ]
=4L(-1)K+Q(2[K 1 K 2 K])1/2 ( K 1 K2 K 1 )
 1 1
X ( KI K2 K ) X(11)KQ+IKK Q _ Q 4X(OO)00 (20.58)
QI Q2 - Q I 2 I 2

Table 20.3. The SU(3)-Symmetrized Operators X(20) and X(02)
x (20)00 = -vi (abab - 2a ta  1 )
X(20)20= -2 (abab +ata l)
V3
X(20)2:t 1= - 2abat: 1
X(20)2:t2= - V2 a lat: 1
X(02)00= Vi (ao'lo-2a l a- l )
-2
X(02)20= -(ao'lO+ala-l)
V3
X(02)2:t 1 = - 2ao'l  1
x (02) 2 :t 2 = - V2 a  1 a  1
Table 20.4. Reduced Matrix Elements ofX(20)
[ 2(21+ 1)(n-l+2)(n+l+3) ] 1/2
<n + 2/IIX(20)0Ilnl)= 3
1/2
[ 1(1+ 1)(21+ 1)(n-l+2)(n+l+3) ]
(n+211IX(20)21I nl )=2 3(21-1)(21+3)
1/2
[ 2(1+1)(1+2)(n+l+3)(n+l+5) ]
(n + 21 + 21I X (20)2I1 nl ) = - (21 + 3)
1/2
[ 21(1-1)(n-l+2)(n-l+4) ]
(n+21-21I X (20)21I nl )= (21-1)
280

COMMUTATION RELATIONS 281
where K is limited to I and 2, and
X(OO)OO= VI T(OO)OO+ 1= iH
(20.59 )
Note that H commutes with all the components of X(II) and in the
same manner as T(OO)OO, and hence may be taken as a generator of U(3).
However, H does not commute with all the components of X(20) and
X(02), since
[X(20)KQ,H] = 1X(20)KQ
[X(02)KQ,H] = jX(02)KQ
(20.60)
Neither do the components of X(II) commute with all those of X(20) or
X(02); indeed,
[X(20)K 1 Ql'X( II )K 2 Q2] =  a(KQ)X(20)KQ
KQ
(20.61 )
and
[X(02)K 1 Ql,X(II)K 2 Q2] = - (-I)K2a(KQ)X(02)KQ (20.62)
KQ
where
a(KQ) = [I + (_l)IK2] (-I)K+Q(2[K)K 2 K]) 1/2 ( I
K 2
I
)
x ( Kl
Ql
K 2
Q2
- )
(20.63 )
Inspection of Eqs. 20.58 to 20.62 shows that the twenty components of
the tensor operators X(II), X(20)), X(02), together with H, form a closed
set under commutation and thus must define a Lie algebra that includes
su(3) as a subalgebra.
EXERCISES
20.3. Show that the 21 operators
Tij = t { at, aj } , P ij = t { at, a] } , Qij = ! { ai' a j }
( i,j = 1, 2, 3 )
(20.64)

282 CASE STUDY I: THE ISOTROPIC HARMONIC OSCILLATOR
close under commutation:
[Tap, T y ( J = Ta(6 py - Typ 6 a(
[TaP'Py(] = Pa(6py + Pay6p(
[ TaP' Qy(] = - Q(p 6 ay - Qyp 6 a(
[Pap, Qy(] = - TfJy 6 a( - Tay8fk - T fk 6 ay - T a (6 py
(20.65)
20.4. Show that the antisymmetric bilinear form 24o
.hj = X;Pj - XjPi (i =/= j)
(20.66)
commutes with all the operators in Eq. 20.64 and thus constitutes a group
invarian t.
20.5. Use the results of the preceding two exercises to show that the 21 operators
in Eq. 20.64 [or equivalently the 21 components of X(II), X(20), X(02), and
X(OO)] generate the Lie algebra associated with the real noncompact group
Sp(6,R).
20.6. Show that the operator
a> =  X(02)K. X(20)K
K
(20.67)
is a scalar under SU(3) and that its matrix elements for harmonic-oscillator
states are
<nlmla>ln ' I' m/) = 8nn,6/1,8mm,(2n2 + 14n + 24)
(20.68)
20.7. Use the above result to construct an operator that is at most quadratic in the
group generators of Sp(6,R) and determine its eigenvalues when acting on
an arbitrary harmonic-oscillator state Inlm).
20.11 A LARGER GROUP FOR THE OSCILLATOR
The group SU(3) is the minimal group that has representations that
completely span the states of individual degellerate levels of the harmonic
oscillator. The operators X(ll) permit a laddering among the states
associated with a given degenerate level, and for this reason SU(3) may be
said to form the degeneracy group of the harmonic oscillator. However, the
operators X(ll) cannot couple different SU(3) representations and hence
different degenerate levels of the oscillator. To ladder between states of
different degenerate levels requires the introduction of operators that lie
outside the degeneracy group.

SUBGROUPS OF Sp(6,R) 283
I I I I I I I I I I
I I I I I I I I I I
10 . . . . .
I I I I I I I I I
I I I I I I I I I
0 I 0 I 0 I 0 I 0
I I I I
I I I I I I I I
8 . I . I . I .
I I I I
I I I I I I I
0 I 0 I 0 I 0
I I I
I I I I I I
6 . I . I .
1 I I I
I I I I I
0 0 0
I I I I
n I I I I
4 . .
I I I
I I I
0 I 0
I
I I
2 I .
I
0
0 2 4 6 8 10
I >
Fig. 20.1. The states with n odd are indicated by diamonds  and those with n even by dots
e. These two sets of states belong to two distinct infinite dimensional Sp( 6,R) representa-
tions. The dashed lines link states belonging to a common SU(3) multiplet.
The preceding exercises show that the 21 components of the tensor
operators X(II), X(20), X(02), and X(OO) generate the Lie algebra
associated with the real noncompact group Sp(6,R) which contains the
degeneracy group SU(3) as a subgroup. The operators X(20) and X(02)
have the property of laddering n by 0 and + 2 units. The eigenvalues
associated with these operators are bounded from below but not from
above. The states of the harmonic oscillator span two infinite-dimensional
unitary representations of Sp(6, R), one containing all the states associated
with n even and the other with n odd.241-243 These two representations are
illustrated diagrammatically in Fig. 20.1.
EXERCISE
20.8. Show that for n-dimensional isotropic harmonic oscillator the degeneracy
group is SU(n), and that this group may be embedded in the real noncom-
pact group Sp(2n,R).

284 CASE STUDY I: THE ISOTROPIC HARMONIC OSCILLATOR
20.11 SUBGROUPS OF Sp(6,R)
The group Sp(6,R) possesses a rich and useful subgroup structure. The
operators X(II) generate the degeneracy group SU(3), while the three
components of X(II)I generate the SO(3) subgroup of SU(3), with
X(II)IO generating SO(2), so that
Sp(6,R):J 8U(3):J 80(3)  80(2) (20.69)
Other nontrivial subgroup structures exist.
If we put
K+ 1 = - X(20)OO, K_ 1 = 1 X(02)OO, Ko=tX(OO)OO= 
(20.70)
we find from Eq. 20.58 that
[K+I,K_I]=+Ko and '[Ko,KI]= + K1
(20.71 )
which corresponds to the Lie algebra associated with the noncompact
group 80(2, I). Furthermore, the generators of 80(3) commute with those
of 80(2, 1), and hence we identify the subgroup structure
Sp(6,R):J [80(2, I):J 80(2)] X [80(3):J 80(2)]
(20.72)
where the first 80(2) subgroup has Ko as its generator and the second Lo.
The Casimir operator for the 80(2, 1) group is readily found as
C=Ko(Ko-l) +2KIK_I (20.73)
Applying it to a harmonic-oscillator state and using the results of Table
20.4, we have
4Ctnlm> = [/(1 + 1) - i ] Inlm> (20.74)
The operators K  1 raise or lower n by 2 units. It follows from Table 20.4
that K _ 1 must yield a lower bound for n = I, while there is no upper bound
for K + I' and hence the 80(2, 1) representations must all be of the type
D + ( - I) introduced in Section 17.8-all the even values of 1 being assigned
to one representation of 8p(6,R), and all the odd values of I being assigned
to the other. Thus restriction of the group 8p(6, R) to its subgroup
80(2, I) X 80(3) resolves the harmonic-oscillator states into two series of
infinite 80(2, I) towers, each characterized by a lowest state with n = I, as
shown in Fig. 20.2.
The introduction of the direct-product group 80(2, 1) X 80(3) into the
harm9nic-oscillator problem is directly associated with the factorization of
the harmonic-oscillator wave functions into a product of radial functions

SUBGROUPS OF Sp(6,R) 285
10 -- --.-- --.-- --.-- --.-- --.
0-- - -0-- - -0-- --0-- --0
8 ----.- - --e-- - -.----.
0----0----0----0
6 -- --.- - - -e - -- -.-
t
n
0--- -0-- --0
4 ----.----.
0-- --0
2 - - - -.
o
o
2
4
6
8
10
I ::-
Fig. 20.2. Infinite 80(2,1) towers of harmonic-oscillator states. Each dot. or diamond <>
represents 21 + 1 degenerate states.
Rnl(r) and angular functions Y1m«(},cf». The radial functions form a basis
for the SO(2, 1) representations, while the spherical harmonics form a basis
for the SO(3) representations. We later exploit this factorization property
to calculate the matrix elements of r k for harmonic-oscillator states, using
the Wigner-Eckart theorem for SO(2, 1) X SO(3).
EXERCISES
20.9. Noting Eq. 20.70 and Table 20.3, show that
K+ = 1(a t 'a t ),
K_ = 1(a.a),
H
Ko=-
2
(20.75)
with
[K+,K_] = -2Ko and [Ko,K:t] = + K:t
(20.76)
20.10. Use Eq. 20.3 to obtain new generators for 80(2, I) such that
K =-lr.r
+ 2'
K =l p ' p
- 2 ,
- I
Ko= - (r'p+p'r)
4
(20.77)

286 CASE STUDY I: THE ISOTROPIC HARMONIC OSCILLATOR
20.13 A FURTHER GROUP FOR THE OSCILLATOR
The group Sp(6,R) covers the states of the harmonic oscillator in two
representations, one covering the states with n even and the other the states
with n odd. The failure of Sp(6,R) to cover the states in a single representa-
tion is directly related to the absence in its Lie algebra of any operators
that ladder n by + 1 units. However, it is evident from Eqs. 20.46 and
20.48 that the operators at and a have the desired property of laddering n
by + 1.
The set of six operators at, a together with the identity operator E form a
solvable Lie algebra whose associated group is commonly referred to as the
quantum-mechanical 18 or Heisenberg groupI9,244,245 and will be desig-
nated as N(3).
A larger group, known as the oscillator group245 and designated here as
Os (3), may be formed by adding to the generators of N(3) the Ham-
iltonian H or simply the number operator N = at · a. This group may be
written as a semidirect product
Os(3) =N(3) H
The group Os(l) has been designated H4 by Miller. 246 ,247
In view of the fundamental importance of the Heisenberg commutation
relations, it is not surprising that the representation theory of these groups
has been the subject of much work} 6, 18, 19,244-249 The unitary representa-
tions, apart from the trivial one, are all infinite dimensional. We do not
pursue them in detail here.
It is evident from the form of Eqs. 20.46 and 20.48 that the operators of
N(3) allow one to move from any state of the harmonic oscillator to any
other state by a series of applications of a and at, and to ladder between
states of different parity, as shown in Fig. 20.3.
20.14 A DYNAMICAL GROUP FOR THE OSCILLATOR
The Heisenberg group N(3) does not by itself provide an adequate
description of the degeneracies of the harmonic-oscillator states, as it
includes neither the degeneracy group SU(3) nor the rotational invariance
group SO(3) as subgroups. Ideally we seek a group that can yield the
energy spectrum and the degeneracies of the levels, and that contains a set
of operators that determine the transition probabilities between states. This
latter property requires that we consider noninvariance groups whose
generators do not all commute with the Hamiltonian of the physical
system. The construction of a group having the foregoing properties would

A DYNAMICAL GROUP FOR THE OSCILLATOR 287
/
/
/
/
10
/
/
/
/
.
/
/
0/
/
/
/
.
/
/
0/
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/
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0/
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/
0/
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/
2 /
/
/
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/
/
0/
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/
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/
/
0/
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0/
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/
4 /
/
/
/
/
/.
/
0/
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/
/
.
/
/
0/
/
/
6 /
/
/
/
./
/
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0/
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8 /
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0/
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0/
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0/
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0/
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0/
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/
I
n
o
2
4
6
8
10
I ::>
Fig. 20.3. The operators of N(3) permit a laddering between states of odd and even n.
permit a complete description of the dynamical properties of the physical
system; it is referred to as the dynamical group of the system. I5 ,250-260
To satisfy the requirements of forming the dynamical group G D of the
harmonic oscillator, we must find a group that includes among its sub-
groups the degeneracy group SU(3), the rotational invariance group SO(3)
of the Hamiltonian, the energy-spectrum-generating group SO(2,1) dis-
'cussed in Section 18.5, and the transition group, which must involve at and
a.
Clearly neither N(3) nor Sp(6,R) separately satisfies the requirements of
a dynamical group for the oscillator. If we consider the generators of
Sp(6,R) defined by Eq. 20.64 under commutation with those of N(3), we
find that
[Tap,a y ] = -apay
[Tap,a] = aaPy
[Qap,a y ] =0
[ Qap, a] = aapy + apay
[Pap,a y ] = -alpy -aZay
[Pap,a] =0
(20.78)

288 CASE STUDY I: THE ISOTROPIC HARMONIC OSCILLATOR
Thus the 21 generators of Sp(6,R), taken with the 7 generators of N 3 , form
the semidirect-product group N (3)  Sp (6, R) and hence
G D =N(3) Sp(6,R)
(20.79)
forms an acceptable dynamical group for the harmonic oscillator.261-263
We note that N(3) Sp(6,R) is not the only possible dynamical group
for the harmonic oscillator, though it is undoubtedly the most convenient.
An alternative dynamical group241,257 can be formed by adding to the
eight generators Tij of SU(3) the seven operators
Too = -H+c
To; = g(H)aJ
T;o = j( H )a;
(20.80)
where
H + c = - j( H )jt ( H) + -!
H +c= - g(H)gt(H) --!
and
rrt - - T
.1 0; - ;0
to give the 15-parameter noncompact group SU(3, 1).
Both dynamical groups have a set of operators that permit a laddering
through all the states of the harmonic oscillator. Thus in principle we can
generate all the oscillator states fronl the ground state 10(0) by a syste-
matic laddering. The allowed oscillator states up to n = 5 are displayed in
Fig. 20.4.
The dynamical group N(3)l!JSp(6,R) contains a rich and useful sub-
group structure. There is the whole series of subgroups N(3) l!JA, where A
is any proper subgroup of Sp(6,R), as well as the groups A by themselves.
In addition the oscillator group Os(3) occurs as a subgroup of N(3)
l<g) Sp(6, R).
20.15' GROUP CONTRACflONS AND THE DYNAMICAL GROUP
inonii and Wigner 26 4--266 have shown that under certain conditions it is
possible to obtain from a given Lie group another nonisomorphic Lie
group by a process of group contraction via a limit operation. The concept
of group contraction has been elaborated upon by Saletan. 267

GROUP CONTRACTIONS AND THE DYNAMICAL GROUP 289
.£
6 I'. 
, '. /"
1 '. ///'
\ '. /'
5 ',.."...,...,...........................,.... \.'....""....".,.  .... :"': , , , . . , . , , , , . . , . , , . , ,
\ ., '/' r \.
\., t /', '.
\ . / I \
4 . , . . . . . . . . . . . . . . . . . . . . . . . , . . . , . . . . , . .. ......,.. -\. . . . . X . ,  ' , . , . r ' , . , , , , ," , , , , , , , , . , , .
, \ ./ I '. I ·
I. . ,
I' ,..f" 1 \1. 1 I
3 .................................:. .;.(. .. \ . .. ..;.. . .. ::"'\ "'!, .......,:......... .. .. .
)./ I. I I I. /
./' 1 '. I ,/ I '. I
2 ................... ..\) :. .. .. . : .. .! . . ..   ."{ .. .. 3..1.. .. .. Y.. .. .. , .. .. .. .. . .
/'\ J '. 1 / I I '. / :, /
/' I I \ . / I I I 'I " /
......... .. -=::..... ..' '1" /..."'.. ,_.. ,/.,.,.\ ,j" ,/,... ," /," ,...",.....,.,.....
/' /' I' I 1 I 1 ' II / ''.f
 '1/ I 11/ I
y \, / ! ,,/ I
2
3
N
4
5
Fig. 20.4. The allowed states of the harmonic oscillator up to N = 5.
The dynamical group N(3)@Sp(6,R) contains the group N(3)@SU(3)
as a subgroup. This group is generated by the fifteen components of the set
of tensor operators X(II),X(OI),X(10) together with the identity operator
E. These are just the operators required to extend SU(3) to SU(4), if we
append an operator that commutes with all the components of X(II).
The generators of SU(4), in Cartesian coordinates, satisfy the commuta-
tion relations
[ Tap, Ty£ ] = Ta£Py - Typa£
(20.81 )
where a,{3,y,E= 1,2,3,4. The operators TaP with a,{3= 1,2,3 generate the
group SU(3). The operator T 44 commutes with all the generators of SU(3),
and Ta4 and T 4p transform as (10) and (01) under SU(3).
If we introduce the set of contracted operators
Pa = limETa4
E:O
( where Pa = aJ)
Qp = 100 ET 4P
E:O
(where Qp = a p )
E = 100 E 2 T 44
E:O
(20.82)

290 CASE STUDY I: THE ISOTROPIC HARMONIC OSCILLATOR
we find that these operators taken with the operators Tap, close on the
group N(3);SU(3) and hence we may regard the group N(3) SU(3) as a
contracted version of the group SU(4).
Further study shows that the dynamical group N(3)\Sp(6,R) is a
contracted version of the group SO(8).
EXERCISES
20.11. If the commutation relations of SO(3) are written as [X I 'X 2 ] = X 3 etc.,
show 266 by use of the substitutions Y 3 = X 3 , Y I = !Xl' Y 2 =!X 2 that the
Euclidean group of the plane E 2 is a contracted version of SO(3).
20.12. ShOW l8 that the Euclidean group E3 is a contracted version of SO(4).
20.16 THE N-DIMENSIONAL ISOTROPIC HARMONIC OSCILLATOR
The preceding results for the three-dimensional isotropic harmonic is-
cillator may be readily generalized to the N-dimensional isotropic oscilla-
tor. The dynamical group becomes N(N)f!i!JSp(2N,R), which is a con-
tracted form of SO(2N + 2). The subgroup schemes of principal interest
are
N(N) {Sp(2N,R):) SO(2, 1) X [SO(N):) SO(3):) SO(2)]} (20.83)
and
N(N)  [Sp(2N,R):) SU(N):) SO(N):) SO(3):) SO(2)] (20.84)
We note for N >3 the appearance of SO(N) as a subgroup.
20.17 TENSOR OPERATORS FOR THE SO(2, 1)XSO(3) SUBGROUP
As noted earlier, the occurrence of the group SO(2, 1) X SO(3) as a
subgroup of the harmonic-oscillator dynamical group is intimately con-
nected with the separability of harmonic-oscillator wave functions into the
products of functions involving the radial and angular variables separately.
In calculating the matrix elements of operators acting on harmonic-
oscillator states, it is desirable to exploit the separability of the radial and
angular variables by using tensor operators symmetrized with respect to
SO(2, 1) X SO(3).
At this point we note that the harmonic-oscillator states Inlm) may be
equivalently labeled under SO(2, 1) X SO(3) by the correspondence
Inlm) = I( Tt)lm)
(20.85)

TENSOR OPERATORS FOR THE SO(2, I)X SO(3) SUBGROUP 291
where
T = ! ( I - -! ) and t = ! ( n + 1 )
(20.86)
with T(T+ 1) the eigenvalue of K 2 , and t that of Ko, with respect to the
state Inlm>.
Let us consider an irreducible tensor operator T(A) transforming as the
finite nonunitary representation (A) of SO(2,1) with components T(M)
(A= -A, -A+ 1,...,A). If K and Ko are generators of SO(2, 1), then for
T(A) to be a tensor operator we must have (cf. Eq. 19.14)
[Ko, T(AA)] =AT(AA)
[K, T(AA)] = + [(A + A)(A + A+ 1) ]1/2 T (AA + 1) (20.87)
-a result that may be proved by induction 140 by showing that the coupled
tensor operator
T(AA) =  <AIA2IAA>T(AIAI)T(A2A2)
AIA2
also satisfies Eq. 20.87.
The second-quanization operators aJ and a q (q = 0, + 1) for fixed q form
the projections + t and -! of a rank-! tensor under SO(2, 1), since 243
[Ko,aJ] = !aJ
[Ko,a q ] = - !a q
[ K +, aJ] = 0
[K+,a q ] =aJ
[K_,a q ] =0
{ K at ] =-a
-, q q
(20.88 )
Starting with these basic nonunitary tensor operators, it is possible to
build up more complex SO(2, 1) nonunitary tensor operators from coupled
products of a and at, using the same coupling coefficients as used for the
corresponding unitary representations of SO(3) in Section 17.14.
It is convenient to consIder double tensor operators T(AA; kq), which
form a basis for the (A) X (k) representations of SO(2, 1) X SO(3). The
reduced matrix elements of such a tensor operator may be found by
applying the Wigner-Eckart theorem to the groups SO(2,1) and SO(3)
separately. Thus for harmonic-oscillator states we have
<nlml T(M; kq) In' I'm'> = < (Tt)lml T(AA; kq) I (T't')/'m'>
= <T't' MI Tt><I'm' kql/m><TIII T(A; k) II T'I'> (20.89)

292 CASE STUDY I: THE ISOTROPIC HARMONIC OSCILLATOR
where the first coupling coefficient is appropriate to SO(2, 1) and is of the
type discussed in Section 17.15, while the second coupling coefficient is a
standard SO(3) Clebsch-Gordan coefficient.
20.18 MATRIX ELEMENTS OF MUL TIPOLE OPERATORS
As a practical example of exploiting the group SO(2, 1) X SO(3) to
simplify calculations, let us consider the calculation of the matrix elements
of the multipole operators 243
Ykq(r) = rky kq ( fJ,cp) (20.90)
for harmonic-oscillator states.
Commencing with the operators aJ, we can produce a double tensor
T(k/2;k) of rank k/2 in SO(2,I) and k in SO(3) whose maximum
component in SO(2, 1) is
T(   ;kq)=Ykq(at)
(20.91)
If we make use of Eq. 20.3 to transform to the generators given by Eq.
20.77 as
K += -tr.r, K _=tp.p, Ko =  (r.p+p.r)
(20.77)
we find that the operators rt and ipt for fixed t are the projections +! and
- 1- of an SO(2, 1) tensor operator of rank t. We may construct from this
new basis tensor operator a new tensor T (k/2;k) of rank k/2 in SO(2, 1)
and k in SO(3) whose maximum component is
- ( k k )
T 2. 2. ; kq = Y kq (r)
(20.92)
where (noting the results of Section 17.10) we have
- ( k. ) _  ( k '. ) k/2 ( 'IT )
T 2",kq - £.J T 2" ,kq K'K 0, - 2,0
(20.93 )
Ie'
and hence
k/2 1/2
L [ k' ] ( k )
Y r = . T -". k
kq ( ) ( k + 2" ) ! ! ( k - 2" ) !! 2 ' q
Ie = - k /2
(20.94 )

MATRIX ELEMENTS OF MULTI POLE OPERATORS 293
gives the multiple operator as an expansion in terms of the double tensors.
The matrix elements of Ykq(r) may be evaluated for harmonic-oscillator
states using Eq. 20.89 to give
k/2 1/2
< nlm l Y ( r )l n'I'm' > =  [ k! ] CT'k/2TCl' kl
kq £..J ( k + 2" ) !! ( k _ 2" ) ! ! t'" t m' q m
K= -k/2
x < Till T(  ; k ) II T'I')
(20.95 )
The matrix elements of the multipole operator will vanish unless
T+ T'   IT- T'I, i.e." 1+ l' -1 k 1/-1'1
1'+/kI/-I'1
(20.96)
(20.97)
and
m' + q = m
(20.98 )
(20.99)
, . 2 '
t +" = t, I.e.," = n - n
These selection rules follow directly from those of the SO(2, 1) and SO(3)
coupling coefficients.
The reduced matrix element in Eq. 20.95 may be determined by noting
that since
1/2
< llm l Y ( r )I /'I'm > = [ k! ]
kq (k+2T'-2T)!!(k-2T'+2T)!!
x CJ' 1! T' C;;, < Till T( k /2; k) II T'I'>
we may write Eq 20.95 as
(20.100)
1/2
[ (k+I'-/)!!(k+I-I')!! ]
nlm Y r n'I'm' -
< I kq( )1 ) - (k + n' - n)!! (k + n - n')!!
C!'k/2,T
x ;, k;t t T <llm/Ykq(r)/I'I'm') (20.101)
C T ' T-T' T
In terms of the notation
<n/lrkln'I') = {XI Rn/(r ) rkR n'l'(r),-2 dr
(20.102 )

