Lecture Notes
in Physics
Edited by J. Ehlers, Munchen, K. Hepp, Zurich,
R. Kippenhahn, Munchen, H. A. Weidenmuller, Heidelberg, and J. Zittartz, Koln
Managing Editor: W. Beiglbock, Heidelberg
63
V K. Dobrev • G. Mack • V B. Petkova
S. G. Petrova • LT Todorov
Harmonic Analysis
on the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory
Springer-Verlag
Berlin • Heidelberg • New York 1977
Authors
V. К. Dobrev
V. В. Petkova
S. G. Petrova
I. T. Todorov
Institute of Nuclear Research and Nuclear Energy
Bulgarian Academy of Sciences
Sofia 1113/Bulgaria
G. Mack
II. Institut fur Theoretische Physik der Universitat Hamburg
Luruper Chausee 149
2000 Hamburg 50/BRD
Library of Congress Cataloging in Publication Data
Main entry under title:
Harmonic analysis on the n-dimensional Lorentz group and. its application to conformal quantum field theory.
(Lecture notes in physics ; 63)
Includes bibliographical references.
1. Lorentz,transformations. 2. Quantum field theory.
3. Harmonic analysis. I. Dobrev, V. KB
II. Series.
QC174.52.L6IB7	530.1’43	77-5339
ISBN 3-540-08150-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08150-X Springer-Verlag New York Heidelberg Berlin
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© by Springer-Verlag Berlin • Heidelberg 1977
Printed in Germany
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2153/3140-543210
PREFACE
Work on conformal quantum field theory and partial wave expansions led the authors to the necessity to master a number of topics in representation theory and to solve some problems whose coverage in the literature was not quite adequate. It soon became apparent that a systematic study of representations and intertwining operators for the pseudoorthogonal group is desirable, which would place our subject in the context of modern harmonic analysis. That led to Part One of this book. We included in Part Two (following the opportune advice of Klaus Hepp) some recent results on tensor product expansions and related physical applications (see М2 D4 - D6 ).
In the course of our work on Part One (particularly during the stay of three of us in Princeton) we had the opportunity to consult a number of experts in the field. At different stages we benefited from discussions with F. Gursey, H. Kraljevic, D. P. Zhelobenko and especially with G. J. Zuckerman. It is a pleasure to express our sincere gratitude to all of them.
One of us (V.K.D.) would like to thank Professor A- S. Wightman for the hospitality extended to him at the Princeton University. Two of us (G.M. and I.T.T.) thank Professor C. Kaysen for his hospitality at the Institute for Advanced Study.
CONTENTS
Introduction	"1
Part One Type I Representations and Intertwining Operators for 0*(2h+1, 1) q
1 Elementary representations of the pseudo-orthogonal group
1.	Group structure. Preliminaries
l.A The group 0(2h + 1,1) and its Lie algebra	ИИ
l.B Subgroups and decompositions	И2
l.C The compactified Euclidean space as a homogeneous space of G qg
l.D Matrix realization of various subgroups of G. Construction of
the Bruhat decomposition	П8
l.E Relationship between the Bruhat and the Iwasawa decomposition The Haar measure	25