294 CASE STUDY I: THE ISOTROPIC HARMONIC OSCILLATOR
where Rn/(r) is the radial part of the harmonic-oscillator eigenfunctions, we
obtain for Eq. 20.101
1/2
[ (k+I'-/)!!(k+I-I')!! ] (k+I+I'+I)!!
<n/lrkln'I') =
(k + n' - n)!! (k + n - n')!! [2 k (21 + I)!! (2/' + I)!! J 1/2
C T'k/2 T
t' t - t' t
X C T 'k/2 T
T' T - T' T
(20.103 )
where we have made use of the fact that the radial integral
<lllrkl/'I')= (k+I+I'+ 1)!![2 k (2/+ 1)!!(2/'+ 1)!!]-1/2 (20.104)
In Section 17.15 we saw that the algebraic forms of the coupling
coefficients C:3 for coupling a finite nonunitary representation D(}I)
to a positive discrete representation D + (}2) to yield positive discrete
representations D+(}3) of SO(2, 1) differ from those of SO(3) by simply a
phase (-I)it-j2+j3. Thus we may rewrite Eq. 20.103 as
1/2
[ (k+I'-/)!!(k+I-I')!! ] (k+I+I'+I)!!
nl r k n' I' -
< I I )- (k+n'-n)!!(k+n-n')!! [2k(2/+1)!!(2/'+1)!!] 1/2
X(-I),-r( 
T'
k/2 ) /( T
t' - t t T
T'
- T'
k /2 ) (20.105) .
T'-T
- t'
where the 3j-symbols are to be taken as the algebraic jorms of the
corresponding SO(3) 3}-symbols.
Let us calculate the matrix elements <n/lrln'I'). In this .case k = 1 and
K = + -!. From Eq. 20.97 we have the selection rule Lli = 0, + I, and from
Eq. 20.98, Lln = + 1. The latter selection rule precludes the possibility of
Lli = 0,. and hence the only possible nonzero matrix elements are < n/l'l
n + l/ + l) and <nllrln + l/ + 1). Consider the case of <n/lrln+l/+I).
Equation 20.86 implies that T' = T + t and t' = t + t, and hence Eq. 20.105
becomes
<n/lrln+l/+1)
2(2;/: 3 1) (-1)'- r( t+_!t
T
! )/( T+l
! , - T!
 :)
(20.106)
t

MATRIX ELEMENTS OF MULTI POLE OPERATORS 295
From the tables of Edmonds lO4 we have
( T+t
-a-!
T
) 1/2
! _ -1 T+a T+a+l
t - ( ) [ (2T+2)(2T+ 1) ]
(20.107)
a
and hence
<n/lrln+l/+1)=
[ ] 1/2
21+3 (_1)2(t-T) ;+/ (20.108)
2(2/+1)
Putting T=t(/-t) and t=!(n+), and remembering that n+1 is nec-
essarily even, we have
1/2
<nllrln+ 1/+ 1)= _ ( 1++3 )
(20.109)
in agreement with the results Shaffer obtained by direct integration. 268
The preceding results are valid for k  O. The corresponding results for
k < 0 are found by first noting that the SO(2,1) tensor operators T(A)
defined by Eq. 20.87 satisfy, with respect to K  and Ko, the same
commutation relations as do the components of the tensor operator T( -
A-I). As a result the matrix elements for k < 0 may be found by making
the substitution k- k-l throughout Eq. 20.105. Thus for <n/lr- 2 In/) we
have k = - 1, and hence
<n 1 Ir- 2 Inl)= 21 1 ( _1)t-T(  T 1 )/ ( ; T 1 )
t T
= 2 (_I)t-T( T T )/ ( ; T )
21 + 1 t t T
2 (20.110)
-
2/+ 1
where we make use of the invariance of the 3j-symbols under the Legendre
reflection k-k-l. We note that if k-k-l, then -!-!, so that
our method fails for the matrix elements of 1/ r.
Under the substitution A-A-l we still have A= -A, -A+ 1,...,A,
and hence the tensor operators T( - A-I) still transform under SO(2, 1) as
a finite nonunitary representation, which, unlike that for T(A), is of the
indecomposable type.

296 CASE STUDY I: THE ISOTROPIC HARMONIC OSCILLATOR
Tne preceding calculations of the matrix elements of r k show how it is
possible to develop selection rules not only on the familiar angular-
momentum quantum numbers but also on the principal quantum numbers
that characterize the radial oscillator functions. The calculation of the
radial integrals involving r k has been given a purely algebraic treatment,
and the Wigner-Eckart theorem fully exploited.
EXERCISES
20.13. Show that 269 ,270
1/2
< nIl r 3 1 n + 31 + 1> = [ (n + 1+ 3) ( n + 1+ 5) ( n -I + 2) 2 - 3 ]
20.14. Investigate the calculation of the matrix elements of pk, where p is the
linear momentum of the oscillator.

21
Case Study II: The Hydrogen Atom
21.1 INTRODUcnON
One of the remarkable results of the early Bohr-Sommerfeld model of
hydrogenic atoms was that the energies of the bound states could be
expressed, in atomic units, by the compact formula
-Z2
En= 2
2n
(21.1)
where Z is the atomic number and n is the principal quantum number
(n = 1,2,...). In the nonrelativistic solution no other quantum numbers are
required. Nowhere in the formula is there an explicit dependence on the
orbital (/,m/) or spin (s,ms) quantum numbers. Every energy level is
2n 2 -fold degenerate, the 2 coming from the twofold spin degeneracy and
the n 2 from the orbital degeneracy.
The Hamiltonian for a hydrogenic atom may be written as
p2 e 2
H=--Z-
2p. r
( 21.2 )
where p. is the reduced mass. In the usual atomic units, and putting p. = 1,
we have in quantized form
v 2 Z
H=----
2 r
(21.3 )
297

298 CASE STUDY II: THE HYDROGEN ATOM
The conventional Schrodinger equation is written as
Hn/sm/Ins = En n/sm/ms
( 21.4 )
where in spherical coordinates (r,(},cf»
n/sm/Ins = R n /( r) Y/ m / ( (}, cf> ) (Jsm s
(21.5 )
with Rn/(r) the solution of the radial equation
d 2 R + 2 dR +2 ( E+ Z ) R- l(l+ 1) R=O
dr 2 rdr r r 2
(21.6)
Here Y/ m /«(}, cf» is the usual spherical harmonic associated with the solution
of the angular equation
I a ( . a Y ) 1 a 2y
--:-- (} a() SIn (} a(} + . 2  + 1 ( 1 + I ) Y = 0
SIn sIn cf> acf>
(21.7)
The functions (Jslns are spinors that determine the spin dependence of the
complete solution.
The energy eigenvalues En are normally calculated from the solution of
the radial equation, either using the theory of special functions, or using
80(2, I) as the spectrum-generating group as in Section 18.6. Since )2, Iz,
S2, and Sz all commute with H, we expect the energies to be independent of
m/ and ms' and would thus anticipate a 2(21 + I )-fold degeneracy for each
energy eigenvalue. However, we find associated with a given principal
quantum number n the orbital eigenfunctions having
I=O.I __ n -} (21.8)
to give the degeneracies portrayed in Fig. 21.1.
The additional degeneracy is surprising and demands an explanation.
This problem was early investigated by Pauli, 27 I Fock!97,198 and Barg-
mann;99 who sought to explain it in terms of a higher symmetry of the
Schrodinger equation for the hydrogen atom.
Since H is independent of time, the energy E is a constant of motion.
While the rotational symmetry of H causes the orbit to lie in a fixed plane,
it does not, by itself, ensure that the orbit is closed. The angular-
momentum vector L is an axial vector that is perpendicular to the plane of
the Keplerian orbit, and we seek an additional constant of the motion that
characterizes the orientation of the major axis in the orbital plane.
In 1926 Pauli 271 showed that the classical Runge_Lenz 272 ,273 vector
Zr
a =pXL--
r
(21.9)

INTRODUCTION 299
10 . . . . . . . . .
. . . . . . . .
8 . . . . . . .
. . . . . .
t 6 . . . . .
n
. . . .
4 . . .
. .
2 .
o
2
4
6
8
10
l
Fig. 21.1. The degeneracies of the hydrogen atom may be arranged as an infinite tower. Each
dot represents the 2/ + I-fold orbital degneracy. The states associated with a given level of the
tower have the same energy eigenvalue.
which occurs as a constant of motion in the classical Kepler problem, may
be expressed as a Hermitian quantum-mechanical operator by writing
A' = !(pXL- LXp) _ Zr
r
(21.10)
A simple derivation has been given by Wulfman,200 and a historical
account by McIntosh. 274
Use of the commutation relations for r, p, and L tediously leads to
[A',H] =0,
L. A' =A'. L=O
and
A'2=2H(L 2 +1)+Z2
(21.11 )

300 CASE STUDY II: THE HYDROGEN ATOM
Furthermore,
[ L;, Lj] = if.ijkLk'
[L;,A;] = if.ijkA"
- 2iHf.ijk L k
[A'j,A J ] = (21.12)
Z2
The presence of H in the final commutator may be avoided by replacing
A' with
A= Z A'
Y -2H
to give two possible commutator algebras:
(21.13)
[L;, Lj] = if.ijkLk
[L;,Aj] = if.ijkAk (E<O) (21.14)
[A;,Aj] = if.jikL k
and
[L;, Lj] = if.ijkLk
[L;,Aj] = if.ijkAk (E>O) (21.15)
[A;,Aj] = - if.;;kLk
where E < 0 for bound states, while E > 0 for continuum states.
The above two commutator algebras are isomorphic to the Lie algebras
so(4) and so(3,1), respectively. In this chapter we explore some of the
properties of the groups SO(4) and SO(3,1) and their relevance to the
hydrogen atom, and then construct the dynamical group 80(4,2).
21.2 SO(4) AND HYDROGEN ENERGY LEVELS
The commutation relations given in Eq. 21.14 may be simplified by
introducing two new operators 53
J=!(L+A) and J'=!(L-A) (21.16)
to give
[J;,] = if.ijkJk' [J/,J;] = if.ijkJ/c
[ J;, J: ] = 0 (21.17)
The three components of J and J' generate the Lie algebra of the group
SU(2) X SU(2), which is locally isomorphic to 80(4).

50(4) AND HYDROGEN ENERGY LEVELS 301
The operators J2 and J,2 are each Casimir operators of SU(2) with
eigenvalues
J2=j(j+l) and J,2=j'(j'+1)
( . ., - 0 1 1 )
] ,J - , 2' ,...
(21.18)
We may form two Casimir operators for 5'0(4), namely,
F=J 2 +J,2 =! (L 2 + A2)
(21.19)
and
G = L. A = J2 - J,2
(21.20)
But for a hydro genic atom (though not for a many-electron atom,32,275) we
have from Eq. 21.11 that L. A = 0, and hence must have j = j', since then
G = O. Thus for a hydrogenic atom
F=2j(j+ 1)
(j = 0, !, 1,. . . )
( 21.21 )
But noting Eq. 21.13 together with Eqs. 21.11 and 21.19, we have
F= ; (L2- 2 A'2)= - ;; -1
Comparison with Eq. 21.21 then gives
2j(j+ 1) = 2 - t
and hence
_Z2
E=
2n 2
( 21.22 )
where n = 2j + 1 = 1,2,3,.... Thus we have arrived at the celebrated Bohr
formula by generating the energy spectrum via the Casimir operators of
SO (4).
The representations of SU(2) X SU(2) are of degree (2j + 1) X (2j' + I),
and since for a hydro genic atom j= j', we conclude that the hydrogenic
levels are n 2 -fold degenerate, with an additional factor of 2 required to
accommodate the twofold spin degeneracy. Thus the orbital degeneracy
group of the hydrogenic atom is SO(4)SU(2)X SU(2). Let us now
consider the building up of the relevant SO(4) representations, and then
determine the matrix elements of the group generators.

302 CASE STUDY II: THE HYDROGEN ATOM
21.3 SPHERICAL TENSORS AND SO(4)
It is convenient to treat L and A as two rank-one spherical tensors with
components
L (I) - L
0-3
L(I)= + -LL_
:tl V2 +
A (I) = + -L A
:tl - V2 :t
( 21.23 )
A (I) - A
o - 3
( 21.24 )
The basic nonzero commutation relations for 80(4) then become
[ L(l) L(I) ] =+L(I)
o ':tl -:tl
l " L (I) L(I) ] =-L(I)
'+ I, - I 0
[A (I) A (I) ] = + L {I)
O':tl -:tl
[A (I) 4 (I) ] = - L (I)
+1' -I 0
[L (I) A (I) ] = + A (I)
O':tl -:tl
[A (I) L ] = + A (I)
o ':tl -:tl
( L (I) A ) ] = + A (I)
:tl' +1 0
(21.25)
If we choose a basis with Ll) and AI) as the two Weyl self-commuting
operators, then we find that L+ A and L- A transform under 80(4) as [11]
and [1 - 1], respectively. Recalling Section 19.13, we label the states in this
basis as [pq], where p = j + j' and q = j - j'. Then the eigenvalues of the
Casimir operators F and G become
Fla[pq ]LM> =! (p + q )2 Ia [pq ]LM> (21.26)
and
Gla[pq]LM>=q(p+ 1)la[pq]LM>
(21.27)
21.4 REDUCED MATRIX ELEMENTS OF A
To calculate the reduced matrix elements of A, we start by calculating
the matrix elements of G=L.A. Use of Eq. 19.228 gives
<LIILIIL>
<a[pq ]LMIL.Ala[pq ]LM>= 2L+ 1 <a[pq ]LIIA Ila[pq ]L>
1/2
=[ ;::; ] <a[pq]LIIAlla[pq]) (21.28)

REDUCED MATRIX ELEMENTS OF A 303
Comparison with Eq. 21.27 leads immediately to
1/2
[ ( 2L + 1) ]
<a[pq]LIIA\la[pq]L)=q(p+ 1) L(L+ 1)
(21.29 )
Since L+A transforms as [11] under 80(4), we have from Eq. 19.81 that
-!(p+q)
<a[pq ]LIIL + A Ila[pq ]L> = (2L + 1) V3 t(p - q)
L
!(p+q) 1
t(p-q) 0
L 1
x <a[pq] IlL + A Ila[pq]>
(21.30)
Upon reducing the 9j-symbol to a 6j-symbol via Eq. 19.66 and then
evaluating the 6j-symbol algebraically, we may express the right-hand side
as
1/2
[ L(2L+l) ] L(L+I)+q(p+l)
= L(L + 1) [(p +q)(p + q+ l)(p + q+ 2)(p - q + 1)] 1/2
x <a[pq]IIL+Alla[pq]>
Making use of Eq. 21.19 then leads to the result
<a[pq] IlL + A lIa[pq]> = [(p + q) (p + q+ 1) (p + q +2) (p - q + 1)] 1/2
The corresponding result for L- A follows in the same manner, and
remembering that the group generators are diagonal in the representations
of 80(4), we conclude that 1l6
<a[pq] IlL + A Ila / [ p' q']>
= aa'pp'qq'[ (p + q) (p + q+ 1) (p + q+ 2) (p + q+ 1)] 1/2
( 21.31 )

304 CASE STUDY II: THE HYDROGEN ATOM
If the ab<?ve result is used in Eq. 21.27, again we readily find that the
reduced matrix elements of A are
1/2
[ ] [ [ (P+L+2)(P-L)(L+I- Q )(L+l+ q ) ]
< pq L+ IliA II pq]L)= ( )
L+l
[ 2L + 1 ] 1/2
<[pq ]LIIA II [pq ]L) = q(p + 1) L( L + 1)
1/2
<[pq]L- 1 1I A lI[pq]L)= _ [ (p+L+ l)(p+ 1  L)(L+q)(L-q) ]
(21.32)
in agreement with the results of Biedenharn,172 apart from a factor arising
from our choice of definition of the 80(4) matrix elements (see Eq. 19.81).
21.5 LADDER OPERATORS IN SO(4)
The components of the spherical tensor L(I) act as ladder operators
taking us through the weight space of the 80(3) representations, but do
not permit a coupling of different representations. Typically we have
Lrll)l a.[ pq] LM> = M I a [pq] LM>
Lna[pq]LM)= +  [(L + M)(L + M+ 1) ]1/2 Ia [pq]LM + I)
(21.33)
The components of the spherical tensor A(I) will ladder not only M but
also L in steps of 0 or + 1. Thus we have
Al)la[pq ]LM>=  <a[pQ ]L' M'IAl)la[pQ ]LM>la[pq ]L' M'>
L'M'
(21.34 )
Application of the Wigner-Eckart theorem together with use ot Eqs. 21.32

LADDER OPERATORS IN SO(4) 305
leads to the results
Adl>la[pq ]LM>= MALla[pq ]LM>
+ [(L- M + 1) (L+ M + 1)] 1/2 BL1a [pq ]L+ 1 M>
- [L2- M2] t/2cLIL-IM> (21.35)
1/ 2
2 1/2 A 1la[pq ]LM>= + [(L + M) (L + M + 1)] ALILM + 1>
+ [(L + M + I)(L + M +2)] 1/2 BL1L + 1 M + 1>
1/ 2
+[(L + M)(L + M-I)] CLIL-IM + I>
( 21.36 )
where
<a[pq ]LIIA Ila[pq ]L>
A -
L- [L(L+l)(2L+I)]1/2
< a [ pq ] L + 111 A II a [ pq ] L>
B L =
[ ( L + 1 ) (2L + 1 ) (2L + 3) ] 1/ 2
<a[ pq]L - IliA Iia [pq ]L>
C -
L - [L(2L _ 1) (2L + 1) ] 1/2
(21.37)
As is to be expected, the components of L and A supply a complete set
of ladder operators that permit us to move throughout the weight space of
any 80(4) representation, though not between different 80(4) representa-
tions. Thus 80(4) may be interpreted as the degeneracy group of the
nonrelativistic hydrogen atom.
EXERCISE
21.1. Starting with the observation that
[A (I) A (I) ] = L(I)
- I' + 1 0
show that 276 ,277
[(L+2)AL+ 1 - LAL]CI+ 1 =0
[(L+ I)A L - (L-I)AL-1]C L =0
- (2L-I)CIC L + (2L+3)Cl+ 1 C L + 1 -Ai =-1
(21.38)

306 CASE STUDY II: THE HYDROGEN ATOM
and use these relations to give an alternative evaluation of the reduced
matrix elements of A.
21.6 BOSON OPERATORS AND SO(4)
Schwinger 278 has shown that it is possible to make a realization of SU(2)
in terms of boson spin operators a;,aJ (i= 1,2), where
[ a. , aj ] =..
, J IJ
(21.39)
Using the Pauli spin matrices
01 = (  ) 02 = (  i )
°3 = ( 
0 ) (21.40)
-1
we can write the three generators of SU(2) as
j,=1 2 at (J.a
, ,
(21.41 )
with
a= ( :: )
at=(at a)
(21.42)
The creation operators at acting on the vacuum state 10> may then be
used to create an arbitrary SU(2) ket vector Ijm>, namely,
I . > = (at)j+m(aOj-m 1 0 >
Jm 1/2
[(j + m)! (j - m)!]
The group SO(4) is locally isomorphic to SU(2) X SU(2), and hence we
can make a realization of the generators of SO(4) by putting
(21.43)
j, = _ 2 1 at(J,a and J: = _ 2 1 bt(J.b
" , ,
(21.44)
where
[ a" a j ] =..
':J IJ
[ b" bt ] ="
, } IJ
(21.45)
and the components of at and a commute with those of b t and b. The
SO(4) ket vectors then become
(at)J+m( at)J-m (bt)J'+m' (b t)J'-m
Ijm,j'm') = 1 2 1 2 1/2 1 0 ) (21.46)
[ (j + m)! (j - m)! (j' + m') ! (j' - m')! ]

DYNAMICAL GROUP OF THE HYDROGEN ATOM 307
The ladder operators for this basis become just
J + =aTa 2
J = bTb 2
J _ =aal
J'- = bbl
(21.47)
The operators L and A introduced earlier may be realized in terms of the
boson operators as
L.=J.+J.'= _ 2 1 ( ato.a+bto.b )
I I I I I
(21.48 )
and
A. =J. - J! = 1 2 ( ato,a- bto,b )
I I I I I
(21.49)
In later work it is convenient to replace the L and A by the antisym-
metric tensors
L.. = - L..
I} "}l
(i=l=j)
(21.50)
where
Lk = Lij = -! (a tOka + btokb)
(i,j,k= 1,2,3 and cyclic permutations)
( 21.51 )
and
-A j =L j4 = --!(atoja-btojb)
(21.52)
Under commutation,
[Lij' L kl ] = i ( jkLjl + i1Lkj + jkLIi + jILjk)
(21.53)
EXERCISE
21.2 Show 2oo ,279 that for fixed n,
(n/mlrln/'m') = (n/mIAln/'m')
(21.54 )
21.7 DYNAMICAL GROUP OF THE HYDROGEN ATOM
So far we have found that the degeneracy group of the nonrelativistic
hydrogen atom is 80(4) and that the energy-spectrum-generating group is
80(2, 1). The states of the hydrogen atom may be labeled by the tradi-

308 CASE STUDY II: THE HYDROGEN ATOM
tional quantum numbers Inlm> associated with the solution in spherical
coordinates or the set In 1 n 2 m> associated with the solution in parabolic
coordinates. 28o In the latter case
n =n l + n 2 + Iml + I
(21.55)
In constructing the dynamical group we must find a group that contains
80(4) as a subgroup and includes operators that ladder n and I.
Let us first consider the operator
r 0 = L56 =! (ata+ btb+ 2)
(21.56)
which obviously commutes with all the generators of 80(4), and has the
form of a simple number operator. The eigenvalues of L56 with respect to
an arbitrary 8U(2) X 8U(2)-symmetrized ket Ijm,j'm'> follow if one ap-
plies L56 to both sides of Eq. 21.46 and uses the commutation properties of
the annihilation and creation operators, remembering that
aiIO>=O
(21.57)
and
[ a. aJx ] = xa Jx - 1
I' I I
(21.58 )
Then we have
L 56 Ijm,j'm'> = (j + j' + I )Ijm,j'm'>
(21.59)
But for a hydrogen atom, j =j' and 2j + I = n (cf. Eq. 21.22), and hence
L56lnlm> = nlnlm>
(21.60)
Thus the eigenvalues of L56 are simply the principal quantum number n.
The operator L 56 , taken with the generators of 80(4), generates the
compact group 80(2) X 80(4). Clearly we need to produce a noncompact
group that includes 80(2) X 80(4) as a subgroup. To this end, consider
two further scalar operators,
T= L = 1 ( ato bt -ao b )
45 2 2 2
(21.61)
and
S=L 46 =  (a t (J2 bt +a(J2 b )
(21.62)
Under commutation,
[L 45 , L 46 ] = iL56
(21.63)

DYNAMICAL GROUP OF THE HYDROGEN ATOM 309
Consider the two operators
N :t: = L45 + iL46
(21.64 )
that is,
N =ato b= - i ( atb t -atbt )
+ 2 I 2 2 1
(21.65)
and
N _ =ao 2 b= i(a 1 b 2 -a 2 b l )
(21.66)
Since
[ L 56 , N :t: ] = + N :t:
(21.67)
we anticipate that N:t: raises or lowers n by + 1. Now consider the ket
Inll) = Ijj,jj)
(21.68)
-that is, with n =2j + 1, 1= n -1. We have
iN +Inll)= (arb -abT)ljj,jj)
( at2j+ Ibt2jbt _ at2jatbt2j+ I )
= 1 1 2 2 ., 1 2 1 10)
'J.
{I ' + l' l' l' I ) I . l' 1 . + 1 . + I )}
= n J "2J + 2,J + !J -"2 - J + "2J - "2,J "2J "2
= ( _ 1 ) n - 1 n [ 1 _ ( _ 1 ) n + l' ] (2/' + 1 ) 1/2
x ( nl2
nl2
nl2
n12-1
I' ) In + II' II
-n+l
(21.69 )
where we have made use of Eq. 19.76 to transform to the nlm representa-
tion and then exploited the symmetry properties of the 3j-symbol. Inspec-
tion of the 3j-symbol and the phase of (- l)n+1' shows that I' is limited to
I' = n - 1 = I. Explicit evaluation of the 3j-symbol finally gives
iN +Inll)= (-I)n 2n V6 (2n+ 1) In+ III)
(21.70)
where 1= n -1.

310 CASE STUDY II: THE HYDROGEN ATOM
Likewise we find that
n
( -1) 2
-iN_ln+lll>= 1/2 1nll>
[6(2n + 1)]
(21.71)
where again 1= n - 1. The above results may be readily verified by the
application of the commutator
[N +, N _ ] = - 2L56
(21.72)
to the ket Inlm>. The matrix elements of L45 and L46 follow from Eq. 21.64
and show that L45 and L46 act as ladder operators on n.
The operators Lij' L;4' L 45 , and L46 permit a laddering from the vacuum
state 10> to any ket Inlm> by means of an appropriate series of operators.
However, this set of operators, together with L 56 , does not close under
commutation. To produce a set closed under commutation requires the
introduction of two additional vector operators M and r with components
M; = L;5 = i[ L;4,L 45 ] = - ! (a to;Cb t - aCo;b)
(21.73)
and
. -I
r; = L;6 = -1[L;5,L 56 ] = 2 (ato;Cb t +aCo;b)
(21.74)
where
c=(
°
-1
)
(21.75)
It is easily verified that the 15 operators Lij' L;4' L;5' L 45 , L 46 , and L56
close under commutation:
[Lab' Lcd] = - i ( gacLbd + gadLcb + gbcLda + gbdLac)
(21.76)
where gab is associated with the metric (- - - - + +). The Lie algebra
formed by the 15 operators is readily found to be that associated with the
noncompact group 80(4,2), which holds invariant the real form
-  gabxaxb
(21.77)
The complete set of generators that yield the boson representation is
collected in Table 21.1. The group 80(4,2) is isomorphic to the group
8U(2,2), which holds invariant the complex form
* + *- *- *
Z I Z I Z 2 Z 2 Z 3 Z 3 Z 4 Z 4
(21.78 )

THE CASIMIR OPERATORS 3m
Table 21.1. Generators of the Boson Representation of 80(4,2)
Lk = Lij = !Eijk (a taka + b takb )
A;=L;4= -!(ata;a-bta;b)
M; = L;s = -! (ata;Cb t -aCa;b)
-i
r. =L. 6 = - ( ata.Cb t +aCa.b )
, , 2' ,
S=L 46 = !(atCb t +aCb)
-i t
T=L 4s =T(a t Cb -aCb)
r O=LS6 = !(a t a+b t b+2)
Since the operators of 80(4,2) permit us to pass from any hydrogenic
state Inlm> to any other state In'l'm'>, we conclude that 80(4,2) must
contain among its representations a single irreducible representation that
covers all the states of the hydrogen atom. Thus 80(4,2) constitutes a
dynamical group for the hydrogen atom. 15, 111,281-289
The matrix elements of the complete set of group generators acting on
hydrogenic states may be calculated tediously, but easily, using the estab-
lished results for Lij' L;4' L 45 , L 46 , and L56 by systematic use of the
commutation relations used to define the components of M and r'. A
complete tabulation has been given by Ferreira 290 and Englefield. 279
21.8 THE CASIMIR OPERATORS
The Casimir operators for 80(4,2) may be taken as l5
C 2 = Lab L ab
C - L ab L cd L ej
3 - Eabcdej
( 21.79 )
C 4 = LabLbcLcdLda
where
L ab = gaaLab
(21.80)
If these operators are realized in terms of the boson operators used to

312 CASE STUDY II: THE HYDROGEN ATOM
construct the irreducible representation appropriate to the hydrogen atom,
we find the eigenvalues
C 2 =6
C 3 =o
C 4 =o
(21.81)
Of course here we have constructed only a single irreducible* representa-
tion of SO (4, 2). The establishment of all the irreducible representations of
80(4,2) and of its covering group 8U(2,2) is a very different, and difficult
problem whose full solution is still incomplete. 29 1-297
21.9 THE 80(4,1) SUBGROUP
The dynamical group 80(4,2) has a very rich subgroup structure. The
subset of the SO( 4,2) generators comprising Lij' L i4 , LiS' and L45 closes
under commutation on the Lie algebra of the de Sitter group 80(4, 1),
which is isomorphic to the noncompact symplectic group 8p(2,2). We have
already seen that L45 connects hydrogenic states differing in n by + 1, and
hence under 80(4,2)80(4, 1) the irreducible representation of SO(4,2)
must remain irreducible. The two Casimir operators and their associated
eigenvalues for this representation of 80(4,1) are
C =L L ab =4 and C =L LbcL Lda=o
2 ab 4 ab cd
(21.82)
The representation theory of 80(4,1) has been the subject of numerous
studies 112 - 116 ,298-307 and will not be pursued here.
The group 80(4,1) has a single irreducible representation that covers
the bound states of the hydrogen atom and generators capable of creating
the complete set of quantum numbers nlm, and is therefore sometimes
referred to as the quantum-number group of the hydrogen atom. 308 The
enlargement of 80(4,1) to the dynamical group 80(4,2) introduces no
additional quantum numbers and leaves the representation space un-
changed. However, going to the larger group introduces additional opera-
tors, which (as we see later) may be identified with interaction operators;
and in particular the larger group contains the dipole operator.
EXERQSE
21.3. Show that the set of operators L;p L;4, L;6, and L46 are closed under
commutation and give an alternative 80(4, I) subgroup of 80(4,2).
*N.B. When we refer to the single irreducible representation of SO(4,2), we mean that
representation appropriate to the description of the H-atom.

FURTHER SUBGROUPS OF SO(4,2) 313
Table 21.2. 8ubgroups of 80(4,2) and Their Generators
SO(2)XSO(4)
SO(4,1)
SO(4,1)'
SO(3,2)
SO(3,1)'
SO(3, 1)"
SO(4)
SO(2, I) x SO(3)
SOl(2, 1) x S02(2, I)
(L S6 ) (LiP L i4 )
LiP L i4 , LiS' L 4S
LiP L i4 , L i6 , L46
Lij' LiS' L i6 , LS6
L.. , L. 6
I} I
L.. , L. s
I} I
L.. , L' 4
I} I
(L4S' L 46 , L S6 ) (Lij)
(N;,Nj) (N;,Nf)o
QThe generators of SOl(2, 1) X S02(2, 1) are defined by Eqs. 21.85 and 21.86.
21.10 FURTHER SUBGROUPS OF 80(4,2)
Many important subgroups of 80(4,2) may be found by a simple
examination of the commutation relations of the group generators (cf. Eq.
21.76) and picking out subsets of generators that close under commutation.
Some of the most important subgroups and their generators are given in
Table 21.2. This list, as we see later, is not exhaustive. There are of course
many possibilities for using l1able 21.2 to form chains of embedded
subgroups, such as
80( 4,2):J 80( 4,1):J 80( 4):J 80(3):J 80(2)
The group 80(2) X 80(4) is the maximal compact subgroup309 of
80(4,2). The 80(4,2) irreducible representation is simply reducible under
reduction to 80(2) X 80(4). The eigenvalues of the generator L56 of 80(2)
are just the principal quantum number n, while the basis states of 80(4)
may be labeled by the quantum numbers nlm. In terms of the 80(4,2)
infinite tower the 80(4) multiplets occur as levels, as shown in Fig. 21.2.
The group 80(3,2) is the de Sitter group associated with de Sitter spaces
of negative curvature. 310 Herrick and Sinanoglu 31l ,312 have used bound-
state hydro genic radial functions to construct a basis for a unitary repre-
sentation of 80(3,2). The classification of the representations of 80(3,2)
has been the subject of many investigations.313-317, The group 80(3,2) is
isomorphic to the group of the two-dimensional harmonic oscillator,
8p(4,R).
The groups 80(3, 1)' and 80(3, 1)" are isomorphic to the homogeneous
Lorentz group, which has the same Lie algebra as the noncompact group

314 CASE STUDY II: THE HYDROGEN ATOM
10 ---.---.---.---.---.---.---.---.---.
---.---.---.---.---.---.---.---.
8 ---.---.---.---.---.---.---.
- - - . - - - . - - -. - - - .- - - .- - - .
SQ(4) multiplets
"<
6 ---.---.---.---.---.
t
n
---.---.---.---.
4 ---.---.---.
- - - .- - -.
2 ---.
o
2
4
6
8
10
I ::-
Fig. 21.2. The states of the hydrogen atom are displayed as an infinite SO(4,2) tower with
SO(4) multiplets as the tower levels.
SL(2,C). The representation theory of the Lorentz group has understand-
ably received much attention. 61 ,276,277,318-329 Note that neither SO(3,1)'
nor SO(3,1)" corresponds to the SO(3,1) group that arose in discussing
the continuum states of the hydrogen atom. In that case SO(3,1) essen-
tially arose as the analytic continuation of the SO(4) degeneracy group of
the bound states. Later we wish to interpret SO(3, 1)" as the physical
Lorentz group.
The groups SO(2,I)XSO(3) and SO (2, I)XSO(2, 1) are associated with
the separation of the Schrodinger equation for hydrogen into spherical and
parabolic coordinates, respectively.
21.11 SO(4,2) BASES AND HYDRO GENIC ATOMS
The enormously rich subgroup structure of SO(4,2) makes possible a
diversity of choices of basis states. The basis for the Hilbert space of the

SO(4,2) BASES AND HYDROGENIC ATOMS 315
hydrogenic states will depend on the choice of operators to be diagonal-
ized. This choice will depend on the physical applications being consid-
ered. Three bases are of particular significance for hydrogenic atoms: the
angular-momentum basis states Inlm), the parabolic states In l n 2 m), and
the scattering states Inl,n - n 2 ,m).
The angular-momentum states Inlm) are associated with the spherical
coordinates and diagonalize the operators L 56 , L 2 , and L 12 - In terms of the
80(4) basis we may choose L I2 and L34 as the two Weyl self-commuting
operators and have
Inlm) = l[n-l0]lm)
(21.83 )
Alternatively we can use the generators of SU(2) X 8U(2) defined in Eq.
21.44 to give basis states l(n-l)/2 m l ,(n-l)/2 m 2 ) which from Eq. 19.76
are related to those of Eq. 21.83 by the transforma,tion
Inlm)= L (-I)m(2/+ 1)1/2( (nI)/2
m.,m2 1
m 2
_1m)
(n-l)/2
X I (n - 1) /2 m l , (n - 1) /2 m 2
(21.84 )
The parabolic states In 1 n 2 m) provide a natural basis for the treatment of
the Stark effect 280 and are directly associated with the existence of a
80 1 (2, 1) X 80 2 (2, 1) subgroup of 80(4,2). The generators of the two
80(2,1) groups may be taken as
80 1 (2, 1 ) :
N: = !(L 46 + L 35 )
N =! (L45 - L 36 )
(21.85)
Nl=!(L 56 +L 34 )
and
80 2 (2, 1):
N =! (L 46 - L 35 )
N;=!(L 45 +L 36 )
(21.86)
N = !(L 56 - L 34 )
It is convenient to introduce the ladder operators
N I = N I+ l 'NI a nd N2 =N 2 +iN 2
:t I - 2 :t 1 - I
(21.87)

316 CASE STUDY II: THE HYDROGEN ATOM
or in terms of the boson operators,
N I = - a t b t
+ 2 I'
NI=-ab'
- 2 I'
N 2 = a t b t
+ I 2'
N =a l b 2
N = -} (aa2 + bib!)
Nj =! (a!a l + bib!)
(21.88)
If the parabolic states are defined as
1/2
In l n 2 m ) = [n l ! (n l + Iml)! n 2 ! (n 2 + Iml)!] arn2+ma!nlbrnl +mb! n2 IO>
(m>O)
(21.89)
1/2
In l n 2 m) = [n l ! (n l + Iml) !n 2 ! (n 2 + Iml)!] arn1tl!nl-mbrnlb!n2-mIO>
(mO)
{21.90)
we find that the operators N, Nj, and L I2 are diagonalized. If the group
generators of 80 1 (2, 1) and 80 2 (2, 1) (Eq. 21.88) are allowed to act on a ket
In l n 2 m), we find that
1/2
N In l n 2 m) = - [(n l + 1) (n l + Iml + 1)] In l + l,n 2 m)
N In 1 n 2 m> = - [n l (n l + Iml)] 1/2 1nl -1,n 2 m)
(21.91)
Nlnln2m)= !(2n l + Iml + 1)ln 1 n 2 m)
and
N In l n 2 m) = [(n 2 + 1) (n 2 + Iml + 1)] 1/21nln2 + I,m)
Nlnln2m)= [n 2 (n 2 + Iml) ]l/2Inln2 -I,m)
(21.92)
Nj In 1 n 2 m) = !(2n 2 + Iml + 1 )ln l n 2 m)
Since L56 = N + Nj we recover the result that
n=n l + n 2 + Iml + 1
The parabolic states may be related to the angular-momentum states by
noting that for the parabolic states, the eigenvalues of the 8U(2) X 8U(2)

SO(4,2) BASES AND HYDROGENIC ATOMS 317
Table 21.3. Hydrogenic Basis States for Parabolic and Spherical Bases
n
Intn2m) basis
Inlm) basis
1 1(00) 1(00)
2 11(0) = a!b1 1
12(0)= -(a1b! -abr)
v'2
1010) = a1b! 1
1210)= -(a1b! +a!b1)
v'2
1(01) = a1b1 1211) = a1b1
100-1)=a!b! 121- I) = a!b}
generatorsJ 3 andJ'3 defined in Eq. 21.44 are just given by
J 3In 1 n 2 m>=! (n 2 + m - n 1 )ln 1 n 2 m>= m 1 ln 1 n 2 m>
and
Jlnln2m>=! (n 1 + m - n 2 )ln 1 n 2 m>= m 2 ln 1 n 2 m>
(21.93)
Putting the resulting values of m 1 and m 2 in Eq. 21.84 gives
Inlm>
= (_I)m(21+1)1/2 ( (n-I)/2
!(n 2 + m - n 1 )
(n-I)/2
!(n 1 +m- n 2 )
_1m )In\n 2 m>
(21.94)
The basis states for n = I and 2 are given in both schemes in Table 21.3.
The scattering states In, n 1 - n 2 , m> diagonalize the operators L S6 ' L 34 ,
and L 12 , and follow trivially from those of the parabolic basis, so they are
not discussed further.
The angular-momentum states Inlm> may be realized in position space
by the wave functions 280
I 2
1/;nlm(r) = Nn/e- r / n ( : ) F( - n + l+ 1,21 + 2,2 : )
(21.95)

318 CASE STUDY II: THE HYDROGEN ATOM
with
1/2
2/+ I [ (n + I) ! ]
N n /= (2/+ 1)!n 2 (n-I-l)!
(21.96)
On the other hand, the parabolic states In l n 2 m) are realized in terms of the
parabolic coordinates (,,,.,)
=r+z
".,=r-z
(21.97)
by the wave functions
Iml/2
",I, (  'n Ih ) = N e imCPe - i< + TJ) 12TJ ( "., )
'rnln2 m , ." 't' nln2 m 2
n
Llml ( i ) L1m l ( 11 )
nl+lml n n2+lml n
(21.98)
where <l> is the azimuthal angle and
( _ I) n 2 -<m- 1 m l )/2 [ ] 1/2
n I! n2!
N = (21.99)
nlnlm n 2 (n l + Iml) !3(n 2 + Iml) !3'1T
EXERCISES
21.4. Show that for the boson representation of SO (4, 2) the Casimir operators for
the SO(2, I) and SO(3) groups contained in the direct-product subgroup
SO(2, I)X SO(3) have the forms
C=L 56 (L 56 -1) -N +N_
(21.100)
and
C'= L3(L3 -I) + L+L_
(21.101)
respectively, with L3 = L 12 , L = L 23 + iL 31 , and N  = L45 + iL46'
21.5. Show that for the above case the eigenvalues of C and C' are identical to
1(/+1), and hence that under SO(2,I)XSO(3) the hydrogenic states span
the representations
D+(-I-I)XD(/)
(21.102)
and the eigenvalues of L56 are
n = 1 + 1,1 + 2,. . .
(21.103)

A COORDINATE REALIZATION OF SO(4,2) 319
21.6. Use the above results to show that eigenfunctions transforming as D + ( -I-
I) may be completely specified by the quantum numbers n,l, and those
transforming as D( I) by I, mi'
21.7. Show that the Casimir operators of the 80 1 (2,1) X 80 2 (2, 1) subgroup of
80(4,2) are
Cl=Nl ( Nl_l ) -NlNl
3 3 + -
(21.104)
and
C 2 =N 2 ( N 2 -1 ) -N2 N 2
3 3 + -
(21.105)
with eigenvalues for parabolic states Inln2m) of
4C 1 Inln2 m )= (I m I 2 -1)l n ln2 m )
4C 2 l n ln2 m ) = (Im1 2 - 1 )lnln2m)
(21.106)
(21.107)
21.12 A COORDINATE REALIZATION OF 80(4,2)
The generators of 80(4,2) may be realized in terms of the six real
variables X; (i = 1,2,... ,6) by writing
Lab=i( gaaxa a: b -gbbxb a:a ) (21.108)
It is readily verified that the Lab satisfy the 80(4,2) commutation relations
given by Eq. 21.76 and hold invariant the real quadratic form  gaa x ;,
The richness of the Lie algebra so (4, 2) can be further demonstrated by
complementing the generators of 80(3, 1)" (L,M) with the four operators
P.=L 4 ,-L 6 .
I I I
( i = I, 2, 3 )
(21.109)
(21.110)
H=L 45 -L 65
Under commutation we find
[P;,lj] =0
[P;,H] =0
[L;,H] =0
(21.111)
[L;, lj] = if.ijkPk
[ M" P. ] =i..H
I J Y
,
[M;,H] =iP;
and thus the ten operators L, M, P, and H close under commutation to
generate the algebra associated with the Poincare (or inhomogeneous
Lorentz) groUp.53,318,320,322,330-332 If the P; and H are allowed to act on the

320 CASE STUDY II: THE HYDROGEN ATOM
variables (X 1 ,X 2 ,X3'XS)' we find that the Pi produce spacelike translations,
while H produces a timelike translation.
The Poincare group is the group generated by infinitesimal rotations (L),
Lorentz transformations (M), space translations (P), and time translation
(H), and has T4  SL(2, C) as its universal covering group. The group
including space, time, and space-time reflections is usually termed the
extended Poincare group. It is apparent from Eq. 21.111 that the Poincare
group has the inhomogeneous rotation group (or Euclidean group) as a
subgroup.
EXERCISE
21.8. Demonstrate that
[M3,P'L] = iHL3 + t(M +P _ - M _P +)
[K+,P.L] =iHL+ + M 3 P + - M +P 3
and hence that the two operators
C 1 =H 2 - P;
(21.112)
and
C 2 = W IL WIL
(21.113)
with
WIL = (P.L,HL+PXM)
(21.114)
are invariants of the Poincare group.
21.13 A PHYSICAL REALIZATION OF 80(4,2)
The generators of SO(4,2) admit many realizations. For the particular
case of the hydrogen atom the set of generators IS, 333
L=rxp
A= !rp2 -p(r'p) - 1 r
M=!rp2-p(r.p)+!r
r=rp
(21.115)

TILTED STATES OF THE HYDROGEN ATOM 321
T=r'p- i
r 0 = ! ( rp2 + r )
8= !(rp2-r)
provides a convenient realization. The Casimir invariants have, in terms of
this realization, the eigenvalues
C 2 =L2+A2-M 2 -r 2+8 2 _ T2-r =-3
C =0
3
(21.116)
C 4 =0
21.14 TILTED STATES OF THE HYDROGEN ATOM
The generators 8 = L 46 , T = L 4S ' and r 0 = LS6 clos under commutation
on the o(2, 1) energy-spectrum-generating sub algebra of so (4, 2). The
SchroJinger equation for the nonrelativistic hydrogen atom is
( p; -  - E )1/1=0 (21.117)
This equation may be expressed linearly in the group generators of
80(2, 1) by first left-multiplying by r = LS6 - L46 to give
[(En -! )L S6 - (En +! )L 46 + 1 ]nlm =0
(21.118)
As we saw in Section 18.3, we may diagonalize either the generator L46 or
LS6 by use of the nonunitary transformation
- 1 . 8L
.1, - e I n 45.1,
't' nlm - N 't' nlm
(21.119)
where N is an as yet undetermined normalizer and the states Inlm) are
tilted states with respect to the 80(4,2) basis states Inlm). The operator L 4S
is consequently termed the tilting operator. 33 4-338
If Eq. 21.118 is left-multiplied by e-;On L 45 and use made of Eq. 21.119, we
obtain
[ (En - t) { cosh9 n LS6 + sinh9 n L 46 }
- (En +!) { cosh9 n L46 + sinh9n L S6 } + 1] t[;nlm = 0
(21.120 )

322 CASE STUDY II: THE HYDROGEN ATOM
The coefficient of L46 vanishes for
On=t 1n ( 21J
(21.121)
to leave
[ { (En - ! ) cosh (} n - ( En + ! ) sinh (} n } LS6 + 1 ] 1/Inlm = 0 (21.122 )
The basis Inlm) diagonalizes the operator LS6 with eigenvalues n to yield
the usual discrete energy levels En = - 1/2n 2 with
(} = -Inn
n
( 21.123 )
so that En < O.
For En >0 we must diagonalize L46 by using a continuous basis IAlm)
with L46 having a continuous spectrum A. In this case our tilted states are
IXim> = 1- e ;/J.L 4 'IAlm > (21.124 )
N
It is important to note that while the basis states Inlm) form a complete set
by themselves, the tilted states form a complete set only if Inlm) and IXlm)
are taken together, this being a direct result of the non unitary nature of the
tilting operation. 109
The basis states Inlm) form an irreducible representation of the sub-
group 80(4,1) or 80(4,2), which describes all bound states in the rest
frame. The continuous basis states IAlm) form an irreducible representa-
tion of the subgroup 80(3,2) and describe all the scattering states in the
rest frame.  36
The physical states are the tilted states liilm), which are certain admix-
tures of the basis states Inlm). The normalization requirement
<nlmlnlm) = 1
( 21.125 )
implies that
<nlml(L s6 - L 46 )lnlm)= 1
( 21.126 )
Applying of Eq. 21.119 and noting that
e-i8nL4s(Ls6 - L46)ei8nL4s = n(L S6 - L 46 )
(21.127)
We obtain the normalization factor N = n. Hence Eq. 21.119 may be
written as
liilm) = .!ei8nL45Inlm)
n
(21.128)

A DILATATION-OPERATOR REALIZATION OF SOI(2, I)XS0 2 (2, I) 323
where On is as in Eq. 21.123. The normalization given by Eq. 21.126 is a
direct consequence of our linearizing the Schrodinger equation in terms of
the group generators by left multiplication by r in Eq. 21.1 i 8.
EXEROSE
21.9. Verify that
e - ifJ n L 4S (LiS - L i4 )e ifJ n L 4S = n (LiS - L i4 )
(21.129)
and
e - ifJnL4s L e ifJnL4S - L
i6 - i6
(21.130)
wi th i = 1, 2, 3.
21.15 A DILATATION-OPERATOR REALIZATION OF 80 1 (2, 1  X 80 2 (2, 1)
Ultimately we would like to be able to translate the electric dipole
operator into a form involving just the group generators and group
elements of SO(4,2) and thereby obtain a complete group-theoretical
description of the transition probabilities. As we have already seen, the
group generators couple states only with dn ='0, + 1. However, it is well
known that dipole transitions can occur over the entire spectrum of n.
Thus we are motivated to find a realization that will display this fact of
observation. To this end we first seek a realization of the generators of
80 1 (2, 1) X S02(2, 1) in position space.
We start by noting the associated Laguerre-polynomial recursion rela-
tions 339 ,340
nl+m+l ( a )
L:;+m+'()= n,+l  a +n,+m+l- Ln+ma)
(21.131)
and
L:; +m-' () = (n,  m )2 ( aa - n, )L:,' +m()
(21.132 )
The associated Laguerre polynomials occurring in the parabolic states
\[;nln2 m (, 11, <J» defined in Eq. 21.98 are of course functions of / nand 11 / n
rather than just  and 11. The ladder operators N and N; given in Eqs.

324 CASE STUDY II: THE HYDROGEN ATOM
21.91 and 21.92 may be realized by writing 28S ,288,289
I ( n ) 2 [ a  IL I2 1 ]
N+=- n+l D n /(n+l)  a -+ 2n +2+nl+1
(21.133)
I ( n ) 2 [ a  IL I2 1 ]
N - = - n -I Dn/(n-O  a + 2n - 2 - n]
and
2 ( n ) 2 [ a 1] IL I2 1 ]
N + = n+ 1 D n /(n+l)'" a." -.,,+ 2n + 2 +n 2 + 1
( 21.134 )
2 ( n ) 2 [ a 1] 1 L I2 1 ]
N - = n-I Dn/(n-O'" a." + 2n - 2 -n 2
where the eigenvalues of IL I2 1 are just Iml.
The factor in square brackets first acts on \[Jnln2 m to change the lower
index of one or the other associated Laguerre polynomial by + 1, but
without changing their functional dependence from one on / nand 1] / n
to one on /(n + 1) and 1]/(n + 1). The operator Dn/(n-z.l) is referred to as
the dilatation operator, and is defined so that
D J( x ) = j( ax )
(21.135)
and in our case yields the desired change in the functional dependence of
the associated Laguerre polynomials. The factor [n/(n + 1)]2 is inserted to
permit the transformation N n n mN n + I n m or N n n n N n n + 1m' as the
1 2 1- 2 1 2 1 2-
case may be.
If the results of Eqs. 21.133 and 21.134 are returned to Eqs. 21.85 and
21.86, it is possible to obtain a realization of all the operators L 34 , L 3S ' L 36 ,
L 4S ' L 46 , and LS6' The commutation relations of these operators with Lij
and L i4 then permit a realization of the remaining operators LiS and L i6 .
EXEROSE
21.10. Verify that the operators N and N realized in Eqs. 21.133 and 21.134,
acting on parabolic states \[Inln2m(,1l,CP), produce the results given in Eqs.
21.91 and 21.92.