2.	Induced representations. Definition and various realizations
2.A Synopsis on the irreducible representations of the orthogonal group	28
2.В Covariant vector-valued functions on G. Definition of
the induced representations	52
2.C The compact picture. К-content of the elementary representations 54
2.D The noncontact picture: x-space realization	57
3.	Further properties of the elementary representations .
3.A Equivalence, irreducibility, completeness	4И
3.B Characters of elementary representations	47
3-C The spherical trace function. character of a subquotient of an elementary representation	53
3.D The principal series of unitary representations	54
3.E Infinitesimal generators and Casimir operators of the elementary representations	57
VI
II. Intertwining distributions and their Fourier transform
4.	Intertwining operators: x-space realization
4	.A Group theoretical definition of the intertwining operators
4	.В The intertwining distributions in the noncompact picture
5.	Momentum space expansion of the intertwining distribution and positivity
5.A Fourier transform of
5.В Harmonic expansion of G (p) Л
5.C Normalization and positivity for nonexceptional representations. Complementary series of unitary IR's
5.D Wightman positivity
III. Properties of elementary representations at exceptional integer points
6. Nondecomposable representations and intertwining differential operators
6.A Subrepresentations of exceptional elementary representations
6.В Intertwining differential operators. Partial equivalence among
('> + the representations
6.C Hermitian forms on invariant subspaces. Exceptional series of unitary representations
6.D Differential identities between hermitian forms for exceptional representations
7. Discrete series of unitary representations
7.A Definition and general properties of the discrete series of S0*(2n,l)
60
64
66
69
75
81
85
89
94
-10-1
-103
VII
7. В. Unitarily induced representations on G/K	107
7. C. Realization of the unitary representation- U* in the
space Zf (NA)	110
S +
7. B. К-invariants. Solution of the eigenvalue problem for the
Casimir operator. The discrete series Ugy	115
7. E. Two-point Green function. Equivalence of	with the
subrepresentation of	acting in B^	-122
8. The Plancherel theorem. Concluding remarks
8. A. Harmonic analysis of the left regular representation of
S0't'(2h +1, 1) for integer h	128
8. B. Harmonic analysis on S0't'(2n,i;. The role of the discrete
series	1J1
8. C, Synopsis on unitary type 1 representations. Summary of
equivalence relations	1J5
Appendix A.	Symmetric tensor representations of S0(n) and
their decomposition in IR's of S0(n-1)
A.1 Harmonic extension of homogeneous polynomial functions
on the light cone	1J8
A.2 S0(n-1) expansion of homogeneous polynomials. The zonal
spherical functions	НД-Н
A.5
A.4
Evaluation of the proportionality constant a(
Vе	1»3
expression for the projection
scalar products in
Berivation of factorized
a, operators
between the
144
A.5 Interior differentiation on the complex cone. Expression
for the convolution of two tensors in terms of homogeneous
polynomials
. 149
Appendix B.	The special cases h=1 and h=y . Relation to the 155
formalism of two by two matrices
B.1 Reduction of the representation % of О* (5,1) into elemen-
VIII
tary representations of SL(2,C)	153
n^s B.2 Vanishing of the projection operators I I for s>1	155
B.5 The structure of exceptional representations for h=1
B.4 Elementary representations of S0'I“(2,'l). The analytic discrete
series	^59
Appendix C, Positivity of the invariant scalar product in the 165 subspace of C^>
C.1 The problem. Asymptotic expansion of f(p,^ ) for p-> 0	165
C.2 Existence of nontrivial positive semidefinite hermitian form
(f,G& f) on c;-	167
Part Two.	Conformal Partial Wave Analysis	173
IV. Clebsch-Gordan expansion of the tensor product of two
unitpry principal or supplementary series representations
9.	The Kronecker product of two elementary representations
as an induced representation on G/MA	175
10.	Construction of the Clebsch-Gordan expansion
10.A.	Clebsch-Gordan kernels	181
10.B.	Application of the Plancherel theorem to the Kronecker
product of two principal series representations	192
10.C.Odd space time dimension 2h	200
10.В.	Analytic continuation in c^ and c^	203
IX
11.	Special cases and further properties of the expansion formula
11.	A. The Clebsch-Gordan kernel for two class I representations.
Symmetry and normalization	206
11.B. Identities for the Clebsch-Gordan kernels at exceptional integer points	2'13
11.C. Tensor product representation and Clebsch-Gordan expansion for distributions	217
V. Dynamical derivation of vacuum operator product expansion in Euclidean conformal quantum field theory
12. Benormalizable models of self-interacting scalar fields.
Dynamical equations for Euclidean Green functions
12.A. A б-dimensional model. Euclidean Green functions. Generating functionals	219