THE ELECTRIC DIPOLE OPERATOR 325
21.16 THE ELECfRIC DIPOLE OPERATOR
A calculation of the probability of electromagnetic transitions requires a
knowledge of the matrix elements of x and p. If we can obtain x and p as a
function of the group generators of SO (4, 2), then the problem of comput-
ing transition probabilities is amenable to a complete group-theoretical
treatment.
Again it is simplest to start with the parabolic coordinates (, 1], cp) with
r= t(+1])
z=t(-1])
We first compute the action of z on \[Inln2m(,1],CP) and then generalize to
the components of x. We start with the recursion relation 339 ,34o
-n +1
zL n m +m(z) = I 1 Ln m +m+I(Z) + (2n , +m+ 1 )Ln m +m(Z)
1 n +m+ 1 1
1
- (n 1 +m)2Ln+m_l(Z)
(21.136)
and proceed to
 - n1.rr I Nnln2m
n tfln,n2 m = n l + Iml + 1 N D(n+ O/ntfln, + In2m + (2n , + Iml + 1 )tfln,n2 m
nl+ln2 m
N
2 n I n 2 m
-(n1+lml) N D(n-1)/n\[lnl-ln2 m
nl-ln2 m
(21.137)
and
1} n 2 + I Nnln2m
n tfln,n2 m = n 2 + Iml + 1 N D(n+ O/ntfln,n2+ 1m + (2n 2 + Iml + 1 )tfln,n2 m
nln 2 +1m
N
( I I ) 2 n I n 2 m
- n 2 + m N D(n-l)/n \[In1n2-1m
nln2- 1m
(21.138)
Recalling the action of N; and N; in Eqs. 21.91 and 21.92, we can write
_ ( (n + 1)2 I (n _1)2 I )
n,n:zm - n Dn/(n-O N + + L34 + L56 + n Dn/(n+ oN -
x \[Inln2 m
(21.139)

326 CASE STUDY II: THE HYDROGEN ATOM
and
( 2 2 )
(n+l) 2 (n-I) 2
1J'/Inln2m = n Dn/(n-O N + - L34 + LS6 + n Dn/(n+ oN -
x \[Jnln2 m
(21.140)
We can obtain expressions for rand z by addition and subtraction (and
generally for x; by rotational symmetry): 28S, 289
x.=
I
2 2
(n+l) . (n-I) .
2n D(n+ O/n(L;s -ZL;6) + L;4 + 2n D(n-l)/n(L;s + ZL;6)
(21.141)
(n+I)2 . (n_I)2 .
r= 2n D(n+O/n(L46+zL4S)+Ls6+ 2n D(n-O/n(L 46 -zL 4S )
(21.142 )
The matrix elements of x; are found by noting that L;s + iL;6 ladder n by
+ I and making use of a complete set of states:
<n'I'm'lx;lnlm>
2
= <n' l'm'ID(n+ l)/nln + 1/' m')<n + II' m'IL i5 - iL i6 lnlm) (n ;n I )
2
(n-I)
+ <n' l'm'ID(n-O/nln -II' m'><n -II' m'IL iS + iL;6Inlm> 2n
+ <n'I'm'IL;4I n1m >
( 21.143 )
The calculation of the matrix elements of x; in terms of the group
generators of 80(4,2) would be complete but for the need to compute the
matrix elements of the dilatation operator. Barut and Kleinert 28S ,289,334
have shown, by consideration of the fiber space of the hydrogenic wave
functions, that the matrix elements of D(n-z.l)/n may be expressed in terms
of integrals over hypergeometric functions. These integrals may be
evaluated recursively and the matrix elements identified with finite

THE ELECTRIC DIPOLE OPERATOR 327
SO(2, 1) transformation matrix elements, to yield finally334
(n'lmID(n:!: l)/nln + 1/m)( n + 1)2/ n = + o8 (n'lmle- i /J...L 4S ln + 1/ m)
SI n'n
(21.144)
where
8n o n = In( :' )
(21.145)
If this result is used in Eq. 21.143, we obtain the important result
(n'I'm'lxilnlm) = tiL n' (n'l' m'le -./J...L 4S L i6 Inlm) + (n'l' m'IL i4 lnlm)
(21.146)
where
- 1 1 n 2 - n,2
w =-+-=
n'n 2n2 2n,2 2n 2 n'2
is the Rydberg frequency for a transition nn'.
We recall that L45 can connect states differing in n by + 1, and hence
e-;(Jn'n L 45 can connect states over the whole spectrum of n, in agreement
with observation. Since under parity x is an odd operator, we will obtain a
nonzero result for Eq. 21.146 only if 1+1' is odd. The matrix elements of
the first term on the right-hand side of Eq. 21.146 vanish for n = n', while
those of the second term vanish for n =t=n'.
In practice it suffices to calculate the matrix elements of z, as we may
obtain the matrix elements of x and y by use of the Wigner-Eckart
theorem, and the entire dependence of the matrix elements on the m
quantum numbers is contained in a 3j-symbol. That is, we calculate
<n'I'mlzlnlm> and use
<n' I'mlzlnlm> = <n' I'mlx o (l)lnlm>
= ( _ 1) /' - m ( I'
-m
1
o
 ) (n'I'llx(1) II nl)
to obtain the reduced matrix element <n'I'llx(l)IInl>, and thence
/' m, ( I'
<n' l'm'\x(l)lnlm> = ( -1) -
P ,
-m
1
p
 )(n'I'lIx(l)lInl) (21.147)

328 CASE STUDY II: THE HYDROGEN ATOM
Table 21.4. Nonzero Matrix Elements of L 34 , L 36 , and L 45 for Hydrogenic
Wave Functions
[ 2 ] 1/2
(l-m+ 1)(I+m+ 1)[ (1+ 1)2_ (n-l) ]
(nl+ ImIL 34 Inlm)=i 2
4(1+ 1) -1
[ (12-m2)[12-(n-l)2] ] 1/2
(n 1-1 mlL34lnlm) = - i 412 -1
1/2
i [ (12- m 2)(n+l)(I+Ij:l) ]
(n j: 11-1 mlL 36 l n lm) = j: 2 1 2
4 -1
1/2
i [ [( I + 1 ) 2 - m 2 ] ( n j: I j: 1 ) ( n j: I j: 2) ]
(n j: 11 + 1 mlL 36 lnlm) = j: 2 2
4(1+ 1) -1
i 1/2
(n j: 11 miL 451 nlm ) = j: 4" [ ( n - I) ( n j: I j: 1) ]
Thus for practical calculation it suffices to have available the hydro genic
matrix elements of just L 45 , L 34 , and L36' Using the results of Ferreira,290
we have the nonzero matrix elements given in Table 21.4. Hence we have
reduced the calculation of the matrix elements of x for hydrogenic wave
functions to a purely algebraic process involving the group generators and
group elements of SO(4,2). The calculation of these matrix elements is
important not only for computing electric dipole transition probabilities)
but also in treating the Stark effect.
The matrix elements of the momentum operator p may be obtained
directly from those of x by use of the familiar result
p;=i[H,x;]
( 21.148 )
Remembering that the eigenvalues of H are just - 1/2n 2 and that H
commutes with all L;4' we have from Eq. 21.146 and 21.148 that
<n' l'm'lpil n1m ) = n' <n'I'm'le-iIJ.,.L45Li6Inlm)
( 21.149 )
Recalling the definition of the tilted states given in Eq. 21.128, we see that
in terms of the tilted states Inlm> we may represent the momentum

GALILEAN BOOSTS 329
operator p by the replacement
PiLi6
( 21.150 )
This completes our representation of the position (x) and momentum (p)
vectors in terms of the group generators and elements of SO (4, 2). In
principle the calculation of electromagnetic transition probabilities and the
Stark effect has been reduced to an algebraic calculation, with a complete
elimination of the need to compute integrals. We note that this accomp-
lishment was possible only by enlarging the group from SO(4,1) to
SO (4, 2).
21.17 GALILEAN BOOSTS
So far we have restricted our discussion to the rest-frame states of the
hydrogenic atom. A tilted rest-frame state (O) may be boosted to a total
momentum p by application of the operator e ipor to give 341
(p) = eipor(o)
(21.151)
In terms of the physical realization of the SO(4,2) generators given in
Section 21.13 we have
r=M-A
(21.152)
and hence
(p) = eiP'(M-A)(O)
(21.153)
The set of six operators L, M - A satisfy the commutation relations
[L i , Lj] = i€ijkLk
[Mi-Ai,-Aj] =0
( 21.154 )
[Li' - Aj] = i€ijk(M k - A k )
which are just the commutation relations satisfied by the group involving
pure rotations and Galilean transformations. 341,342 Hence we shall refer to
the operators M - A as Galilean boosting operators 286 and e-p'(M-A) as the
Galileo boost operator that takes a system from a rest frame to a moving
frame.
As a simple illustration of the application of the Galileo boost operator,
consider the transformation of the rest-frame hydrogenic equation
[ (En - t ) L56 - ( En + ! ) L46 + 1 ]nlm (0) = 0
(21.118)

330 CASE STUDY II: THE HYDROGEN ATOM
to a moving frame. If we premultiply by eip'(M-A) and use Eq. 21.151 to
boost (O) to (P), we have
[( En -! - !p2) L56 - (En +! - !p2) L46 + p.r' + 1 ]nlm(P) = 0 (21.155)
where we have made use of the results 286
eip.(M-A)L e-ip.(M-A)=L - p .r'+l p 2 ( L - L )
56 56 Z 56 46
( 21.156 )
and
eip.(M-A)L e-ip.(M-A)=L - p .r'+.l p 2 ( L - L )
46 46 2 56 46
(21.157)
The coefficient of L46 may be reduced to zero by performing a rotation
in the 4-5 plane using the tilting operator e -i9L45 with
o =! In (p2-2En)
(21.158)
and noting that L45 commutes with p .r'. Then we have
[ (p2 - 2En) 1/2 L56 +p. r+ 1 ] t¥n1m (p ) = 0
(21. \59)
The term involving p' [' may be eliminated by rotating the states with the
operator e- IA ' M , where
Pi
tanh Ai = 1/2
(p2 - 2En)
( i = 1, 2, 3 )
( 21.160 )
leading to recovery of the rest-frame energy. In terms of the moving
Galilean frame, the energy E is related to the rest-frame energy En =
- 1/2n 2 by
p2
E=En+ 2:
(21.161)
the second term being just the kinetic energy of the hydrogenic atom as a
who/e. Hence the introduction of the Galilean booster allows us to go from
the Schrodinger equation for the hydrogen atom at rest to a nonrelativistic
equation of motion for the hydrogen atom treated as a single entity.

LORENTZIAN BOOSTS 331
EXERCISES
21.11. Use the Campbell-Hausdorff formula (Eq. 3.26) together with the 80(4,2)
commutation relations (Eq. 21.76) to show that
e {5 L j6 e - ;5 = Lj6 cosh  - L56 sinh 
e;5L56e-;5= L 56 coshA j - Lj6cosh
and for k=l=j,
(21.162)
(21.163)
e ;Lk5 L e -;Lk5- L
;6 - ;6
(21.164)
21.12. Use the above results to derive Eq. 21.160.
21.13. Show that in the presence of an electromagnetic interaction the replace-
ments (atomic units)
pp - a ,
EE --r.A 0
(21.165)
in Eq. 21.155 leads to additional interaction terms of the form 343
-(i . I -c..Ao(L56 - L 46 ), fp.a(L 56 - L 46 ), and - fa 2(L56 - L 46 )
(21.166)
21.18 LORENTZIAN BOOSTS
The three operators Lij (L) form the generators of the rotation group
SO(3). This group may be enlarged to the homogeneous Lorentz group by
the introduction of the three additional generators L i5 (M) which may be
associated with pure Lorentz transformations.
If the four-momentum in the rest frame is
pit = M ( 1,0, 0, 0 )
(21.167)
then under a Lorentzian transformation to a moving frame characterized
by velocity components Vi (i= 1,2,3) the four-momentum becomes
Pit = M( cosh,  sinh)
( 21.168 )
where M is the total mass of the atom and 344
i=tanh-l ( ; )
are the rapidity parameters of special relativity with  = ((. Ql/2.
( 21.169 )

332 CASE STUDY II: THE HYDROGEN ATOM
The rest-frame relativistic wave function (O) may be boosted to the
momentum of Eq. 21.168 by the Lorentzian boosting operator d'M to give
() = ei'(O)
= eiE'Mei9L4(O)
( 21.170 )
(21.171)
The operators (L and M) of the Lorentz group all commute with L 46 ,
and hence L46 is a Lorentzian scalar and
ei.ML46eiE-M= L46
( 21.172 )
The operators r It = (L 56 ,r) form a Lorentzian four-vector, and
r It
i'M L ie-M _ ,JJ
e 56 e - M
(21.173 )
Thus th typical equation
(aL 56 + bL 46 + c )(O) =0
(21.174)
encountered in the nonrelativistic hydrogen atom may be Lorentz boosted
to give the relativistically covariant wave equation
(a'r pi' It + bL 46 + c )() = 0
(21.175)
where a' = a / M.
The above equation contains generators of 80(4,2) that may be realized
in terms of infinite-dimensional unitary matrices or finite-dimensional
nonunitary matrices. In the former case we must consider infinite-
component wave functions, and in the latter finite-component wave func-
tions as in the Dirac wave equation. 202
21.19 INFINITE-COMPONENT WAVE EQUATIONS
Infinite-component wave equations were first considered in 1932 by
Majorana 345 ,346 and rediscovered and generalized by Gel'fand and Yag-
lom 276 ,277,347 in 1948. Majorana introduced the infinite-component wave
equations of the type
(Pltrlt-K)(p) =0
(21.176)
primarily as an attempt to produce a relativistically covariant wave equa-
tion that avoided the negative masses that appeared to be plaguing the
Dirac four-component relativistic wave equation. Interest in Majorana's

INFINITE-COMPONENT WAVE EQUATIONS 333
wave equations was quickly lost when the Dirac equation was reinterpreted
shortly thereafter. 202
In recent times there has been a great revival of interest in the Majorana
infinite-component wave equation and its generalizations. 15,284,287,348-358
This revival has largely come about from the desire to construct wave
equations appropriate to the description of the observed mass spectrum of
the hadrons and in consideration of the possibility that hadrons are
composite structures having infinitely many excited states. 308,359 In the
absence of detailed models of hadron structure and interactions, it is
natural that great attention has been devoted to the mass spectrum of the
excited states of the much-studied hydrogen atom. 285-289,334,360-362 Here
we shall limit our attention almost entirely to the mass spectrum of the
hydrogen atom.
Majorana's treatment of infinite-component wave equations was limited
to wave functions spanning infinite-dimensional representations of the
Lorentz group. The equation
(plLflL - K )(p) = 0
(21.177 )
is linear in the group generators of SO (4, 2). Its mass spectrum may be
readily determined by transforming to the rest frame:
(ML56-K)(O)=0
(21.178)
Since the eigenvalues of L56 are just n, we obtain the mass spectrum
K
M=-
n
(21.179 )
which implies that the mass decreases with increasing n, so that states of
higher n are more stable than those of lower n. In this sense the mass
spectrum is inverted with respect to that found for hydrogen.
Under the substitutionPILPIL -(lIL' we have
[flL(P IL -aIL) - K ](p) =0
( 21.180 )
giving rise to the electromagnetic interaction term flL, where flL is identi-
fied as the current operator. As a result the four-vector flL is frequently
referred to as the algebraic vector current operator. 308
The Majorana equation, extended to SO(4,2), is clearly unsuitable as a
source of realistic mass spectra, and as a result considerable effort has
been devoted to constructing, admittedly in a somewhat ad hoc manner,
infinite-component wave equations of greater generality. These extended

334 CASE STUDY II: THE HYDROGEN ATOM
Fig. 21.3. Interaction vertex for electromagnetic interactions.
wave equations normally involve the representation of the interactions (in
our case electromagnetic interactions) in terms of a four-vector current
operator jp, constructed from the generators of the dynamical group and
the momentum operators
Pp, = Pp, + p;,
,
qP, = Pp, - Pp,
(21.181)
where Pp, and P are the momenta of the ingoing and outgoing particles, so
that qP, is the momentum carried by the participating photon, as illustrated
in Fig. 21.3.
We have already seen that four-vector currents that include just the
algebraic four-vector current rp, lead to an inverted mass spectrum. This
defect may be remedied by adding to the algebraic current" operator
nonalgebraic or convective currents 308 ,337 that are proportional to the
momentum p. The inclusion of convective current operators not only leads
to a mass spectrum that increases with increasing n, but also yields, in
contrast with the Majorana equation, magnetic moments of the correct
sIgn.
Particular attention has been given to the study of the so-called minimal
linear conserved currents, 337,363 which are linear in both the group genera-
tors and the momenta pp,. The simplest minimal linear conserved current
for SO(4,2) may be taken as 356
jp, = aIr p, + a 2 Pp, + a 3 Pp,L 46 + a 4 Lp,pqP (21.182)

INFINITE-COMPONENT WAVE EQUATIONS 335
The normalization of the tilted states requires that 308 ,364
<nlmIJoln' 1m) = nn,q
( 21.183 )
for all n, where q is the charge associated with the tilted states. The usual
conservation of currents requires that
<Ii' 1m; p'lq,jltliilm; p) =0
( 21.184 )
The currents qlt and LItP PP are not conserved 337 and hence are absent from
Eq. 21.182. The term a 4 Lp.P q P is always conserved and does not contribute
to the mass spectrum or to the charge, though it does contribute to the
magnetic moment. As a result the last term of Eq. 21.182 is frequently
omitted, leaving just the current
J it = aIr It + a 2 P It + a 3 P lt L 46
( 21.185 )
The four-vector current of Eq. 21.185 has found extensive application,
not only to the hydrogen atom, but also to the derivation of the mass
spectra, magnetic moments, and form-factors of hadrons. 308,337,364--368 The
parameters a; of Eq. 21.185 are not entirely arbitrary. The normalization
requirement of Eq. 21.183 with the current of Eq. 21.185 implies that
q=  (a)n ooshO n +2m n a 2 +2m n a 3 n sinh On )
N n
( 21.186 )
where m n is the mass of the composite system, N n is the normalization
constant for the tilted states, and f)n is the tilting angle. It is important to
note Eq. 21.181, which leads to the appearance of the factors of 2 in Eq.
21.186.
The conservation of currents, Eq. 21.184, implies that
m n <iiIJoe;MI Ii') = mn,<lil eiE'MJol Ii')
(21.187)
which leads after tedious manipulation to the mass equation 337
-I [
2 2 2a 2 ya 2
m n = (2a 3 + 7 ) a) + 2fJa 3 + 2-;;;-
1/2
+ ( ( ai+2fJa 3 +2 Y:2 f -4( 13 2 + :: )( a+ : ) ) ] (21.188)

336 CASE STUDY II: THE HYDROGEN ATOM
where
f3 = a 2 m n tanh8n + a 3 m;
(21.189)
-I 2
'Y = alm n ( cosh8n) + a 2 m n
(21.190)
and
8 = . nh - I ( f3 - a 3 m ; )
n SI n 2
'Y - a 2 m n
(21.191)
which gives the mass spectrum as a function of the six parameters
(a I' a 2 , a 3 , 8n' f3, 'Y).
The mass spectrum obtained in Eq. 21.188 was derived as a direct
consequence of the conservation of the current. An identical mass
spectrum may be obtained by considering the infinite-component wave
equation
(jP - f3L 46 - 'Y )(p) = 0
( 21.192 )
with the current as given in Eq. 21.185. For later convenience we shall
interpret P_ as the total momentum of a composite particle, as in Eq.
21.168, and  (P) as its wave function. We may transform Eq. 21.192 to the
rest frame by a Lorentzian boost as in Eq. 21.170 to give
[( a l L 56 + a 2 m + a 3 mL 46 )m - f3L 46 - 'Y ](O) =0
(21.193)
The rest-frame wave equation may now be diagonalized with respect to L56
by the tilting operation
(O) = ei9nL4(0)
( 21.194 )
to yield
[alm n cosh8 n + (a 3 m; - (3) sinh8n]n = - (a 2 m; - 'Y) (21.195)
with
8n = tanh -I ( _ a 3 m; - f3 )
alm n
( 21.196 )
which implies that
( 2 2 ) ( 2 2 ) -1/2
cosh8 n = alm n alm n - a 3 m n - f3
(21.197)
sinh On = - (a 3 m;- /.n [ a:m; - (a 3 m;- p)2] -1/2
(21.198 )

EXAMPLE OF HYDROGEN 337
and hence we obtain the mass spectrum for bound states as
[ 2 2 ( 2 ) 2 ] 1/2 ( 2 )
lXlm n - lX2mn-f3 n=- lX2mn-Y
( 21.199 )
which is equivalent to that of Eq. 21.188.
The corresponding formula for the scattering states follow by diagonaliz-
ing Eq. 21.193 with respect to L46 and putting
(In=tanh- I ( - aln )
lX3mn - f3
(21.200)
to give
[ aim;- (a 3 m; - /3 )2]p2= (a2 m ;- y)2
(21.201 )
where 11 is the continuous eigenvalue of L46'
EXERCISE
21.14. Show that the Lie algebra of 80(4,2) possesses two four-vectors fit
=(L;6,L 56 ) and r = (L;4,L 45 ), and that any linear combination of fit and
f may be transformed by a unitary transformation into a term involving
just fit" As a result, confirm that terms linear in f'lt may be excluded from
the currentjlt given in Eq. 21.182.
21.20 EXAMPLE OF HYDROGEN
The hydrogen atom provides an interesting example of the application
of infinite-component wave equations, producing a highly accurate relatilv-
'istically covariant wave equation in which (in contrast with the Dirac
wave equation) neither the Coulomb potential nor the relative coordinates
are explicitly present.
We start with the infinite-component wave equation36362 of Eq. 21.192.
(jJ1P J1 - f3L 46 - Y )(p) =0
with the current as in Eq. 21.185:
jJ1 = lXlr J1 + lX 2 P J1 + lX3PJ1L46
and identify (P) as the wave function of the whole atom and PJ1 the total

338 CASE STUDY II: THE HYDROGEN ATOM
momentum of the atom. Transformation to the rest frame yields the mass
spectrum given by Eq. 21.188 as a function of the five coefficients a;, {3,
and y.
The coefficients (a.;,{3, y) are chosen so that the resultant equations for
the mass spectrum, normalization, tilting angles, matrix elements, and so
on yield the correct nonrelativistic limits. With those considerations in
mind we find 361
a} = 1,
-a
a 2 = 2m l '
1
a 3 = 2m l '
m 2 -m 2
{3= 2 }
2m}
a(mf+mi)
y=-
2m}
(21.202)
which upon insertion into the mass equation (Eq. 21.188) yields
( 2 ) }/2
m;=mf+m +2m l m 2 1- 2 a 2
n +a
(21.203 )
where a is the fine-structure constant.
If the binding energy Bn is defined as
Bn=m n -m l -m 2
( 21.204 )
we obtain from Eq. 21.203
B B 2 ( 2 ) -1/2
1+2+ n = 1+
p. 2m}m 2 n 2
(21.205)
where
m}m 2
p.=
m l +m2
(21.206)
is the so-called reduced mass.
For small a the right-hand side of Eq. 21.205 may be expanded in
powers of (a/n)2. Retaining terms to order (a/n)4, we obtain
- a 2 p.
Bn = 2n 2
3a 4 JL
8n 4
a 4 JL
8n 4 (m l +m 2 )
(21.207)
The first term is readily recognized as the nonrelativistic Schrodinger
result, while the second term is the Dirac fine-structure term 280 for levels
with n = j + t to order (a / n)4. Putting m l = mp and m 2 = me and remember-

EXAMPLE OF HYDROGEN 339
ing that mp»me' we may write the last term as
me a 4
E =---
b m 8 4
P n
which is identified with the first-order energy shift due to motion of the
nucleus (the recoll energy).280
We may readily verify that for small a and mpoo we obtain from Eq.
21.205 exactly the result of the Dirac equation for the n = j +! levels. Thus
our mass equation contains not only the Dirac spectrum but also the recoil
corrections, and hence for the n = j +! levels we obtain a more accurate
result than the corresponding one from the Dirac equation. A similar result
has been attained using the so-called eikonal approximation. 37373
The mass spectrum given by Eq. 21.203 is, unlike the Dirac equation,
valid even when (a/n)2> 1. Thus the infinite-component wave equation
leads to a solution even for strong-coupling interactions, a result that has
been exploited by Barut in his dyonium model for hadrons. 15,374,375
Recoil effects are particularly important in the spectrum of positronium,
where JL=me/2. For small a we have, to order (a/n)4,
Bn=-me( 4a2 +  :: +...)
which is close to the usual perturbative result for the 1= n - 1 levels of
parapositronium 376 when n is large.
The mass spectrum obtained in Eq. 21.203 yields only the levels with
n = j + t. The calculation of the mass spectrum including spin is possible
only by adding further terms to the four-vector current, as has been done
by Barut and Baiquni. 361 ,362 Indeed, Baiquni 362 has produced a current
that yields a mass spectrum including not only the fine-structure and recoil
terms, but also part of the Lamb shift.
(21.208)
(21.209)
EXERCISES
21.15. Show that if the four-vector currentj/t has a2=0, then with
a. = 1,
a2 = 0,
1
a3 = -,
2ml
m - m?
/3= 2m. '
y = - m2 a (21.210)
the mass spectrum becomes 360
( 2 ) 1/2
222 a
m,-;=m.+m2+2m.m2 1- n 2
(21.211 )
and hence is imaginary for (a / n )2 > 1.

340 CASE STUDY II: THE HYDROGEN ATOM
21.16 Verify that the coefficients ai' fj, Y chosen in Eq. 21.202 yield the correct
tilting angle and normalization in the nonrelativistic limit.
21.21 A FINITE-DIMENSIONAL REALIZATION OF 80(4,2)
The generators of 80(4,2) may also be realized in terms of the four-
dimensional Dirac y-matrices. 111,250,376 The y-matrices 103
It - ( 0 I 2 3 ) - ( 0 )
y = y,y,y,y = y,y
(21.212)
are defined in terms of the two-dimensional Pauli spin matrices a as
yo=( 1)' y=(__) (21.213)
and satisfy the anticommutation relations
ylLyP + yPylt = 2g ltP
where glt P is the space-time metric tensor (1, - 1, -'1, - 1):
( 21.214 )
goo = 1,
gji = - 1,
glt P = 0 (I-L =F p )
(21.215)
and
glt P = glt P
(21.216)
The indices of the y-matrices may be raised or lowered by noting that
Ylt = glt P yP
( 21.217)
to give
YO=yO and Yk= _yk
(21.218)
We also require the use of the matrix
y5 = yOyly3
(21.219)
which anticommutes with ylt. The inverse of y5 may be obtained from Eq.
21.217 as
Y5=Y3Y2YlYO
(21.220)
We may obtain a realization of the generators of 80(4,2) from the set of
matrices
Ya = ( Y I' Y 2' Y 3' - Y 5' Yo' - if )
(21.221 )

A FINITE-DIMENSIONAL REALIZATION OF 80(4,2) 341
by considering the 15 operators
I
lab = 2 Y a Yb
(a,b = 1,2,3.4_ 5 = 0,6, a <b)
(21.222)
which under commutation satisfy the same relations as found for the
boson realization of 80(4,2) in Section 21.7. (y./e use lab for the generators
of this realization to distinguish them from the generators Lab of the boson
realization.) The commutators of the lab are evaluated by noting that
Y; = - 1 Y= 1 Y; = - 1 (21.223)
and
Y/ = - Yi yJ = Yo Y = - Y5 (21.224 )
Thus the set of 15 operators lab generate a four-dimensional realization
of SO(4,2). As expected, for a noncompact group, this representation is
nonunitary. The Casimir operators for this representation follow from use
of Eq. 21.79 to give the eigenvalues
c 2 =¥
c 3 =0
C =0
4
(21.225)
showing that this representation is quite distinct from that found for the
boson representation.
The generators may be identified as earlier with
jk = Ii} = !iYiYj (spin)
OJ = Ij4 =  Y 5 Yj (analog of the Lenz vector)
I
m i = li5 = 2 YiYO (pure Lorentz transformations)
r p. = lp.6 = ty p. ( p. = 1,2,3,4) (four-vector algebraic current operator)
t = 145 = -  Y 5YO (the tilting operator)
s = 146 = - ty 5 (the Lorentz scalar operator)
All of the subgroup structures encountered earlier may be constructed
and their subgroup content determined. Thus for the spin group generated

342 CASE STUDY II: THE HYDROGEN ATOM
by 'i) we find
12==j(j+l)
(21.226)
and hence under the reduction SO(4,2)SO(3) we obtain the spin value
j= t twice. Likewise for the SO(4) subgroup we have the two Casimir
operators
12+a 2 = £
1 · a = - i iy 0
(21.227)
and hence we obtain two two-dimensional representations of SO(4) dis-
tinguished by the eigenvalues of Yo, and irreducible under SO(3). Similarly
the homogeneous Lorentz subgroup has
1 2 - a 2 = 0
1 · a = - i iy 5
(21.228)
and hence has two distinguishable two-dimensional representations.
The rest-frame states of the SO(4,2) representation may be labeled by
the eigenvalues n,jU+ I), and m of the set of commuting operators '56,1 2 ,
and '12 respectively. Thus a typical basis state will be designated as Injm],
where we use a square bracket to indicate that the four-dimensional
representation is nonunitary. A parity operator P= Yo that commutes with.
'i) and anticommutes with the m j may be introduced to label the parity of
the rest-frame states. The eigenvalues of '56' 1 2 , '12' and P are readily found
to be
n= + t
J '_ I
-2
m= + !
(21.229)
P= +
and hence the basis states may be taken as the set of four spinors
It, t,!, +]=
I
o
o
o
o
o
I
o
1- t,!,!, - ]=
1 1 1 _ 1 + ] -1
2'2, 2, -
o
I
o
o
o
o
o
I
(21.230)
I 1 1 1 ] -
-2'2'-2'- -
We notice that the parity operator has the same sign as n and hence does
not supply an additional label.

REFORMULATION OF THE DIRAC THEORY OF THE ELECTRON 343
EXERCISES
21.17. Show that the operators 378
I - _1* - I * *
ab- ab- 2 Ya Yb
( 21.231 )
form a second inequivalent four-dimensional representation of SO (4, 2).
21.18. show3 78 that under parity the generators of SO (4, 2) transform as
PlabP = gaagbb1ab
(21.232)
while under charge conjugation
CI C -1 = ( - 1 ) 8a'2+ 8 a ,o+ 8 b ,2 + 8 b ,ol
ab ab
(21.233)
21.22 REFORMULATION OF THE DIRAC THEORY OF THE ELECTRON
Barnt 377 has taken the basic states above as the starting point for a
reformulation of the Dirac theory of the electron. We sketch only the
barest details here. We first note that the operator
c=;y1.yO=;( _°(72 ) (21.234)
has the properties of the usual charge-conjugation operator,
C-1=-C=C t
(21.235)
and acting on a basis state I njm] gives
Clnjm]=2ml-nj - m]
(21.236)
which suggests that states having opposite sign of n are of opposite charge.
The requirements of current conservation demand that
[nljoln] = en (21.237)
and
(p - p)j = 0 (21.238)
The rest-frame states may be boosted to momentump=m(cosh,sinh

344 CASE STUDY II: THE HYDROGEN ATOM
by the boosting operation
Injm + ,p]= ei.mlnjm + ]
= (COSh t+. (   ) sinh H )Injm + ]
(21.239)
Using this result with Eq. 21.238 establishes for current conservation that
m n ,[ n'lioln,p] = m n [ n', - plioln].
(21.240)
If we make the additional requirement that the current be parity
conserving, then we must have
ip. = erp' + alJWqP (21.241)
The last term is always conserved and does not contribute to the charge.
Thus if Eq. 21.241 is used to define io in Eq. 21.237, we are forced to the
conclusion that the charge e must be taken as positive for states with
positive n and negative for states with negative n. If this result is now used
in Eq. 21.240, we have
(m n ,n,2 - m n n 2 ) [n'ln,p] =0
(21.242)
and hence
m n , = m n
(for all n, n')
(21.243)
from which we may conclude that the mass of the particle and antiparticle
are the same.
Thus by going to the group 80(4,2) rather than the Lorentz group
80(3, 1), we find it is possible to describe the properties of a particle-
antiparticle system in terms of a single irreducible representation where the
sign of the energy appears as an internal quantum number in the rest
frame, and hence to develop a single-particle theory of particle-antiparticle
systems with the boosting operator boosting both particles and antipar-
ticles to positive energies. Charge conjugation and parity then appear as
inner automorphisms of the group 80(4,2). It is indeed remarkable that
the same group introduced as the dynamical group of the orbital states of
the hydrogen atom reappears as the group of the Dirac matrices.
21.23 THE HYDROGEN ATOM WITH SPIN
The existence of both the boson representation with generators Lab and
the Dirac representation with generators lab provides a natural means of

THE L:UNFORMAL GROUP AND 80(4,2) 345
constructing a linear space for describing the states of a hydrogen atom
with spin by allowing the generators Lab to act on the infinite-dimensional
orbital space and the lab to act on the finite spin or space. The wave
function for the whole atom may then be labeled as sa(p)' where s labels
the infinite-component orbital eigenfunctions and (1 the four-spinors. The
momentum-boosted wave function sa(p) may then be related to the
rest-frame basis states \fIsa (0) by
sa(P) = e i €.(M+m)ei(OL 4s +4> 1 4s)\fIS(1(0)
(21.244 )
Barnt and Baiquni 361 ,362 have used wave functions of just this form in
describing the properties of the hydrogen atom with spin in terms of the
current-algebra formulation.
21.24 THE CONFORMAL GROUP AND 80(4,2)
We noted in Section 21.12 that the generators of the physically impor-
tant 10-parameter Poincare group (the inhomogeneous Lorentz group)
may be formed by adding to the six generators (L,M) of the homogeneous
Lorentz group the four generators Pp. (p.=1,2,3,4) of space-time transla-
tions.
The Poincare group may be enlarged to the 15-parameter conformal
group 379-383 of Minkowski four-space by considering the group of
coordinate transformations that leave invariant the form
ds 2 = (dXO)2 - (dXl)2 - (dX2)2 - (dX3)2 =0
(21.245)
In terms of the group generators we may enlarge the Poincare group to the
conformal group by adding five new generators. The first is the generator
of scale transformations or dilatations D, where
X'p. = pxp.
(p>O)
(21.246)
The remalmng four generators Kp. are associated with the so-called
special (or proper) conformal transformations that are essentially associated
with a change of scale from point to point and thus amount to space-time-
dependent dilatations. The special conformal transformations may be
written as the product of an inversion 383
k 2 p.
I: x'p. = x (21.247)
( xpx P )

346 CASE STUDY II: THE HYDROGEN ATOM
followed by a translation
T: x""" = x''''' - a""
(21.248)
and then another inversion
k 2 x "
I: x'"'''' = ,.,.
(x p " x"P)
(21.249)
to give the resultant coordinate transformation
x"" - a""x 2
x' ,.,. =
o(x)
(21.250)
where
o ( x ) = 1 - 2a p x p + a 2 x 2
(21.251 )
Where a 2 = apa p and x 2 = XpX p.
The generators of the conformal group may be realized as differential
operators acting on the Minkowski space y writing 38o ,383
Translations: P = i (21.252)
,.,. ax
,.,.
Lorentz rotations:
M"" = i( x,. a: v - Xv a:,. )
(21.253)
Dila tations:
D=ixP
ax p
(21.254 )
Special conformal transformations:
K . (2 p a 2 a )
,.,. = I x,.,.x ax,.,. - x ax,.,.
(21.255)
Under commutation we then find that 383
[p,.,.,p p ] =0
[P A , M,.,.p] = i( g,.,.APp - gPAP",,)
[P,.,., D] = iP,.,.
[P,.,., K p ] = 2i( gJU'D - M,.,.p)
[D,M,.,.p] =0 [D,K,.,.] = iK,.,.
(21.256)
[M,.,.p, Mop] = i( g,.,.pM po + gpoM,.,.p + gp.aMpp + gppMo,.,.)
[K,.,.,K p ] =0
[K A , Mp.A] = i( gp.AK p - gPAK,.,.)

THE CONFORMAL GROUP AND SO(4,2) 347
If in terms of the generators of 80(4,2) we write
P p. = Lp.6 + Lp.4
Kp. = Lp.6 - Lp.4
D = L64
IL = 1,2,3,5
(21.257)
M p." = Lp."
we find exactly the commutation relations of Eq. 21.256 and hence may
conclude that the group 80(4,2) is locally isomorphic to the conformal
group.
The conformal group first entered physics in a demonstration by Bate-
man and Cunningha11l 38 4-386 (c. 1908) that Maxwell's equations of the
electromagnetic field are covariant not only under the Poincare group, but
also under the larger conformal group.
Problems arise once we consider conformally invariant equations of
motion for massive particles and for Maxwell's equations including
sources. This difficulty can be realized by noting that [Pp.' K,,] does not
commute with the momenta Po and hence conformal transformations do
not take momentum eigenstates into momentum eigenstates. Under the
action of the dilatation operator D we readily find from Eq. 21.256 that
Pp.pp. becomes
e i8D p pp.e- i8D = e 28 p pp.
p. p.
(21.258)
and hence Einstein's rest mass VPp.pp. cannot be an invariant under
dilatations. Indeed, exact dilatation symmetry is possible only if the mass
spectrum is continuous or if all masses are zero. Thus generally we would
expect conformal symmetry to be broken.
Barut and Haugen 383 have recently suggested a possible way out of the
scale-invariance dilemma by noting that the Newtonian mass m is not
invariant under Lorentz transformations and takes on continuous values,
and that Lorentz invariance was maintained in relativity by defining the
relativistically invariant rest mass mo' They suggest that instead of taking,
under conformal transformations, continuously varying rest masses mo, we
should attempt to define a new mass moo that is conformally invariant. In
this manner Barut and Haugen have been able to develop new forms of the
Maxwell equations with source terms and new equations of motion that
are conformally invariant.

348 CASE STUDY II: THE HYDROGEN ATOM
EXERCISE
21.19 Show that the 80(4,2) realization of the conformal group contains two
Poincare subalgebras.
21.25 CONCLUDING REMARKS
The case study of the hydrogen atom has taken us through vast tracts of
recently explored applications of both compact and noncompact groups
and their associated algebras. Our treatment is by no means exhaustive,
and the subject has still many unexplored areas. The implications of much
of this work for elementary-particle physics continue to be of great
interest, and the extension of the methods outlined here to many-particle
systems constitutes almost virgin territory. Is there a connection between
the conformal group as the group of space-time and scale transformations
and as the dynamical group of the hydrogen atom and related problems?
The question must be not only asked but answered.

22
Case Study III: Fermions and Shell
Structure
22.1 INTRODUCfION
As our final case study we consider the group properties of the states
and interactions associated with atomic and nuclear shells. In these cases
the Pauli exclusion principle is operative, and it is appropriate to work in
terms of fermion annihilation and creation operators.
Briefly, our mode of attack will be first to construct the basic states of a
given shell from fermion annihilation and creation operators, and then to
use the same fermion operators to obtain a realization of the generators of
a supergroup that can encompass all the states of the shell in a single
irreducible representation. The supergroup, constructed from fermion
operators, will of necessity be compact. The subgroup structure of the
supergroup is then investigated by forming appropriate linear combina-
tions of its generators. The supergroup and its subgroups are then used to
classify the complete set of eigenfunctions associated with the given shell.
If required, the symmetrized eigenfunctions may be constructed as par-
ticular linear combinations of the basic states of the shell.
Next the relevant interactions are expressed as linear combinations of
the fermion operators. These interactions so expressed are then
symmetrized with respect to the representations of the same group as used
to symmetrize the states.
349

350 CASE STUDY III: FERMIONS AND SHELL STRUCTURE
With the states and the interactions symmetrized according to a com-
mon group scheme, it becomes possible to exploit fully the Wigner-Eckart
theorem to calculate matrix elements of the symmetrized interactions
placed between the symmetrized states. The selection rules for the matrix
elements follow those for the relevant Wigner coupling coefficients.
Furthermore, we can frequently use the Wigner-Eckart theorem to set up
simple relationships between various sets of matrix elements. Finally, with
the introduction of quasi-spin it becomes possible to encase the depen-
dence of the matrix elements of the symmetrized interactions on the
number of particles in a single coupling coefficient.
22.2 STATES OF A FERMION SHELL
Normally, for reasons of simplicity and brevity, we consider an identical
fermion ii-coupled shell involving (2) + 1) fermion configurations} N with
N = 0, 1,2, . . . ,2} + 1. Extensions to mixed shells with configurations of the
type }N 1 }N 2 , to LS-coupled shells, or to shells involving two types of
fermions (e.g., protons and neutrons) create few extra problems.
The states associated with a single-particle orbit of the ii-coupled inde-
pendent-particle shell model 387 may be labeled by the set of quantum
numbers a = n}m. In general we use Greek letters to designate these trios of
quantum numbers. The anti symmetrized N-particle ii-coupled states may
be represented as normalized determinantal product states, which in the
second-quantization picture may be regarded as being created by a
sequence of N fermion creation operators acting on the vacuum
state:41, 166, 388
a!aZ.. .aLIO> = {a,p,...,w}
(22.1 )
Taking the adjoints of Eq. 22.1, we have
<Ola w " .apaa = {a,p,...,w }.
( 22.2 )
The fermion annihilation and creation operators obey the usual rela-
tions,202
{aa,a p } = { a!,aZ} =0
( 22.3 )
and
{ a!,a p } = a,p
(22.4 )
The first anticomrnutation relations ensure that the states are antisym-

STATES OF A FERMION SHELL 351
metric. The requirement that the states be orthonormal is met if in addition
to Eq. 22.4 we take
aalo>=o and <Ola!=O
(22.5)
Finally, the Pauli exclusion principle necessitates that we take
aa=atat=O
a a a a
(22.6)
Any state of the identical fermion configuration j N can be represented
by a product of N creation operators acting on the vacuum state. Since
m = j,j -1,..., - j + 1, -j
( 22.7)
there can only be 2j + 1 distinct creation operators (am) associated with a
given nj-orbit. The Pauli exclusion principle expressed via Eq. 22.6 limits
the occupancy of a given nj-orbit to a maximum of N = 2j + 1 identical
fermions.
The number of states, , associated with a given j N configuration will
be just
(2j+ I)!
 =
N N! (2j + 1 - N) !
(22.8)
The number of states,  s, associated with a given shell nj will be equal to
the sum of the numbers of states of the configurationsjN (N=O,...,2j+ 1),
which is readily found to be
CYY _ 22j+ I
" S -
(22.9)
It is this complete set of s states that concerns us in the construction of
the supergroup.
EXERCISES
22.1. Show that Eqs. 22.4 and 22.5 are necessary if <Ola,BaaaJa!IO) is to reproduce
the result
f {a,/3 }. { y,f.} dT = 8ay8,BE - 8 aE 8,By
22.2. Show that the operator  Ea!aE acting on an arbitrary N -particle state
aJaJ.. . a 10) simply yields the eigenvalue N and hence may be regarded as a
number operator.
22.3. Verify EQs. 22.8 and 22.9.

352 CASE STUDY III: FERMIONS AND SHELL STRUCTURE
22.3 THE SUPERGROUP
We are now in a position to construct the supergroup with the 2 2j + I
states of the shell spanning a single irreducible representation. We may
also construct the generators of the group, which constitute a complete set
of ladder operators that allow us to ladder from an arbitrary N-particle
state I<PN> to any other N' -particle state I<PN' > of the shell (N .B. N is not
necssarily equal to N'.)
A typical N-particle state of the shell will be created by an operator
pt = at at. . · at
N 01./3 W
(22.10)
acting on the vacuum state to give
I<PN> = ptlO>
( 22.11 )
In the special case of the j2 j + I configuration there is just one state
1<P2j+ I> = rrtlo>
(22.12 )
where
rrt= a :t a.t ... a:t .
:Jm im-I :J-J
(22.13 )
We seek an operator Qcp'cp that has the property of annihilating an
N -particle state I<PN> and creating a new N' -particle state I<PN' > . Judd 41 has
shown that the operator can be taken as
Qcp'cp = p, rrtrrp N
(22.14 )
where <P and <P' may range over all of the 2 2j + 1 states of the shell. The total
number of distinct operators is readily seen to be just 24} + 2.
We now attempt to use the operators Qcp'cp to construct a closed Lie
algebra. Following the Cartan-Weyl prescription, we first select the 2 2j + 1
operators
Hcp=Q#
(22.15 )
which are obviously self-commuting. Furthermore,
[H cp' Qcp'CP" ] = (8cpcp' - 8cpcp" ) Qcp'cp"
(22.16 )
The Lie algebra may be identified with that of Cartan's An' where here
n = 2 2j + I. If we restrict ourselves to transformations that preserve the
orthonormality of the states, then we may identify the supergroup as the
unitary group U(2 2j + 1).

TWO IMPORTANT SUBGROUPS 353
For an arbitrary state of the shell we have
H 4>1</>'> =  #,1</>'>
(22.17)
and hence the weights associated with any state will involve 2 2j + 1 - 1 zeros
and one unit. The highest weight will have the unit leading, and hence we
conclude that the 2 2j + 1 states of the J-shell span' a single irreducible
representation labeled {100... O} = {I} of the supergroup U(2 2j + 1). Similar
conclusions have been reached by Moshinsky and Quesne. 389
EXERCISE
22.4. Show that the supergroup for identical fermions in LS-coupling is U(7 4/ + 2 ).
22.4 TWO IMPORTANT SUBGROUPS
We can find a subgroup of U(2 2j + 1) in which the representation {I} of
U(2 2j + 1) remains irreducible. This new group contains a subgroup that
supplies a representation that can be spanned by all the states of the jN
configurations with N odd, and another representation that can be spanned
by all the states with N even.
These two important subgroups of U(2 2j + 1) can be most conveniently
derived by considering the set of (2j + 1) (4; + 3) distinct operators aJ, aa'
ataJ,a!ap,aaap, which is closed under commutation. The appropriate Lie
algebra is found by first taking the set of 2j + 1 self-commuting operators
Ha=![aJ,aa]=aJaa-! (22.18)
and notin that
[Ha,aZ] = apaZ
[Ha' a p ] = - apap
(22.19)
and
[ Ha' aZa] = ( aP + ay ) aZa
[Ha' apa y ] = - ( aP + ay ) apa y
[Ha,aZa y ] = (aP - ay )aZa y
(22.20 )
I t is evident that the roots are all of the form + e; or + e; + e j , with
i = 1,2, . . . ,2j + 1, and hence the relevant Cart an algebra is B 2j + l' which is
the Lie algebra associated with the group SO (4j + 3).
The eigenvalues of an arbitrary Ha acting on an arbitrary N-particle
state of the shell are either + 1: or -!, and hence the weight of an

354 CASE STUDY III: FERMIONS AND SHELL STRUCTURE
arbitrary state is just [ + ! + !. .. + !], where all possible combinations of
the 2j + 1 signs arise. Each weight corresponds to a distinct weight, and as
the highest weight is [!!.. . !] we conclude that the complete set of states of
the shell spans the spin representation [t!...!] of SO( 4j + 3). Thus under
the restriction U(2 2J + 1)SO(4j + 3), we have
{lJ[t!...!]
(22.21 )
An important subgroup of SO(4j+3) can be found by removing from
the generators of SO( 4j + 3) the set of (4j + 2) annihilation and creation
operators aa,a!. It follows from Eqs. 22.19 and 22.20 that we are left with
the Cartan algebra D 2J + I' which is the Lie algebra associated with the
group SO(4j+2). Since we have deleted the operators aa,a!, the residual
operators can connect only states that differ in 0 or + 2 fermions, and
hence the states with N odd and N even must occur in different representa-
tions of SO(4j + 2). It follows from Eqs. 22.8 and 22.9 that there are just 2 2J
even-N states and 2 2J odd-N states. The weights of the states of the shell
are all of the type [ + t + ! . .. + !], with those of even N involving an even
number of minus signs and those of odd N an odd number of minus signs.
Thus we conclude that under SO(4j+2), the states with N even span the
irreducible representation [i!... i!], and those with N odd span the
irreducible representation [t! ...! - !]. Hence under U(2 2J + 1)SO(4j + 3)
SO(4j+2)
{I} [ 11 1 ] [ 11 11 ]+[ 11 1 1 ]
 22"'2  22"'22 22"'2 -2 ·
( 22.22 )
EXERCISE
22.5. Show that for the LS-coupled configurations IN, the complete set of states
of the I-shell can be described by the group scheme
U(2 4 /+ 2 ):J 80(81 + 5):J 80(81 +4).
(22.23 )
22.5 A UNITARY SUBGROUP
The introduction of the group SO(4j + 2) allowed us to assign the states
with N even and N odd to its two basic conjugate spin representations. If
we discard from the generators of SO( 4j + 2) the operators a!aJ and aaa/3'
we are left with just the 4jU+ 1) generators Ha and a!a p (a=l={3), and since
[Ha,aJa y ] = (8 a /3 - 8ay )aZa y
we now have roots of the type e; - e J , which characterize the Cartan

TENSOR OPERATORS AND ANNIHILATION AND CREATION OPERATORS 3SS
algebra A 2j associated with the group SU(2j + 1). We may avoid the
presence of fractional weights by working with U(2j + 1) rather than
SU(2j + 1). This amounts to using
H '- t
a - aaaa
(22.24)
rather than ![al,aa]'
The generators alaa of U(2j + 1) cannot connect states of different N,
and hence the states of different configurationsjN must belong to different
representations of U(2j+ 1). The action of a Ha' on an arbitrary state ofjN
must result in an eigenvalue of 1 or 0, depending on whether a occurs in
the state or not. The maximal weight for the states of jN must involve N
units and 2j+ I-N zeros, and hence under the reduction SO(4j+2)
U(2j+ 1) we must have
[ ! ! .. . t ! ]  { O} + { 1 2 } + · . . + { 1 2j + 1 }
( 22.25 )
and
[ ! t. · . t - ! ]  { 1 } + { 1 3 } + · · . + { 1 2j }
( 22.26 )
As usual, we suppress the null weights and collect together the units.
22.6 TENSOR OPERATORS AND ANNIHILATION AND CREATION
OPERATORS
Before proceeding further with the investigation of subgroup structure,
let us pause and consider the tensorial properties of the fermion operators
a}m and a jm with respect to the generators of SO(3).
The generators of SO(3) may be taken as the familiar angular-
momentum operators
N
J i = }: jia
a=l
( a, i = 1,2,3)
( 22.27)
Operators of single-particle operators of the type
N
F= }: fa
a=l
( 22.28 )
may be expressed in second-quantized form. as
F= }: al<alfl p>a p
a,/3
( 22.29 )