12.B. Graphical notation.	li-	and	2i-kernels	220
12.C. Dynamical equations.	Stress	energy	tensor. Ward identities 22"1
12.D. A more realistic model	224
13. Invariance and invariant solutions of the dynamical equations.
Conformal partial wave expansion for the Euclidean Green functions 1J.A. Euclidean conformal invariance of the equations	225
1J.B. Conformal invariant 2-	and	J-point functions	227
13.C. Skeleton diagram expansion	230
13.D. Conformal partial wave	expansion	232
13.E. Further expansions	234
I2*-. Implications of the dynamical equations. Pole structure of conformal partial waves 14.A. Poles in the conformal partial waves implied by the vertex bootstrap equations	2J8
14.B. Pole structure of the n-point partial waves. Expression for the residues	240
14.C. Basic conformal covariant tensor fields. Analyticity assumption 241 15. Derivation of an operator product expansion for vacuum expec-tation values
X
15.A Another form of the conformal expansion involving a Minkowski momentum space integral. The Q-kernels	244
15.В The vacuum operator product expansion	249
15.	C	Wightman positivity for the 4-point function	253
16.	The problem of crossing symmetry. Concluding remarks
16.	A	Crossing symmetry and duality	254
16.В A crossing symmetry representation for the 4-point function	256
16.и	Summary and discussion	257
Appendix B. Proof of lemma 1o.3.	259
Appendix E,	A summation formula involving ratios of Г -functions 260
Appendix F.	Partial Pourier transform of VCx^x^x ) and related formulas	262
P.l Pourier transform in x^	262
P.2 Derivation of Eq.(15.56) for the conformal partial wave 263
Appendix G.	Identities between Q and / functions for
partially equivalent representations	267
References	268
Figures 1,2, 3	278 - 280
INTRODUCTION
The generalized Lorentz (or de Sitter) groups 0(N,1) are the most important simple noncompact Lie groups used so far in physics. Interest in their representation theory was recently revived, when it was realized that group theoretical methods can be used for a non-perturbative analysis of conformal invariant quantum field theory and its Euclidean Green functions (see, e.g., [M6, M7, М2, T5, D4-D6] and Chapter V of this book.)
The Euclidean conformal group for 2h space-time dimensions is 0(2h + 1, 1). (The case 2h = 3 may be of interest in the study of critical phenomena in statistical mechanics, -cf.[P2].)
On the other hand, these groups belong to the family of real (or split) rank one semi-simple Lie groups, which served as a starting point and are the most developed part of the Harish-Chandra theory of induced representations and harmonic analysis on semi-simple Lie groups [Hl,H2,W3,W2 ] . It is not an accident that this theory originated in the work of Bargmann [Bl] and Gel'fand and Naimark [G3,G2]who studied the examples of the lowest rank pseudo-orthogonal groups 0(2,1) and 0(3,1). The representations of the de Sitter group 0(4,1) have also been the object of a special investigation (see, e.g., [D3, T1,K3,K7] and references to earlier work cited there). A study of the unitary irreducible representations (UIR) of 0(N,l) for arbitrary N was attempted in [H3, 03] using chiefly infinitesimal methods. However, contrary to authors' claims
2
the classification obtained there is not complete, since it omits the so called class I complementary series which incorporates the entire complementary series in the well known special case of the ordinary Lorentz group 0(3,1). (It so happens that this omitted series contains precisely the representations used in the physical applications [M2,D6] .) Recently (while the present work was in progress -cf. [D4] ), Thieleker [T2,T3] gave a more comprehensive treatment of the representation theory of 0(N,l) (and its double covering Spin (N,l)), based on the abstract approach of Harish-Chandra. (We disagree however with his result concerning one of the exceptional series of unitary representations — see the discussion in section 6. C below).
After this work was completed (and circulated in a preprint form) we became acquainted with a new preprint [Gl] on this subject. It contains, in particular, an infinitesimal form of the intertwining operators which is complementary to the global treatment, presented here. An (incomplete) study of the unitary irreducible representations of the general pseudo-orthogonal group 0 (p,q) is also presented in [F6].
The aim of Part One is to give a global construction of a class of so called type I (or symmetric tensor) representations of 0(N,l) and to study associated quantities (characters, intertwining
3
operators), which are suitable for analytic computations that are needed in physical applications. Often the results are extracted from the general theory of Harish-Chandra supplemented by the Knapp-Stein and Schiffmann study of intertwining operators [K3,K4,S1]. (The method of constructing complementary series of unitary representations by exhibiting the range of positivity of intertwining operators was first used for the Lorentz groups of lower dimensions [G2,B3 ] . The relevance of the work of Knapp and Stein fK3,K4] for conformal quantum field theory was already pointed out by Koller [K5 ] .) We emphasize this point, because it refutes a widespread prejudice among physicists that modern mathematical theories do not provide useful tools for analytical computations. We will feel rewarded, if the present lecture notes will help to bridge the existing gap between mathematicians and physicists studying and applying the theory of group representations.