356 CASE STUDY III: FERMIONS AND SHELL STRUCTURE
and hence we may write
J =  a!<al}1 p>a p
a,/3
( 22.30 )
Using the above result, we immediately find that
[Jz,aj] =ma}m
[ J :t' aj!n] = [} () + 1 ) - m ( m + 1) ] 1/2 aj!n:t I
(22.31 )
and hence conclude that 390 the 2} + 1 operators aj form the components
of a spherical tensor operator aJ of half-integral rank}.
The operators a jm do not, as they stand, form the components of a
spherical tensor operator. An operator Cl jm that does form the components
of a spherical tensor a j may be formed by defining
_ j-m
a jm = ( - 1 ) a j - m
(22.32)
22.7 A COUPLED TENSOR OPERATOR
It is convenient to introduce a coupled tensor operator
(aJ.a j2 )JM=  <mlm2IJM>ajm.Clj2m2
mlm2
(22.33)
Use of the anticommutation properties of fermion operators readily leads
to
( t ) ( _ I ,\it+j2-J ( t ) _ (2 ' 1 ) 1/2   (2234)
a j .a j2 JM +) a j2 a j . JM- 'JI + u jd2 uJO u MO .
In the case of identical fermions with}1 =}2 we will normally abbreviate
the coupled product to just (at a)JM'
The commutator [(a t a)k.q.' (at a)k 2 q2] may be evaluated by use of Eq.
22.33 to decouple the fermion operators, followed by application of the
basic fermion anticommutation properties. Finally the fermion operators
may be recoupled, also using Eq. 22.33, to establish the result:
1/2 2'
[(ata)k.q., (a t a)k 2 q2] =  (2K + 1) [(2k l + 1) (2k 2 + 1)] ( -1) :J
KQ
X<Q,Q2I K Q>( :'
k 2
J
 ) [1_(_1)k l +k 2 +K](a t a)KQ (22.35)

CLASSIFICATION FOR THEj-j SHELL 357
22.8 FURTHER SUBGROUPS
The above commutator is a useful aid in finding further subgroups. We
first note that the (2) + 1)2 operators of (ata)KQ having K =0, I,... ,2) can be
taken as the generators of the group U(2} + I). The scalar operator (ata)oo
commutes with all (at a)KQ' and its deletion leaves the generators of
SU(2) + I).
The commutator in Eq. 22.35 will vanish unless k l + k 2 + K is odd, and
hence the set of () + 1)(2} + I) operators (at a)KQ with K odd must be closed
under commutation, generating in fact the Lie algebra associated with the
group Sp(2) + I). Hence Sp(2} + I) is a subgroup of SU(2} + I).
The three components of the operator (ata)KQ with K= I form the
generators of the group SO(3), while the component with K = I and Q = 0
generates a subgroup SO(2).
We have now demonstrated the subgroup structure
U(2 2j + I ):) SO(4}+3):) SO(4}+2):) U(2}+ I)
:) SU(2}+ I):) Sp(2}+ I):) SO(3):) SO(2).
(22.36)
22.9 CLASSIFICATION FOR THEj=  SHELL
The group-subgroup structure just derived finds practical use in the
classification of the states of ii-coupled atomic or nuclear shells involving
identical fermions} 2, 204, 387 The classification is complete for shells with
j < !. Here we consider the classification of the states of the j = t shell.
From Eq. 22.36 we see that the appropriate group structure is
U(256)SO( 17)SO( 16)U(8)Sp(8)SO(3)SO(2) (22.37)
We omit the group SU(8), as it provides no additional labeling of the
states.
Under U(256)SO(17)SO(16) we have
{I} [1 111111 1] [ 1111-11-11 ]+[ 1 1 11111 1 ]
 2222222 2  2222Z2Z2 2222222 - 2:
(22.38)
with the N even states spanning the [!ttttttt] representation of SO(16)
and the N odd states the [tt!t!tt -!] representation. Under SO(16)
U(8),
[tttttttt ]{O} + {12} + {1 4 } + {1 6 } + {1 8 }
(22.39)

358 CASE STUDY III: FERMIONS AND SHELL STRUCTURE
Table 22.1. Classification for the j = 1 Configurations under
U(8):) Sp(8):)SO(3) for N <4
N U(8) Sp(8) SO(3)
0 {O} (0000> [0]
I {I} (1000> []
2 {1 2 } (1100> [2] + [4] + [6]
(0000> [0]
3 {13} (1110> [j] + [] + [J]+ [¥]+ [¥]
(1000> [1]
4 {14} (1111> '[2] + [4] + [5] + [8]
(1100> [2] + [4] + [6]
(0000> [0]
and
[!!!!!!!-!]{I} + {1 3 } + {1 5 } + {1 7 }
( 22.40 )
The states of a givenjN configuration span the irreducible representation
{IN} of U(8). The familiar particle-hole symmetry of thejN andj2 J +I-N
configurations follows from the equivalence under SU(2j + I) of the repre-
sentations {IN} and {1 2 J+I-N} of U(2j+ I).
The decomposition of the representations {I N} of U(8) into those of
Sp(8) may therefore be accomplished by the methods outlined in the
Appendix, to give (N <4)
{IN}=<IV)
v
(22.41 )
where the summation ranges over the values v = N, N - 2,... ,and the small-
est value of v is 0 or I according as N is even or odd. Later we identify v
with Racah's seniority number. 8 , 166, 167
The reductions Sp(8)SO(3) may be most readily performed using the
plethysm method outlined in the Appendix, or they may simply obtained
from published tables. 31 The complete classification of the states of the
j= 1 shell for N <4 under U(8)Sp(8)SO(3) is given in Table 22.1.
EXERCISES
22.6. Enlarge the description of the annihilation and creation operators ajm' a)n to
the set atmJm, a;mJm, where t is the isotopic spin and mt its z-projection, and
show that the states of the nucleon configurations involving protons and

SENIORITY 359
neutrons in a givenj-shell may be classified under the group schemes
U(4 j +2):) SO(8j + 5):) SO(8j +4):) U( 4j + 2):) Sp( 4j + 2)
:) SU T (2) X [Sp(2j+ 1):) SO(3):) SO(2)]
(22.42 )
or
U(2 4J + 2 ):) SO(8j +5):) SO(8j +4)
:) SU T (2) X [ U(2j + 1):) Sp(2j+ 1):) SO(3):) SO(2)]
(22.43)
where S U T (2) describes the isotopic spin (T, M T) of the N -particle states.
22.7. Show, in a similar manner, that for identical fermions in an LS-coupled
I-shell, two possible group structures are l166
U (2 4 / + 2) :) SO (8 1+ 5) :) SO (8 1+ 4) :) U ( 41 + 2) :) Sp ( 41 + 2 )
:) SU s (2) X [SO(21 + 1):) SO(3):) SO(2)] (22.44)
and
U(24/+2):) SO(8/+ 5):) SO(81 +4)
:) SU S (2) X [U(2/+ 1):) SO(2/+ I)SO(3):) SO(2)]
(22.45 )
22.8. Show that for the special case of the f-she11 9 (I = 3), it is possible to make use
of the exceptional group G 2 in the chain SO(7):) G 2 :) SO(3). (Hint: Show
that the components of the operators (ata)K with K = 1 and 5 close under
commutation. 166,388
22.10 SENIORITY
The concept of seniority was first introduced by Racah 8 , 166, 167 and is
intimately' connected with the symplectic symmetry of shell-model states.
Table 22.1 shows clearly that states involving different numbers of par-
ticles may have the same seniority number v and the same SO(3) content.
Thus v = 0 occurs for N = 0, 2, and 4. The v = 0, N = 2 state can be
visualized as being formed by the coupling of a pair of particles in the j = !
orbital to form a state () zero angular momentum, while the N = 4, v = 0
state involves the coupling of two such pairs. For a given N there will be
(N-v)/2 pairs involved. Such a situation is physically realized when
short-range or pairing forces are dominant 387 , 391-396 -for example, in
nuclei and superconductivity.

360 CASE STUDY III: FERMIONS AND SHELL STRUCTURE
22.11 THE QUASI-SPIN FORMALISM
Seniority is of considerable mathematical significance even when the
physical significance is tenuous (e.g., in atomic shells, where the long-range
Coulomb forces dominate), because it leads to simple relationships be-
tween matrix elements within jN involving states of seniority v and those
within the jV configuration. Formerly these relationships were derived
tediously and rather indirectly. The introduction of the quasi-spin formal-
ism 210--212,388-390,395-408 has made it possible to encase the N-dependence
of matrix elements into a single Wigner coupling coefficient.
Let us introduce the three operators 395 , 397,400
Q+=t V (2j+l) (atat)oo
Q_=-t V (2j+l) (aa)oo
Qz= - tV(2j+ I) [(ata)oo+ (aat)oo]
( 22.46 )
Under commutation we find
[ Qz, Q j: ] = + Q j:
[ Q +, Q - ] = 2 Qz
(22.47)
These commutation relations are equivalent to those satisfied by the spin
operators S j:' Sz that generate the spin group SU s (2), and hence we may
conclude that Q j:' Qz generate a quasi-spin group S U Q (2) and form the
components of a quasi-spin Q.
The components of Q are a subset of the generators of SO( 4j + 2) and
furthermore commute with the generators of Sp(2j + I); hence we have the
alternative group structure for the j-shell
U(2 2J + 1):) SO( 4j + 3):) SO( 4j + 2):) SUQ(2)
X [Sp(2j+ I):) SO(3):) SO(2)]
(22.48 )
We must now determine the quasi-spin of the various states of the
j-shell. To this end we note that Qz can be thrown into the form of the
number operator to give the eigenvalues of Qz as
QzljNvJM>=MQljNvJM>
where
M Q = -!(2j+ I-N)
( 22.49 )

QUASI-SPIN CLASSIFICATION OF STATES 361
It follows from Eq. 22.46 that Q + and Q _ are creation and annihilation
operators for a pair of particles coupled to zero angular momentum and
hence cannot change the seniority v of a state. This result is also obvious
from the fact that the components of Q commute with the generators of
the symplectic group Sp(2} + 1) and hence cannot couple different repre-
sentations by the seniority number. It is equally apparent that Q + and Q_
raise or lower N by 2 units, and for N = v we must have
Q_ljVvJM>=O
(22.50)
which implies that
Q 2 Ij V vJM> = [Q+Q_ + Qz( Qz -1) ] I} VvJM >
= Qz( Qz -1)lj v vJM>
= Q( Q+ 1)ljV v JM>
(22.51)
Noting Eq. 22.49 leads us directly to the result that the quasi-spin Q is
given by
Q=!(2j+l-v)
(22.52 )
Thus the quantum numbers Q,M Q of the quasi-spin group SU Q (2) carry
the same information as Nand v. We may regard Q as measuring the
distance from the center of the shell to where a given seniority state occurs,
while M Q measures the filling up of the shell, also with respect to the
center.
22.12 QUASI-SPIN CLASSIFICATION OF STATES
The introduction of the quasi-spin group allows the states of the j-shell
to be classified into a number of quasi-spin multiplets, each involving
2Q + 1 sets of states associated with 2Q + 1 different values of N. The
states belonging to a given quasi-spin multiplet will all have the same
seniority number.
The branching rules for SO(4j + 2)SUQ(2) X Sp(2j + 1) may be readily
found using the results of the Appendix to give
1 2' 2'+2
[ !! . . . !! ]  < 11 . . . 11> + · · · + :I < 110. . .0> +:1 <00. . .0>
[ t t . . . t - t ]  2 < 11 . . . 10> + · · · + 2j - 1 < 1110 . . . 0> + 2j + 1 < 10 . . . 0>
(22.53 )

362 CASE STUDY III: FERMIONS AND SHELL STRUCTURE
Table 22.2. Quasi-spin Classification for the j = ! Shell
U (256) SO (17) SO (16) Q M Q N v Sp(8) SO(3)
0 0 4 4 (1111) [2] + [4] + [5] + [8]
I L H2 (1100) [2] + [4] + [6]
[11111111] 2 8
21211211
I 6
2 0 4 0 (0000) [0]
-I 2
{I} [11111111] -2 0
21111211
 I :f 3
']
1 (1110) [!]+[]+[J]+[¥]+[ li ]
2
- 
2
3 7
[11111!1_1] '2
22222 2 2 5
I (1000) []
_1 3
2
_1 I
2
where we indicate the quasi-spin multiplicity as a left superscript. The
quasi-spin classification of the states of the j =! shell is given in Table
22.2.
22.13 QUASI-SPIN FOR ANNIHILATION AND CREATION OPERATORS
The annihilation and creation operators ajm and ajt", can be regarded as
the two components of an elementary operator of quasi-spin rank Q = !,
since under commutation
[ Q + , ajt",] = 0 [ Q + , a jm ] = ajt",
[ Q _ , ajt", ]  ajm [ Q _ , a jm ] = 0
[ Qz, ajt", ] = ! ajt",
[ Qz, a jm ] = - -!a jm
( 22.54 )
and thu ajt", and ajm transform as the M Q =! and - t components of Q,

SYMMETRY CLASSIFICATION OF OPERATORS 363
respectively. This fact can be conveniently displayed by combining at and
a into the components of a double tensor a(q}) with q = -!.
Irreducible tensors of well-defined quasi-spin can then be constructed
from products of the elementary tensors a(qj) by standard vector-coupling
techniques. It follows that any interaction that can be expressed in terms of
products of annihilation and creation operators can be expressed as a
linear combination of operators of well-defined quasi-spin. This latter
construction is essential if the Wigner-Eckart theorem is to be fully
exploited in the quasi-spin space.
The double tensor
x ( Kk)= (a(qjh(qj») ( Kk'
provides a simple example of a compound tensor of well-defined quasi-
spin rank K. If X(Kk) acts between states of well-defined quasi-spin, we
have from the Wigner-Eckart theorem in the quasi-spin space
<aQMQIX'1TKkl) la' Q' }\fb>
= (_I)Q-M Q ( Q
-M Q
K
Q' ) <aQ11X(Kk)lla'Q'> (22.56)
M q
Q
'IT
Since (from Eq. 22.49) M Q = - t(2j + 1- N) and Q is independent of N,
we have in Eq. 22.56 the entire N-dependence of the matrix elements of
X'1TKk) encased in the single 3j-symbol. This result is valid for any quasi-
spin operator of rank K.
All the usual special formulas concerning the N -dependence of matrix
elements for the j-shell can be obtained from Eq. 22.56. Thus the familiar
particle-hole conjugation relationships follow directly if one notes that Eq.
22.56 implies that
<aQMQlxri:k)1a'Q'MQ>
= ( -1 )Q+Q'+K-2M Q <aQ - MQIXoKk)la' Q' - M Q >
(22.57)
22.14 SYMMETRY CLASSIFICATION OF OPERATORS
The full exploitation of the Wigner-Eckart theorem requires that all
interactions of interest be expressed in terms of symmetrized irreducible
tensor operators that have well-defined transformation properties with
respect to the same groups used to describe the transformation properties

364 CASE STUDY III: FERMIONS AND SHELL STRUCTURE
of the basis states. Generally we proceed by first expressing the interac-
tions as linear combinations of products of the annihilation and creation
operators used to construct the states. These linear combinations are then
expressed in terms of linear combinations of the symmetrized tensor
operators.
The construction of the symmetrized irreducible tensor operators is
effected by first determining the transformation properties of the operators
a(qj). This step may be performed by examining the commutation proper-
ties of the 4j + 2 components a) of a(qj) with respect to the generators of
the classificatory groups, using the results of Section 19.3. This was done,
for example, for the quasi-spin group in Section 22.13. Proceeding in this
way we find that the 4j + 2 components of a(qj) transform under SO( 4j + 2)
as the [10...0] representation. Noting that q = !, we readily conclude that
under SO(4j + 2)SUQ(2) X Sp(2j + 1),
[ 10 . . . 0 ]  2 < 10 . . . 0> + 2 < 10 . . . 0> ( 22.58 )
with the 2j + 1 components of at and a each spanning a < 10. . .0> represen-
tation of Sp(2j + 1). Since j labels the irreducible representations of SO(3)
and those of SO (2), we have now a complete classification of the
symmetry properties of all the components of a(qj) according to the quasi-
spin group scheme (Eq. 22.48) used to classify the states.
Under the unitary group scheme of Eq. 22.36 we find that the 2j + 1
components of at transform under U(2j+ 1) as the {10...0} representa-
tion, while the 2j+ 1 components of a transform as {OO...O -I}. On
restriction to Sp (2j + 1), both sets transform as < 10. . .0>.
Having determined the transformation properties of the elementary
operators at and a (or a(qj», we may construct compound irreducible tensor
operators by forming appropriate linear combinations of products of at
and a, using the standard coupling techniques of Chapter 19.
The irreducible tensor operators involving bilinear products of at and a
are of particular importance in describing one-particle interactions such as
the spin-orbit interaction. Since the components of both at and a transform
in the quasi-spin scheme as 2< 1 O. . .0>, it follows that the bilinear products
must span the representations.
1 < 10. . .0> X 2< 10. . .0> = 1, 3 < 110. . .0> + 1, 3 <20. .. 0> + 1,3<00.. .0> (22.59)
If we resolve this Kronecker square into its symmetric and antisymmetric
components, we find, respectively,
2 < 10 . . . O>@ { 2 } = 3 < 20 . . . 0> + 1 < 110 . . . 0> + 1 < 00 . . . 0>
2<10...0>@ {12} = 1<20...0>+3<110...0>+3<00...0>
( 22.60 )
(22.61 )

SYMMETRY CLASSIFICATION OF OPERATORS 36S
and hence the bilinear products ata t and aa may be resolved into sym-
metric and antisymmetric tensors.
The bilinear operators involving at at or aa will couple states differing in
N by 2, while the operators involving ata will be diagonal in N. Not all the
operators belonging to the representations on the right-hand side of Eq.
22.59 will yield nonzero matrix elements. In either case the states of the bra
and the ket must transform under SO(4j + 2) as [11... !1] (N even) or
[1 -1 ... -! - !] (N odd), and hence the operators will yield' nonzero matrix
elements only if they occur in the reduction of the representations con-
tained in the Kronecker product 409 ,31 [for SO(4}) only]
[ ll l+I ]X[ 11 1+1 ]
2Z'''2-! 22"'2-2
= [00. . . 0] + [ 110. . .00] + . .. + [ 11. . . 1 + 1 ] (22.62)
where + is taken throughout. Under the reduction SO(4}+2)SUQ(2)X
Sp(2} + 1) we find,409,31 again using results derivable from the appendix,
that
[00.. .OO]I<OO.. .0)
[110.. .00]1<20.. .0)+ 3<00.. .0)+ 3<110.. .0)
[ 11110. . .00]  1 <00. . .0) + 1 < 110. . .0) + 1 <220. . .0)
+ 3<110.. .0) + 3<20.. .0)+ 3<2110.. .0)
+ 5 <00. . .0) + 5 < 110. . .0) + 5 < 11110. . .0)
Since a(qj) transforms as [10...0] under SO(4j+2), it follows that for an
n-fold product of a(q))'s we need only consider those representations occur-
ring on the right-hand side of Eq. 22.62 that have no more than 2n units.
In the case of bilinear products this restricts us to just [00...0] and
[110. . .0]. Hence the only quasi-spin-symmetrized operators involving at a t,
aa, or ata and yielding nonzero matrix elements will be those transforming
under SU Q (2)x Sp(}+ 1) as 1<00...0), 3<00...0), 1<20...0), or
3<110...0).
Since under Sp(2} + 1)SO(3) we have 31
<00. ..O)[O]
<20. . . O) [ 1 ] + [3] + · . · + [2}]
<II0...0)[2] +... + [2}-1]
(22.63)
( 22.64 )
we further conclude that the operators X(Kk) defined in Eq. 22.55 with k
odd transform as 1<20...0) and hence are diagonal in seniority v and

366 CASE STUDY III: FERMIONS AND SHELL STRUCTURE
independent of the particle number N, while those with k even (k*O)
transform as 3< 110...0> and satisfy the seniority selection rule Llv = 0, + 2.
The same sort of analysis may be made for operators constructed from
products of an odd number of operators. In this case the matrix elements
will vanish between states differing by an even number of particles, and
nonzero elements can arise only for states differing by an odd number of
particles. In this case we must consider the Kronecker product [for SO(4v)]
J1I 1.1 ]X[ 1.1 1 I ]
1.22"'2 2'''2-2
=[10...0]+[1110...0]+... +[111...10]
(22.65)
22.15 INTERALnON OF PARTICLES IN A CENTRAL FIELD
Let us consider the interaction of a pair of particles in a central field.
For simplicity we take a spin-independent interaction V(lr l -r 2 1) and make
an expansion in terms of Legendre polynomials of COSWl 2 to obtain 387
00
V( Irl - r 2 1) =  Vk( r l , r 2 )Pk( COSWI2)
k=O
(22.66)
where v k (r I' r 2) is a function of the radial variables alone, and W 12 is the
angle between the vectors r l and r 2 referred to a common origin. In the
case of the Coulomb interaction, 39
00 rk
1 ""' <
V(lr,-r 2 1)= I _ 1 = L.J k+' Pk(COSW 12 )
r l r 2 r>
k=O
(22.67)
where r < is the lesser and r> the greater of r I and r 2'
Making use of the spherical-harmonic addition theorem,39 we find
00
V(lr l -r 2 1)=  vk(rl,r2)(Ck)'Ck»)
k=O
(22.68)
where the components of C(k) are as determined in Exercise 19.2. A typical
matrix element within the j2 configuration may be written as
E(j2J) = <j2JMI V(lr l -r 2 1)lj 2 JM>
=  <iJMII(Ck)'Ck))II/JM>Fk
= fkFk
k
(22.69)

INTERACTION OF PARTICLES IN A CENTRAL FIELD 367
where
F k = {'XO oo v( r"r 2 )Ri r ,)2 Rj(r2)2rir dr,dr 2
(22.70)
is a generalized Slater radial integral 387 and
(2k + 1 )!k = <ill c (k) Ili)2<i 2 JMI (Vk). vi k ») li 2 J M)
(22.71 )
Here we define the tensor operators V(k) in terms of their reduced matrix
elements as 388
<illv(k)lli)= Y 2k+ 1
(22.72)
The reduced matrix elements of Ok) may be evaluated using Eqs. 19.41
and 19.222 to give l04
( 2j + 1 + k )
<iIIC(k)lli)=2( _1)k/2 [ (2(=)! ] '/2 (:'!-k-  ) ! (22.73)
'J 'J , k , k ,
2 '2'2'
with k limited to the even values such that
2i - 1  k  O.
For the particular case of i = 1 we find
(22.74 )
<1:IIC(O)lIt)2 =8,
< t II C ( 2 ) II t ) 2 = 2 8 1
< 1: II C ( 4 ) II t ) 2 =  ,
<11Ic(6)111)2= 
(22.75)
The reduced matrix elements of (at a)(k) may be readily evaluated using
Eq. 22.33 and the result compared with Eq. 22.73 to establish the re-
lationship
(ata)k) = - v q (k)
(22.76)
and hence V(k) has the same group transformation properties as (at a)(k) . We
may conclude from the results of Section 22.14 that for k even (k*O), V(k)
transforms under Sp(2i + 1) as the <1100...0) representation, and for k = °
as it transforms the <00...0) representation.
The two-particle matrix elements involve scalar products of the y<k)
tensor operators. The symmetry types arising will be restricted to the

368 CASE STUDY III: FERMIONS AND SHELL STRUCTURE
symplectic symmetries arising from the Kronecker product < 110. . .0) X
<110...0).
To be specific, let us consider the case ofj=!. Under Sp(8) we have 31
< 1100) X < 1100) = (0000) + <1100) + (1111) + <2000) + <2110) + <22(0)
(22.77)
Since the interaction is a scalar in SO(3) (i.e., it transforms like a J = °
state) we need only consider those representations on the right that under
Sp (8)SO(3) contain J = ° states. Specific calculation 3 ] shows that J = °
arises once in <00(0) and twice in (2200) only. Thus it must be possible to
write Eq. 22.69 in a linear combination of four operators e; (i=0,1,2,3)
having well-defined symplectic symmetry, with
fkFk =  e;E i
i
(22.78)
where eo and e] both transform under Sp(8) as (0000), and e 2 and e 3 as
(2200) and the E; are certain linear combinations of the F k integrals.
The e; operators may be constructed by first noting that the scalar
operator
 (2k+ I) -1/2(v(k)'v(k»)«1100)k<1100)kl<0)0) (22.79)
k even
will transform under Sp(8) as the <0) representation. For the case of
interest, <0)= (0000) or <22(0). Noting Eq. 22.71, we may wrIte our
operators e;<o) in terms of the fk'S as
( 2k + 1 ) 1/2
e;(a) = N<o)  <ill c k lIi)2 fk< <I lOO)k<I lOO)kl<a)O) (22.80)
where N <0) is a convenient normalization factor.
The Sp(8)::::> SO(3) isoscalar factors may be determined by the methods
of Chapter 19. In particular, Eq. 19.201 leads to
< <1 lOO)k<l lOO)kl<OOOO)O) = y2;; 1
(22.81)
where k=2, 4, and 6 (cf. Eq. 22.64). In addition we must determine the
two sets of isoscalar factors «ll00)k<1100)kl<2200)aO) and
< (1100)k<ll00)kl<2200)bO), where a and b serve to distinguish the two
J = ° states that arise in the reduction of (2200) under Sp (8)SO(3).

INTERACTION OF PARTICLES IN A CENTRAL FIELD 369
Table 22.3. Some Sp(8)::::> SO(3)/soscalar Factors
( (0000)0(0000)01(0000)0) = 1
«1100)2(1100)21(0000)0) = 
«1100)4(1100)41(0000)0) = Vi;
«1100)6(1100)61(0000)0) = Vi
«1100)2(1100)21(2200)0 0 )= - Vl;
«1100)4(1100)41(2200)0°)= Vj;
«1100)6(1100)61(2200)0 0 )=0
«1I00>2(1I00>21(2200>bO>= - 
«IlOO>4(1I00>41(2200>bO>= - 
«1I00>6(1I00>61(2200>bO>= 
These isoscalar factors are necessarily orthogonal to those of Eq. 22.81 and
form an orthonormal set. Furthermore, we are free to prescribe an ar-
bitrary separation of the a and b states. Let us choose to put
«11(0)6<1100)61<2200)00)=0, in which case we obtain the set of isosca-
lar factors listed in Table 22.3. It is important to note that once the
separation of the a and b states is fixed, this convention must be used in all
succeeding calculations.
Table 22.4. Sp(8)::::> SO(3)-Symmetrized Scalar Two-Particle Operators
eo = /0
el = i (/0 + 105/ 2 + 77/ 4 + 51f6 )
e2 = 3:5 (27/ 2 -11/ 4 )
e3 = llfl( 15/ 2 + 11/4 - /6)

370 CASE STUDY III: FERMIONS AND SHELL STRUcrURE
Table 22.5. Linear Combinations E k of the F k Integrals
r..o _ F o 25 F 2 5 F 4 125 F 6
J:!J - - - 567 - 231 - 11583
E 1 - 49
- 567
E 2- 2 P 2 2 P 4
- 1617 - 5929
E 3 - 10 F 2 + 2 F 4 100 F 6
- 436S9 17787 - 1656369
With the isoscalar factors determined, we may make use of Eq. 22.80
with 22.75 to calculate the form of the operators ej<o>, making convenient
choices for the various N<o)' When this is done, we obtain the results of
Table 22.4. The appropriate linear combinations E k of the radial integrals
F k then follow from equating coefficients in Eq. 22.78 to give the results
shown in Table 22.5, in "agreement with those of Edmonds and Flowers.2 3o
Note that in forming the operator e l , we have added teo to it to simplify
the matrix elements of e l . Indeed, it may be shown 230 that the matrix
elements of e l are simply related to those of Casimir operators for SU(8),
Sp(8), and ,80(3). Of course, adding teo to e l leaves the Sp(8) symmetry
unchanged.
The matrix elements of the e j operators may be determined for (t)2 by
using Eq. 19.226, leading to the results 230 shown in Table 22.6. The
operators eo and e l are scalars in Sp(8) and hence diagonal in the
symplectic or the seniority scheme. It follows from Eq. 22.63 that for a
two-particle operator the representation <2200> can have quasi-spin Q = 0
only, and hence e 2 and e 3 are also diagonal in the seniority nunlber. Thus
we can predict the vanishing of all off-diagonal matrix elements in the t
shell for all e j . Thus for scalar two-body interactions in the t shell,
seniority is automatically conserved. For values j:>! the scalar two-body
interaction gives rise to operators having quasi-spin Q*O and hence leads
to nonconservation of the seniority number.
Table 22.6. Matrix Elements of E[(1)2J]
E[ (1)2 J=O] =£O+4El
E[()2J=2]=EJ
E[(1)2 J =4]=£O
E[ ()2 J=6] =£0
+ 99E 2
+ 143E 3
- 55E 2
+ 143E 3
- 154E 3

INTERACTION OF PARTICLES IN A CENTRAL FIELD 371
The methods just outlined can be readily applied to shells involving
LS-coupled states, spin-dependent interactions, and the like. The aim is
always the same-to expand the interactions in terms of operators sym-
metrized with respect to the same group as the states, and then to use the
Wigner-Eckart theorem to the fullest extent possible. 41o
EXERCISES
22.9. Show that to symmetrize the scalar two-body interaction in the j =  shell it
is necessary to construct an operator e(11110) transforming as <11110) under
Sp(10).
22.10. Show that the ODerator e(11110) is of quasi-spin rank Q=2 and hence will
lead to nonconservation of the seniority number.

Appendix: Schur Functions and Young
Tableaux
A.I INTRODUCfION
The introductory chapters in this book have placed heavy emphasis on
the use of the roots associated with Lie algebras and of weights to describe
many properties of group representations, such as the reduction of
Kronecker products. These methods followed the traditional Cartan-Weyl
approach as modified by Dynkin. While there is no difficulty, in principle,
in enumerating the weights of representations, in practice it becomes a
tedious and cumbersome process for all but the simplest representations.
An alternative approach to the theory of the classical compact groups
which complements the earlier work of Cartan and Lie has been developed
by D. E. Littlewood 26 ,224,225 as a natural consequence of Schur's original
thesis 25 on the properties of invariant matrices. Littlewood's treatment
circumvents the study of weights by considering the properties of special
functions of the roots of the matrices that characterize the elements of the
continuous group. These functions, known as Schur functions, or simply as
S-functions, have been used by Littlewood to find relatively simple formu-
las relating the characters of representations of the unitary, symplectic,
orthogonal, and rotation groups. The basic idea is to be able to express the
characters of these groups in terms of S-functions, and reciprocally to be
able to express S-functions in terms of the group characters. Once these
two problems are solved, the problem of determining branching rules
under various group restrictions is readily solvable. The reduction of the
Kronecker products then reduces to the reduction of products of S-
functions.
372

S-FUNCfIONS 373
We have given an extensive treatment of the use of S-functions
elsewhere;31 here the main results are sketched, and the reader referred to
the literature for further details.
A.2 S-FUNCfIONS
In this section we briefly outline some of the relevant mathematical tools
and notation required in the subsequent development of this appendix.
A. Partitions
A set of r positive integers whose sum is n is said to form a partition of n.
An ordered partition 411 is one where the integers are ordered from largest to
smallest. All partitions henceforth are so ordered, with the notation that a
Greek letter denotes a general partition so: (A) = (A 1 ,A 2 ,... ,Ar)' A Young
diagram 30 is associated with each partition. It is a graph of r rows, with Ai
dots (or squares) in the ith row, and each row left justified. A conjugate
partition is formed Py interchanging the rows and columns of the graph
and is denoted by (A), for example,
(322) - 
(331) - EfE
f"'tt,J
and (322) = (331). Some partItIons will be self-conjugate, for example,
r--..I
(21) = (21). Partitions with repeated parts will often be written with a
superscript denoting the number of times the part occurs, for example,
(322) = (322).
Frobenius' notation is sometimes used. The leading diagonal of the
Young diagram is defined as the one that starts in the top left-hand comer.
For each square on the diagonal we write down the number of squares to
the right of it, and below this, the number of squares beneath it. Thus
"
"
(5321) -
4
"
"-
and (2 2 ) - rn 1
o
1 0 "
"
3 1
(5321) = ( ;
: )
(22) = (:)

374 APPENDIX: SCHUR FUNCTIONS AND YOUNG TABLEAUX
B. Symmetric Functions
A symmetric function of k variables a i is one that is unchanged by any
permutation of the variables. Two types of symmetric functions are rele-
vant here.
(a) Mononomial symmetric junctions. If (A) is a partition, we define the
mononomial SA such that
S = aAlt0,2. . . a
A  } 2 r
( A.I )
where the summation is over all different permutations of the a's. For
example, if k = 3,
SOl) = a 1 a 2 + a 2 a 3 + a 3 a}
S - 2 2 2
(2)-a 1 +a 2 +a 3
(b) Homogeneous product sums. The homogeneous product sum h n is de-
fined as the sum over all of the mononomials SA' where A is a partition of
n.
h n =  Sp
p
( A.2 )
Thus
h 2 = S(2) + SOl)
c. S-Functions
If (A) is a partition of n, the S-function {A} is the determinant of the h;'s,
defined as follows:
{A} = Ih\-s+tl
( A.3 )
sand t being subscripts for the row and column, respectively. We extend
the definition of the homogeneous product sums to include ho = I and
hi=O for i<O.
Thus
h3 h4
{321} = hI h 2
o ho
hs
h3 = hlh2 h 3 - h - h] 2h 4 + hl h S
h]

OUTER S-FUNCTION MULTIPLICATION 375
D. The Symmetric Group
Under the operations of the group, the symmetric group Sn with n!
elements is split into classes p with kp elements. Each class is the complete
set of conjugates of a given element. The irreducible representations of the
group may be placed in a one to one relation with the partitions of n.
Because conjugate matrices have the same trace or characteristic, there
exists a unique number XA), the characteristic for a particular class of a
representation. The set of characteristics of a representation is known as
the character of the representation. 28,30 If we define a function
k
{ A } =  X o..>S
£.J n! p p
p
( A.4 )
we may prove 412 that it is in complete correspondence with the representa-
tions of Sn'
Litlewood26 has shown that this definition is entirely equivalent to the
earlier definition of the S-function, so we have a complete isomorphism
between operations on S-functions and operations on symmetric group
representations. The duali ty 30 that II exists between Sn and GL(n) paved the
way for Littlewood to express many of the properties of compact groups in
terms of S-functions. It is useful to note that the representations {A} of the
symmetric group Sn are of degree
r
II (Ai-Aj+j-i)
f{A}=n!j>il
II (Ai + r - i) !
i= 1
(A.5)
where {A} is a partition of n into r parts. A formula more suited to hand
calculations has been given by Robinson. 3o
EXERCISE
A.1 Show that the representations {631}, {42212}, and {2I s } of S10 are of degrees
315, 567, and 9, respectively.
A.3 OUTER S-FUNCTION MULTIPLICATION
The product of two S-functions defined on different sets of variables,
corresponding to the product of representations of different symmetric

376 APPENDIX: SCHUR FUNCTIONS AND YOUNG TABLEAUX
groups, is known as the outer or ordinary multiplication of S-functions.
The S-functions appearing in the product
{ A } { IL} =  r AILP { V }
( A.6 )
where r AILP is the number of times {v} arises in the product, are those that
can be built by augmenting the graph of {A} with ILl symbols a, IL2 symbols
{3, IL3 symbols "I, and so on in that order and in the ways specified by the
following three rules:
1. No identical symbols appear in the same column of the graph.
2. If we count the a's, {3's, y's, and so on from right to left starting at the
top, then at all times while the count is being made, the number of a's
must be not less than the number of {3's, which must be not less than the
number of y's, and so on.
3. The graph obtained after the addition of each symbol must be regular.
The principal part of the product is the term obtained when the
partitions are simply added, i.e. the partition {AI + ILl' A 2 + IL2""''A; +
ILi" · · }. It corresponds graphically to putting all the a's in the first row, the
{3's in the second, and so on. The other terms in the product may then be
produced systematically by removing the last element and trying it on the
next lower line, then the next, and so on. When the last element will fit
nowhere else, the second to last element is also removed and fitted in the
same fashion; if no place is found, the third to last element is removed,
and the process repeated. The final calculation may be checked dimension-
ally by use of the equation 30
(m+n)!
f{A}f{IL} n!m! =rAILJ{v} (A.7)
For example,
{ 21 X 21 }
EP  ex
:> .
{3
 ex+
:>
{3
a a +W+W+
ex
{3
++  +
[E0 ex
ex {3
ex
ex
{3

S-FUNCfION DIVISION 377
and hence
{21}' {21}= {42} + {41 2 } +2{321} + {31 3 } + {3 2 } + {2 3 } + {2212}
Extensive tables of outer S-function products have been published. 31
A.4 S-FUNCTION DIVISION
Frequently the algebra requires the evaluation of the sum of S-functions
{p} that when multiplied by a particular S-function {JL} give a particular
S-function {A}, the coefficient of {p} being the coefficient of {A} in the
outer product. Hence we define the (outer) division of S-functions {A}
/ { JL} to be
{ A } / { JL } =  r /LVA { P }
v
( A.8 )
where r /LVA is the same as the coefficient in the outer product:
{ JL } { p } =  r /LVA { A } .
A
The evaluation of the quotient is somewhat easier than that of each
product, thus considerably simplifying the calculation. 413
We have the graph of {JL} and wish to know all possible ways of adding
elements to form the graph of {A}, given the rules for the product. To
perform the division, draw the graph for {A} with squares and fill up the
top left-hand corner with the graph corresponding to {JL}. Graph {JL)
must fit entirely inside {A} or the result will be null. The remaining squares
are then labeled by a's, {3's, y's, and so on, row by row, starting at the top
left, as given by rules I and 2 for the product, and also with
3 . The symbols must not decrease when reading left to right across a
row; that is, there must not be an a to the right of a {3 and so on.
4. The resultant S-function must be ordered.
( A. 9 )
As a typical example, we have for {321} / {2}
WCDW+W+W
and hence
{ 321 } / { 2 } = { 31 } + {2 2 } + {21 2 }

378 APPENDIX: SCHUR FUNCfIONS AND YOUNG TABLEAUX
Again, an extensive tabulation of S-function division exists}}
The operation of S-function division may be readily shown to satisfy the
relations
({A} + {IL})/ {v} = {A} / {v} + {IL}/ {v}
({A}/{ IL} )/{v} =( {A}/{v} )/{ IL} = {1l}/{ IL}/{V}
{A} / ( { IL} + {v } ) = {A} / { IL} + {A} / {v }
{A} / { IL} / { v } = {A } / ( { IL } { v } ). ( A.I 0 )
A.5 INNER MULTIPLICATION OF S-FUNCTIONS
The resolution of the Kronecker product of two irreducible representa-
tions {A}, { IL} of Sn into irreducible representations {v} of Sn is isomorphic
to the terms arising in the inner multiplication of the corresponding S-
functions:
{ A } 0 { IL} =  g AJU' { V }
v
( A.II )
where gA/LV is the number of times {v} appears in the reduction of the inner
product. Various relations among the coefficients are of importance:
g A/LV = g i..p.v
( A.12 )
( A.13 )
gA/LV = gpAv = gAV/L
The evaluation of the inner product of S-functions is much more difficult
than that of the outer product and is not treated here. Systematic methods
for the evaluation of inner products have been developed by Butler and
King. 227 ,414,415 Extensive tabulations have been given by Butler 31 and by
Vanagas. 410
A.6 CHARACTERS OF GROUPS AS S-FUNCfIONS
The orthogonal, rotational, and symplectic groups of degree n all occur
as subgroups of Un' Littlewood 225 has expressed the characters of the
orthogonal and symplectic groups in terms of S-functions:
[ A] =  ( - I )P /2 { A } / { )' }
y
( A.14 )
<A> =  ( - I )P/2 {A} / {  }
R
( A.I5)

REDUCfION OF THE NUMBER OF PARTS OF AN S-FUNCTION 379
where (y) and tS) are partitions of p and occur in the Frobenius series
(y): 1,( r:l ),( r:l S:1 ),... (A.16)
(8): 1,( r1 ),( r;l, s:1 ),... (A. I?)
The character theory for the rotation groups is essentially the same as for
the orthogonal groups except when the group dimension is even (n = 2v)
and Ap =1= O. In most of these exceptional cases, it is necessary to resort to
the method of difference characters, 26,416 although in the particular cases
of the groups R4 and R6 it is possible to use simpler methods,417 as is
discussed later.
The exceptional group G 2 occurs as a subgroup of R7 and is important
in the classification of the states of electrons or nucleons in equivalent
orbitals. The character theory of G 2 is discussed in a later section.
A.7 REDUCTION OF THE NUMBER OF PARTS OF AN S-FUNCTION
Under the operations of the restricted groups, an S-function defined on
n variables, where n = 2v or n = 2v + I, and having more than v parts is
equivalent to a series of S-functions on the same n variables but not having
more than v parts. 225
The S-function is expressed in the form
{ r + A I' r + A 2' . . . , r + , r - J.Lp' r - J.Lp - I' . . . , r - J.L I }
( A.I8 )
if n = 2v, or in the form
{ r + AI' r + A 2 , . · . ,r +, r, r - J.Lp' r - Jlv _ I' . . .,r - J.LI } (A.I9)
if n = 2v + 1.
Ignoring a possible change of sign for some transformations, this S-
function is independent of r and will be denoted {A: J.L}. It is expanded to
give a series of S-functions by using the relation
{ A : J.L } =  ( - 1 )p ( {A } / { a } ) ( { J.L } / { a } )
( A.20 )
where the sum is over all S-functions {a} that are partitions of p. This
relation is used as often as necessary to reduce all terms to ones of no more
than v parts.
Two special cases of this equivalence relation are often useful. In n

380 APPENDIX: SCHUR FUNCTIONS AND YOUNG TABLEAUX
variables, for unitary transformations, we have
{ A } = {A 1 - An' A 2 - An' . .. ,OJ.
( A.21 )
Ignoring the change of sign when n = 2v for transformations of negative
determinant only, we have also
{A} = {AI-An,AI-An-I""'O}.
( A.22 )
This latter relation gives the well-known particle-hole correspondence.
A.8 BRANCHING RULES
Under restriction to a subgroup, the characters of a group decompose
into a sum of characters of the subgroup.225 For the unitary group in n
variables, the characters are expressed as S-functions of up to n parts, but
for the restricted groups on n variables, of only v parts. Prior use of
relation (A.20) allows us to use the following relations without producing
nonstandard symbols.
(a) The Orthogonal Group. For this group,
{A}({A}/{a})
ex
( A.23 )
where the sum is over all S-functions of even parts only:
{ a } = {O}, {2 }, { 4 }, {22 }, { 6 } ,. . .
( A.24 )
and where the terms of the division are taken as orthogonal group
characters, that is,
({A}/{a})[/L]
(b) The Symplectic Group. For the symplectic group the result is the same
as Eq. A.23 apart from the replacement of {a) by {a}.
(c) The Rotation Group. For odd dimensions the characters are the same
as for the orthoon-1 group, but for even dimensions the characters with
/Lv =1= 0 decompose into two conjugate characters:
[ /L l' · · · , #Lv - l' /Lv ] , = [ /L l' . . . , /Lv - l' /Lv] + [ #L l' . . . , #Lv - l' - #Lv] . ( A.25 )
(d) The Exceptional Group G 2 . The group G 2 is a proper subgroup of the
seven-dimellsional rotation group, and Judd has derived the branching

KRONECKER PRODUCTS FOR THE CONTINUOUS GROUPS 381
rules by using the infinitesimal-operator approach, to yield the result 170
[WIW2W3] (i - k,j + k)'+ (j - k -I,i - j) (A.26)
where the sum is over all integral values of i,j, k satisfying the relations
w 1 >i>W2>j>w 3 >k> -w 3
The relation
(U 1 U 2 ) = - (u 2 -1,u 1 + 1)
is used to remove characters that do not give regular representations of G 2 .
A.9 KRONECKER PRODUcrS FOR THE CONTINUOUS GROUPS
Since we may express the characters of the unitary, symplectic, and
orthogonal groups in terms of S-functions, we may reduce the Kronecker
products of these groups in terms of the outer product of S-functions and
then perform the appropriate branching to get back the characters of the
group.
Kronecker products for G 2 are obtained in the same manner JY a
two-stage process, expressing the characters of G 2 in terms of those of R 7 ,
thence in terms of S-functions, and so on. The expression of the characters
of G 2 in terms of those of R7 is performed by noting that in the reduction
R7-G2' [U 1 U 2 0] contains (U 1 U 2 ) as the term of highest weight. The terms of
lower weights may be systematically removed by subtraction.
For example, in the case of (21), we may derive
[ 21 0 ]  ( II ) + (20) + (21 )
[200](20)
[110](ll)+(10)
Hence (21)=[210]-°[200]-[110]+[100], which may be expressed In S-
functions to give
(21) = {21 } - {2} - { 11 } + {O}
For even-dimensional rotation groups, products in only two groups have
been separated satisfactorily, the groups in four and six dimensions. 417 In
six dimensions, Littlewood 225 has shown that the group is isomorphic with
the four-dimensional unitary group, and the correspondences
[abc] { a + b,a - c, b - c} (A.27)
{ pqrs }  [ t (p + q - r - s ), 1- ( p - q + r - s ), ! (p - q - r + s ) ] (A.28)

382 APPENDIX: SCHUR FUNCTIONS AND YOUNG TABLEAUX
may be established, allowing us to obtain the products easily. For the
four-dimensional rotation group there is a 2: 1 homomorphism with the
double binary full linear group, and correspondences:
[a,b]{a+b} {a-b}'
{p} {q}'[!(p+q),!(p-q)]
( A.29 )
( A.30 )
may be obtained.
We may readily deduce that the separation of the Kronecker product in
R4 is given by
s t
[a,b][c,d] =   [a+c-a-l3,b+d-a+lJ]
a =0 {3=0
( A.31 )
where s is the lesser of a + band c + d, and t is the lesser of a - band c - d.
EXERCISES
A.2 Show that
[ 42] = { 1 2 } + {2 } - {2 2 } - {31 } - { 4 } + { 42 }
<42)= {42} - {31}
A.3 Show that under U(5)SO(5),
{ 2 2 1 }  [ 10] + [21 ] + [22]
and under U(6)Sp(6),
{2 2 1 } <1(0) + <III) + <210) + <221).
A.4 Establish that for Sp(6),
<Ill) X <210) = <110) + <2(0) + <211) + (220) + (310) + (321)
and for SO(7),
[ III ] X [210] = [ 110] + [ III ] + [200] + [210] + 2 [211 ] + [220]
+ [221] + [310] + [311 ] + [321 ]
A.tO OUTER PLETHYSM OF S-FUNCTIONS
For compact groups involving different-dimensional spaces or products
of different spaces we make use of Littlewood's outer plethysm of S-
functions,26 which amounts to forming symmetrized powers of S-functions

OUTER PLETHYSM OF S-FUNCTIONS 383
under outer S-function multiplication. The plethysm {A} @ { JL} may be
associated with the branching rule for U(n)U(m), where {JL} is an
S-function appropriate to an irreducible representation of U(n), and {A}
corresponds to an irreducible representation of U(m) of degree n, which
defines the embedding of U(m) in U(n).
In practice the outer S-function plethysm may be used whenever the
embedding of a compact group H in another compact group G is defined.
The key theorem 31 ,417 is:
lbeorem A.I
If under the restriction GH the unary character tIt decomposes as
fl1 -fat+fPt+"'+wt
then the character tAt of G decomposes into the characters { p t of H
according the characters of H contained in the outer-product plethysm
[fat +fPt +... +wt ]@tAt
(A.32)
The characters of G and H are expressed in terms f S-functions, and
the outer plethysms evaluated to give S-functions pertaining to H, which
are then expressed in terms of the characte s of H to yield the final result.
The fundamental problem is thus to be ble to evaluate arbitrary outer
S-function plethysms.
The operation of plethysm is distributive on the right with respect to
addition, subtraction, and multiplication. It is assumed that the operation 
precedes ordinary multiplication; thus the following rules may be deduced:
A@(BC) = (A@B)(A@C) =A@BA@C
( A.33 )
(A@B)@C=A@(B@C)
( A.34 )
( A.35)
A@(B + C)=A@B + A@C
The operation is not distributive on the left with respect to addition,
subtraction, or multiplication. Littlewood 26 ,224 has derived the additional
rules that complete the definition of the algebra:
(A+B)@{A) =rMPA(A@{ JL} )(B@{v})
( A.36)
r
(A - B) @ {A} =  ( - 1) r MPA (A @ { JL } ) (B @ { P }) (A.37)
(AB) @ {A} =  glL PA (A @ { It} ) (B@ { v } ) (A.38)

Table A.l. Some Typical Outer S-Function Plethysms
{2}  {2} = {2 2 } + {4}
{2}{12}= {31}
{ 3 }  { 2 } = { 42 } + { 6 }
{ 3 }  { 1 2 } = { 3 2 } + { 51 }
{21}{2} = {2 3 } + {31 3 } + {321} + {42}
{21}  {12} = {2212} + {321} + {3 2 } + {41 2 }
{2}  {3} = {2 3 } + { 42} + {6}
{2}  {21} = {321} + { 42} + {51 }
{2}  { 1 3 } = {3 2 } + {41 2 }
{ 4 }  { 2 } = { 4 2 } + { 62 } + { 8 }
{ 4}  { 1 2 } = {53} + {71 }
{31 }  {2} = {3 2 1 2 } + { 42 2 } + { 431 } + { 4 2 } + {51 3 } + {521 } + {62}
{ 31 }  { 1 2 } = { 3 2 2 } + { 421 2 } + { 431 } + { 521 } + { 53 } + {61 2 :
{ 2 2 }  { 2 } = {2 4 } + { 3 2 12 } + { 42 2 } + { 4 2 }
{2 2 }  { 1 2 } = { 32 2 1 } + { 431 }
{ 2 }  { 4 } = {2 4 } + { 42 2 } + { 4 2 } + { 62 } + { 8 }
{ 2 }  { 31 } = {32 2 1 } + { 42 2 } + { 431 } + { 521 } + { 53 } + ! ':2 } + {71 }
{ 2 }  { 2 2 } = { 3 2 12} + { 42 2 } + { 4 2 } + { 521 } + { 62 }
{ 2 }  { 21 2 } = {3 2 2 } + { 421 2 } + { 431 } + { 521 } + { 53 } + { 61 2 }
{ 2 }  { 1 4 } = { 431 } + { 51 3 }
{ 3 }  { 3 } = { 4 2 1 } + { 52 2 } + { 63 } + {72 } + {9}
{3}  {21} = {432} + {531} + {54} + {621} + {63} + {72} + {81}
{3 }  { 1 3 } = {3 3 } + {531 } + {63 } + {71 2 }
{21}{3} = {32 2 1 2 } + {32 3 } + {3 2 1 3 } + {3 2 21} + {3 3 } + {41 S } + {4213)
+ 2 { 42 2 1 } + { 431 2 } + { 432 } + { 4 2 1 } + { 521 2 } + { 52 2 } + { 531 } + {63 }
{21}{21} = {2 4 1} + {321 4 } +2{32212} + {32 3 } + {3 2 13} +3{3 2 21} +2{4213} +3{42 2 1}
+ 3 { 431 2 } + 3 { 432 } + { 4 2 1 } + { 51 4 } + 2 { 521 2 } + { 52 2 } + 2 { 531 } + { 54 } + {621 }
{21 }  { 1 3 } = {2 3 1 3 } + {32 2 1 2 } + {32 3 } + {3 2 1 3 } + {3 2 21 } + {3 3 } + { 421 3 } + { 42 2 1 }
+2{4312} + {432} + {4 2 1} + {521 2 } + {52 2 } + {531} + {61 3 }
{ 5 }  { 2 } = { 64 } + { 82} + { 10 }
{5}  { 1 2 } = { 52 } + {73 } + {91 }
384

INNER PLETHYSM OF S-FUNCTIONS 385
This last result allows one to extend the key theorem (Eq. A.32) to include
the case where H is actually a product of groups defined on different
variables.
Much effort has gone into the evaluation of outer S-function plethysms,
and it would be out of place to examine them here. A substantial
simplification of the usual methods 31 has been made by Butler and
King. 226 Tables of outer plethysms have been prepared that cover must
cases of practical interest.31, 206, 207,410 A short list is given in Table A.I.
Inspection of Table A.l shows that
{2}  {21 } = {51} + { 42} + {321 }
and since under U(15)U(5), we have {1}{2}. We deduce that {2I}
{51}+{42}+{321}. For 0(5) we have the typical plethysm
[ 21 ]  { 2 } = ( {21 } - { 1 } )  { 2 } (from Eq. A.14)
= {2I } @ {2 } - {21 } { 1 } + { 1 } @ { 1 2 } (from Eq. A.37)
= { 42 } + {31 3 } + { 321 } + {2 3 } - {31 } - {2 2 } (from Table A.l )
+{212}+{12}
These S-functions may be reduced to two or fewer parts, using the method
of Section A.7 and then Eq. A.23, to obtain the 0(5) characters:
[21 ]  { 2 } = [00] + [ 10] + [ II ] + [20] + [21 ] + 2 [22] + [30] + [31 ] + [32]
+ [ 40] + [ 42 ]
EXERCISES
A.5 Show that if [1][2] under SO(5)SO(3), then
[21][1] + [2] + [3] + [4] + [51
A.6 Show that if <1>[] under SD(6)SO(3), then
<12>[2] + [4].
A.tt INNER PLETHYSM OF S-FUNCfIONS
The outer plethysm of S-functions leads to substantial simplification in
the calculation of branching rules for compact groups with embedded
continuous subgroups. The inner plethysm of S-functions is of importance

386 APPENDIX: SCHUR FUNCTIONS AND YOUNG TABLEAUX
in the corresponding problem of the branching rules associated with the
embedding of a finite group in a compact continuous group, such as for
example the embedding of the crystallographic group in SO(3).
The inner plethysm of S-functions essentially involves symmetrized
powers of S-functions under inner S-function multiplication. 418 As with
inner products, the practical computation of inner plethysms raises a
number of difficult problems. In recent times significant progress has been
made by Butler and King,227,415 who have also made a useful tabulation.
The inner plethysm has proved useful in the introduction of the su-
pergroup concept to ligand-field theory.189
A.12 MACHINE CALCULATION OF S-FUNCfION PROPERTIES
The calculation of S-function properties is highly adaptable to machine
computation, and this fact represents one of the great strengths of the
S-function methods. An elaborate computer program for carrying out the
various manipulations required for all the standard S-function properties
has been developed by Butler,31,414 allowing many group properties to be
obtained by a simple interrogation of the computer. Butler has published
an extensive tabulation of the properties of compact groups,31 which
effectively abolishes much of the labor associated with making group-
theoretical calculations.
Undoubtedly the next step will be to construct a similar program to
evaluate the coupling coefficients for arbitrary chains of compact groups
and thus to be able to fully exploit the Wigner-Eckart theorem. It appears
at the time of writing that this important step will soon be realized.

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380. G. Mack and A. Salam, "Finite-Component Field Representations of the Conformal
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Author Index
Numbers in italics refer to pages on which the complete references are listed.
Ado, I. D., 93, 71
Agrawala, V. K., 232, 246, 395
Akhiezer, A. I., 339, 340,404
Aldrovandi, R., 213, 393
Alisauskas, S. J., 100, 390
Anderson, R. L., 217, 394
Ardalan, F., 312, 400
Arima, A., 360, 405
Armstrong, L. L., 201, 296,393, 399
Aydin, Z. Z., 339, 403
Bailey, W. N., 200, 393
Baiguni, A., 333, 337, 338, 339, 345,403
Baird, G. E., 250, 261,396
Bander, M., 311, 399
Bandzaitis, A. A., 145, 199,200,201,219,
231,391
Bargmann, V., 39, 143, 194,196,197,236,
286,298,314,389,392,395,398
Baru t, A. 0., 1, 39, 45, 143, 149, 168, 1 70,
176,178,196,216,287,311,312,320-2,
324,326,331,333-5,337-40,343,345-7,
387,389, 392, 398-400, 402-4
Bateman, H., 347, 404
Behrends, R. E., 1, 387
Belinfante, J. G., 232, 246, 395
Beltrametti, 6. G., 141, 142,391
Berestetski, V. B., 339, 340,404
Bethe, H. A., 5,308,315,317,338, 339,
389,399
Biedenharn, L. C., 99,141,196,201,203,
213,219,235,240,250,261,304,391,
393-4, 396
Birkhoff, G., 43, 389
Bisiacchi, A., 287, 398
Bivins, R., 199, " 57,393
Blasi, A., 141, 142,391
Bluman, G. W., 217,394
Boerner, H., 14, 389
Bohm, A., 287, 312, 398, 400
Bordarier, Y., 232,395
Bornzin, G. L., 320, 339,402, 403
Bose, A. K., 94, 223, 224, 390
Bose, S. K., 313, 401
Boyer, C. P., 312, 400
Braathen, H. J., 319,401
Brezin, E., 339, 403
Briggs, J. S., 231, 395
Brink, D. M., 231, 395
Britten, W. E., 345, 404
Budini, P., 287, 288, 398
Butler, P. H., 126, 138,232,242,246,248-
51,301,377-9,381,383,385-6,388,
390, 395-7, 405
Carmeli, M., 314, 401
Cartan, E., 2, 46, 58, 61, 123, 131, 168,
170,388,390,392
Carter, R. W., 87, 90, 390
Casimir, H., 54, 390
Castel, B., 231,395
Cetlin, M. L., 250, 396
Chac6n, E., 38, 389
Chevalley, C., 2,11,14,87,94,174,388-90
Chih-ta, Yen, 168, 392
Condon,E. U., 3,4,249,250,267, 366,
388
Cordero, P., 213, 216-7, 343,393-4,404
Corrigan, D., 321, 334-5, 402-3
Coxeter, H. S. M., 100,390
407

408 AUTHOR INDEX
Crubellier, A., 283,397
Cunningham, E., 347, 404
Cunningham, M. J., 201, 249, 262,393,
396-7
Dam,H.van,196,235,393
Derome, J-R., 232, 246-7,395
Devine, S. D., 6, 389
Dirac, P. A. M., 243, 269, 313, 332-3, 350,
396,401
Dixmier, J., 149,312,392
Doebner, H. D., 288, 398
Dreitlin, J., 1,387
Dymus, S. A., 217, 394
Dynkin, E. B., 2, 73, 101, 103-4, 115, 123,
136, 223-4, 388
Eckart, C., 4, 196, 218, 392
Edmonds, A. R., 145, 199, 200, 219, 242,
260,265,367,370,391,396-7
Ehrman, J. B., 313,401
Eisenhart, L. P., 31, 389
EI-Baz, E., 231, 395
Elliott, J. P., 1, 268, 271-2, 350, 389, 397
Englefield, M. J., 307, 311, 399
Erdelyi, A., 200, 393
Esteve, A., 312, 400
Evans, N. T., 313, 401
Fano, U.,219, 274,394
Feneuille, S., 283, 397
Ferreira, P. L., 213,311,328,393,399
Ferretti, I., 201, 203,393
Fescbach, H., 323, 325, 402
Flato, M., 345,404
Flowers, B. H., 1, 30, 260, 357, 360, 370,
387,389,397,404
Fock, V. A., 236, 298, 395
Foldy, L. L., 319,401
French,J.B.,359,404
Freudenthal, H., 14, 131, 389
Frobenius, G., 2, 388
Fradkin, D., 332, 402
Fronsdal, C., 1, 39, 143, 149, 196, 286,
311, 329, 330, 333, 387, 389, 398-9
Fulton, T., 345, 404
Furlan, P., 217, 394
Galindo, A., 219, 394
Gantmakher, F. R., 168, 169,392
Gel'fand, I. M., 143, 250,305,314,332,
391,396,399,402
Ghirardi, G. C., 217, 343, 394, 404
Ginocchio, J. N., 360, 405
Goshen, S., 282, 397
Graev, M. I., 143,391
Griffith, J. S., 219, 227, 246,394,396
Gruber, B., 141,391
Giirsey, F., 313, 400
Hall, G. G., 2,388
Halpern, F. R., 314, 319,401
Hamermesh, M., 246, 263, 357, 396
Harish-Chandra, 314,401
Harvey, M., 268,397
Haskell, T. G., 288, 296, 398-9
Hassitt, A., 263, 397
Haugen, R. B., 345-7, 404
Hausner, M., 21, 46, 51, 170, 172, 389
Hecht, K. T., 248, 254, 261, 265,360,396,
405
Helgason, S., 11, 14, 170, 172-4,389
Helmers, K., 359, 360,404
Hemenger, R. P., 360, 405
Hermann, R., 95, 390
Herrick, D., 313,401
Hill, E. L., 214, 394
Hoffman, W. C., 1, 388
Hojman, S., 216-7,394
Holland, D. F., 38,389
Holman, W. J., 201, 203,393
Hope, J., 254, 397
Hull, T. E., 207, 393
Hwa, R. C., 283, 288, 397
Ibrahim, E. M., 248, 385,396
Ichimura, M., 360, 405
Infeld, L., 207, 393
Innes, F. R., 360, 405
Inonii, E., 288, 290, 398
Itzykson, C., 286,311,339,398-9,403
Jacobsen, N., 21, 51, 62,389
Jaffe, L., 313, 401
Jahn, H. A., 1,254,263,387,397
Jakimov, G., 247,396
Jauch, J. M., 214,394
Joos, H., 319, 401
Jucys, A. P., 100, 145, 199-201,219,231,
390-1,395

Judd, B. R., 5,218-9,251,262,350,352,
358, 360, 365, 367, 381,389, 394, 404-5
Kadyshevsky, V. G., 339,403
Kaempfer, F. A., 329,402
Kaufman, B., 1,388
Kerman, A., 359, 360,404
Kihlberg, A., 149, 312, 392
Killing, W., 46, 390
King, R. C., 251, 378, 385-6,397, 405
Kleine.rt, H. M., 287,311,321-2,324,326,
333-5,398-9, 402-3
Klimyk, U., 219, 227, 250, 395, 397
Komen, G. J., 333, 403
Konuma, M., 100, 390
Koster, G. F., 219, 394
Kumei, S., 217, 394
Kuriyan,J.G., 143, 149, 178,391
v
Kursunoglu, V., 331,402
Lafoucriere, J., 231, 395
Lanik, J., 208, 212, 213,393-4
Lardy, P., 168,392
Lawson, R. D., 360,405
Lee, W., 1, 387
Lenz, W., 298,399
Levinson, I. B., 231, 395
Levy-Leblond, J-M., 329, 402
Lezuo, K. J., 100,390
Lindblad, G., 149, 178, 195,392
Lipkin, H. J., 282, 397
Littlewood, D. E., 2,116,126,251,372,
375,378-383,386,388,397,405
Macfarlane, M. H., 360, 405
McIntosh, H. V., 299, 399
MacLane, S., 43, 389
MacMahon, P. A., 373, 405
Mack, G., 345, 346, 404
Magnus, W., 200, 393
Majorana, E., 332, 402
Mal'cev, A. I., 223, 395
Malin, S., 333-5, 402-3
Malkin, I. A., 287, 311, 399
Man'ko, V. I., 287, 311, 399
Martin, C., 312, 400
Massot, J. N., 231, 395
Maurin, K., 19,389
Mehta, C. L., 94, 131, 223,390
Melsheimer, 0., 288, 398
AUTHOR INDEX 409
Merzbacher, E., 179, 182,392
Messiah, A., 145, 209, 211, 340,391
Metropolis, N., 199, 257,393
Miller, W., 1, 208, 286, 388
Minlos, R. A., 305, 314, 332, 399
Montgomery, D., 19, 131, 151,389
Morita, K., 313, 400
Morse, P. M., 215, 323, 325,394, 402
Moser, W. O. J., 100,390
Moshinsky, M., 38, 250, 261-2, 269, 283,
291-2,353,356,360,389,397,404
Mukunda, N., 39,143,149,178,314,389,
391,401
Murai, Y., 312, 400
Murnaghan, F. D., 2, 38, 388
Nagel, B., 149, 178, 195,392
Nagel, J. G., 142,391
Naimark, M. A., 305, 314, 332,399
Nambu, Y.,311, 333,399, 402
Newton, T. D., 149,312,392
Nono, T., 1, 388
Nuyts, J., 283, 288, 397
Oberhettinger, F., 200,393
O'Raifeartaigh, L., 141,219,391,394
Osvjannikov, V., 217,394
Pajas, P., 314, 401
Pang, S.C., 248, 254, 265,360,39405
Parikh, J. C., 360, 405
Parker, R., 313,401
Patera, J., 94, 196, 223-4,390, 392
Pauli, W., 30, 48, 236, 298, 300, 319,389,
399
Perelomov, ,'\. M., 141,391
Phillips, E. C., 149, 178, 322, 392
Pollack, R. D., 50, 390
Ponomarev, V. A., 314,401
Pontryagin, L. S., 8,41,43, 151, 174,389
Popov, V. S., 141,391
Pyatetskii-Shapiro, I. I., 143,391
Quesne, C., 283, 291-2,353,356,360,397,
404
Racah, G., 1,2, 55, 62, 99, 101, 130, 141,
219,221,233,242,248,254,261-2,266,
274,358-9,387,390,394,397,404
Raczka, R., 45, 168, 170, 172, 389

410 AUTHOR INDEX
Resnikoff, M., 248, 396
Richardson, A. R., 375,405
Riordan, J., 197,393
Robinson, G. de B., 2,263,375-6,388
Rohrlich, F., 345,404
Rose, M. E., 269, 397
Rosen, J., 141-2, 391
Rotenberg, M., 199,257,393
Rudzikas, Z. B., 145,391
Riihl, W., 314, 401
Runge, C., 298,399
Salam, A., 1, 345-6, 387, 404
Saletan, E. J., 288, 399
Salpeter, E. E., 308, 315, 317, 338-9,399
Sankoff, D., 196, 392
Sannikov, S. S., 149, 201,391,393
Sansone, G., 323, 325,402
Santhanam, T. S., 141,391
Satchler, G. R., 231,395
Savukynas, A. J., 145,391
Schreier, 0., 160,392
Schur, I., 2, 372, 388
Schwartz,J.T., 21,46,51,170,172,389
Schwarz, F., 312,400
Schwinger, J., 306, 399
Sciarrino, A., 314, 401
Shaffer, W. H., 295, 399
Shalit, A. de, 350, 357, 359, 366-7,404
Shapiro, Z. Ya., 305, 314, 332,399
Sharp, W. T., 232, 246-7,395
Shima, K., 100, 390
Shortley, G. H., 3,4,249,250,267,366,
388
Simon, J., 345, 404
SinanogIu, 0., 313,401
Sirota, A. I., 168, 170,392
Slater, L. J., 200, 393
Smith, P. R., 114, 218, 390
Solodovnikov, A. S., 168, 170, 392
Solomon, A. I., 216, 394
Sona, P. G., 312, 400
Speiser, D., 151,392
Srivastava, P. K., 94, 131, 223,390
Sternheimer, D., 345,404
Stone, A. P., 141,219,225,390
Stoyanov, D. Tz., 333,402
Streater, R. F., 286, 333, 398, 402
Strom, S., 149, 312,392, 400
Sudarshan, E. C. G., 143, 149, 178, 391
Swart,J. J. de, 248,396
Szpikowski, S., 30, 360, 389, 405
Tahir Shah, K., 142, 391
Takabayasi, T., 333,402
Talman, J. D., 1, 39, 286, 290,388
Talmi, 1.,350,357,359,366-7,404
Thomas, L. H., 149, 312, 392
Tiemble, A., 312, 400
Tilgner, H., 286, 398
Todorov, I. T., 333, 339,402-3
Toller, M., 314, 401
Tricomi, F. G., 200, 393
Trifonov, D. A., 287,398
Tripathy, K. C., 335, 403
Ui, H., 201, 291, 393
Umezawa, M., 141,390
Vanagas, V. V., 231, 371, 378, 385,395,
405
Verde, M., 201, 203,393
Vergados, J. D., 248, 396
Vilenkin, N. 1.,1,143,286,387, 3Q1
Vries, E. de, 232, 395
Vries, H. de, 14, 131,389
Wada, M., 100, 390
Wadzinski, H. T., 74,390
Waerden, B. L. van der, 1, 2, 66, 70, 196,
387
Wallace, A., 151, 392
Wang, Kuo-Hsiang, 201, 203,393
Weyl, H., 1,2,61,100,135,196,271,387-
8,390
Wieringen, H. van, 254, 397
Wigner, E. P., 1,4, 54, 92-3, 193, 196, 201,
218-9,232,246,288,314,318,387,393,
398,401
Winternitz, P., 314,401
Witten, L., 345,404
Wooten, J. K., 199, 257,393
Wulfman, C. E., 217, 232, 299, 307,394,
396
Wybourne, B. G., 2, 5, 112, 114, 126, 149,
196, 218, 229, 232, 236, 248-9, 251,
288, 296, 301, 303, 312, 358, 360, 368,
373, 377-8, 381, 383, 385-6,388, 390,
392, 395, 398-9, 405

AUTHOR INDEX 411
Yaglom, A. M., 332,402
Yamanouchi, T., 1, 387
Yao, T., 312,400
Young, A., 2, 373, 375,388
Young, Kiang-Chuen, 201, 203,393
Zanten, A. J. van, 232, 395
Zhelobenko, D. P., 250, 396
Zinn-Justin, J., 339, 403
Zippin, L., 19, 131, 151,
389

Subject Index
Abelian algebras, 44
Abelian groups, 22, 44, 55
Abelian ideal, 44, 49
Abstract representations, 92
Adjoint representations, 44, 95
Adjoint tensor operators, 243
Affine gro up, 9
Algebraic vector current operator, 333
Analytic-continuation, 203
Antisymmetric tensors,S 3
Arcwise connected, 157
Automorphisms, 43
Basic representations, 113
Boson operator realizations, 181, 269, 306
Branching rules, 380
Building-up principle, 251
Campbell-Hausdorff formula, 16
Cartan's criterion for semisimplicity, 46
Cartan matrix, 77
Cartan su balge bra, 61
Cartan-Weyllabels, 124
Cartan-Weyl normalization, 65
Casimir operators, 53, 55,139,178,277,
311
Central field, 366
Chevalley basis, 87
Classification of j = 7/2 shell, 357
Compact Lie groups, 55,
Com pact spaces, 155
Com plex extensions of real Lie algebras, 45
Complex space, 9
Computation of weights, 102
Conformal group, 345
Conformal transformations, 345
Connected spaces, 155
Connectivity, 18
Continuous group, 8
Continuous matrix groups, 10
Continuous topological group, 162
Contragredient representations, 94
Convective currents, 334
Coordinate realization of SO(4,2), 319
Coset spaces, 164
Coupled tensor operators, 263
Coupling coefficients, 195, 222
Coupling coefficients for SO(2,1), 196
Coupling coefficients for SO(3), 196
Coupling coefficients for SO(4), 236
Co-weights, 88
Degeneracy group, 282
Derivations of Lie algebras, 50
de Sitter group, 312
Dilatation operator, 323
Dimensions of representations, 134
Dirac ')'-ma trices, 340
Dirac theory of electron, 343
Direct product,S 6
Direct sums,S 2
Discrete topology, 151
Discrete topological groups, 162
Disjoint pieces, 18
Dynamical groups, 286, 307
Dynkin diagrams, 76
Electric dipole operator, 325
Elementary representations, 116, 1 79
Endomorphisms, 43
Enumeration of roots, 83
Equivalent representations, 93
Euclidean groups, 32, 39, 49, 54
Euclidean space, 9, 152

414 SUBJECT INDEX
Euler angles, 34
Exceptional groups, 127
Exceptional Lie algebras, 71
Faithful representation, 93
Fermions, 349
Finite groups, 8
Finite transformations, 37, 190
Galilean boosts, 329
General Linear groups, 11
Generation of fmite elements, 33
Global properties of Lie grou ps, 150
Graphical representation of root vectors, 65
Grou p characters, 378
Group contractions, 288
Group postulates, 7
Hausdorff spaces, 154
Homeomorphisms, 153
Homogeneous spaces, 166
Homomorphisms, 42
Homotopic paths, 156
Homotopy group, 159
Hydrogen atom, 297
Ideal, 44
Identity representation, 223
Indecomposable representations, 93
Infinite component wave equations, 332
Infinite discrete groups, 8
Infinitesimal group generators, 20, 27
Infinitesimal operators, 25
Infinitesimal operators of SO(3), 28
Infinitesimal rotations, 23
Infinitesimal transformations, 23, 27
Inner S-function multiplication, 378
Inner S-function plethysm, 385
Invariants, 142
Invariant topological subgroups, 164
Invariant subgroups, 56
Isomorphisms, 42
Isoscalar factors, 248
Isotropic harmonic oscillator, 211, 268
Kepler problem, 212, 299
Killing form, 46, 55
Kronecker powers, 114
Kronecker products, 109, 381
Ladder operators in SO(4), 304
Ladder operators in SU(3), 278
Lie algebras, 21, 40
Lie algebras of rank 2, 67
Lie algebras of rank Q > 2, 70
Lie algebra of SO(4), 48
Lie groups, 19
Locally compact spaces, 156
Locally connected spaces, 155
Lorentzian boosts, 331
Lorentzian space, 9
Majorana equation, 333
Matrix exponential functions, 14
Matrix properties, 10
Ma trix represen ta tion, 92
Maximal ideal, 44
Metric spaces, 154
Metric tensor, 46
Mixed continuous groups, 18
Morse potential, 215
Multipole operator matrix elements, 292
Natural topology, 152
Nilpotent Lie algebras, 51
Noncompact generators, 195
Noncompact Lie groups, 55
Num ber opera tor, 351
Orthogonal groups, 12
Oscillator group, 286
Outer S-function multiplication, 375
Outer S-function plethysm, 382
Parametrization of group elements, 17, 19
Partitions, 373
Positive roots, 73
Positive weights, 98
Proper ideal, 44
Properties of roots, 59, 62
Quasi-spin formalism, 360
Quotient groups, 164
Quotient spaces, 166
Racah's factorization lemma, 240
Rank, 58
Real and complex representations, 93
Realization of su(I,I), 208
Real Lie algebras; 167

Real space, 9
Reciprocity, 248, 253
Recoupling coefficients, 230
Regular matrix groups, 8
Representation space, 92
Representations of SO(2,1), 143, 183
Represen ta tions of SO(3), 143
Root vector diagrams, 66
Root vectors, 66
Runge-Lenz vector, 298
Schur functions, 372
Second quantization, 269, 350
Selection rules, 227
Semidirect sum, 49, 52
Semisimple Lie algebras, 46
Semisimple Lie group, 56
Seniority, 359
S-function division, 377
Simple Lie algebras, 46
Simple Lie groups, 56
Simple roots, 73
Solvable Lie algebras, 50
Spectrum generating algebras, 207
Spinor representations, 121, 180
Standard Cartan-Weyl form,S 7
Structure constants of Lie groups, 21,
31
SU(3) tensor operators, 272
Subalgebras, 43
SUBJECT INDEX 415
Subgroups, 55
Supergroups, 349
Symmetric functions, 374
Symmetric group, 375
Symmetry classification of operators, 363
Symmetry of roots, 60
Symplectic groups, 13
Tensor operators, 220
Three parameter Lie groups, 176
Tilted states, 321
Tilting operator, 209, 321
Topological groups, 161
Topological manifold, 167
Topological product, 153
Topological spaces, 151
Topological subgroups, 163
Topological subspace, 152
Transformation of basis, 41
Two-dimensional rotation group, 22, 28
Unitary groups, 11
Unitary representations, 95, 186
Universal covering group, 174
Universal covering spaces, 160
Weight spaces, 66, 98
Weight vectors, 98
Weyl reflection group, 100
Wigner-Eckart theorem, 225