On the other hand, it should be pointed out that there is no general theory of unitary irreducible representations (IR's) of semi-simple Lie groups; the harmonic analysis on such groups (which is well developed) does not solve the problem, since not all the unitary IR’s actually appear in the harmonic expansion of the regular representations. In the case of the generalized Lorentz group, however, there are several simplifying features (in
4
addition to it being of split rank one), which make the classification of all irreducible representations (both unitary and nonunitary) manageable. First of all, the orthogonal groups, SO(n), are multiplicity free; in particular, each elementary representation of G = SO't(N,l) (to be defined in Chapter I below) contains any IR of the maximal compact subgroup К = SO (N) at most once. To formulate the second property, we need to introduce the notion of the (minimal) parabolic subgroup NMA of G, which (in our case) comprises a group (NM) conjugate to the Euclidean Poincare group and the (one dimensional) group of dilatations A. (Sec. IB). The "translation" or alternatively the special conformal transformation subgroup N, which would be in general, just nilpotent, here is actually abelian. As a result, the harmonic analysis on MAN is easily effected, with the help of (Mackey's straight-forward generalization of) Wigner's celebrated study of the Poincare group. We use this observation to analyze the positivity constraints on the bilinear forms that define the scalar products in the various unitary representation spaces (Secs. 5C, 6C and Appendix C). These bilinear forms (whose kernels play the role of physical propagators in the applications) are obtained (by a possible restriction and a change of normalization) from the Knapp and Stein intertwining operators (Sec. 4). In the physicist language the harmonic analysis on MAN amounts to first taking the ordinary Fourier transform of the intertwining kernel (or
5
of the x-space expression of the conformal propagator for tensor fields) and then expanding the result in 30(211-1)^ projection operators (that is, projection onto states of definite spin, SO (2h - 1) being the stability subgroup of the "momentum" p) (Sec. 5). This analysis also allows us to investigate the Wightman positivity condition for the 2-point function (Sec.5D). Mathematicians follow usually a different path: viz. harmonic analysis on K, which is rather complicated even in the case of the generalized Lorentz group (cf. [T3,T4]). Besides the Knapp and Stein intertwining operators there are intertwining differential operators which exhibit partial equivalences among some exceptional elementary representations with the same infinitesimal character (Secs. ЗА and 6A,B, cf. [Z5,G1] ).
It is noted, that one of the four families of (type I) exceptional (integer) points in the representation space is related to the discrete series of analytic representations of the Minkowski space conformal group S0(2h, 2)/2^ considered in [G10,02,R2] . In the special case 2h = 3 the same family is shown to contain the Harish-Chandra discrete series of 0(4,1) (Sec.7). The explicit p-space expression for the intertwining kernels at integer points are used to establish some differential identities between these kernels. These identities become important in physical applications, when one goes over to a Laplace transform on G, in order to derive asymptotic expansions. They are responsible for the cancellation
6
of certain kinematical singularities in such expansions, which come from poles of representation functions of second kind.
The results centered around the harmonic analysis of the intertwining kernels and their interrelations at integer points, form the core of Part One. For the benefit of the physicist reader, we also review the relevant mathematical background and give explicit.realizations of some often used abstract constructions.
The reader who is only interested in the physical applications envisaged so far may omit Sections 2.C, З.В, 3.C and 7.
Section (8.C) contains a synopsis on unitarity, irreducibility and equivalence properties of elementary representations (which can be read independently). Notations for often encountered subgroups of G are collected in Table 1 (Section l.B).
7
Chapter IV is devoted to the explicit construction of the tensor product decomposition of two unitary representations of G^ = o'(2h+l,l) which belong either to the principal series or to a class I (scalar) supplementary series representation. The Clebsch-Gordan kernels for the tensor product expansion under consideration are constructed in Sec. 10A. The Plancherel formula for the Kronecker product of two principal series representations is derived in Sec. 10B for even space-time dimension (integer h). The derivation uses results of part I. The result is extended to supplementary series representations by analytic continuation in Sec. 10D. Some of the results are extended to half integer h in Sec. 10 C; in particular, it is shown that discrete series components do not appear in the tensor product decomposition of two class I representations.
In Sec. 11 we establish some identities among analytically continued Clebsch Gordan kernels at exceptional (integer) points.
The results of Chapter IV extend earlier results derived in [D5 ] .
An effort was made in the last few years to exploit the conformal invariance of the Gell-Mann - Low limit theory for some Yukawa type interactions in order to obtain non-perturbative information for Green functions and operator products at short distances (see, e.g. [M9, P1,M7,D2,S8,H6,M6,M1O,T4,P3,M2,D4,L2,M3,B6,G9,F5, M5,S3,F4,F1,F3,S4,F8,G10,R2,M8,K6,O2j ). In particular, Mack [М2] showed for a model of a self-interacting scalar
8
field that the conformal partial wave expansion of Euclidean Green functions allows to diagonalize and solve the set of renormalized dynamical equations [F7,S6] for that model. It was noted [M1O,P3,M2] that the so-called bootstrap equations for the 3-point functions imply the existence of real poles in the conformal partial waves as functions of the dimension. The remaining integral equations of the model lead to some factorization properties for the residues at these poles [М21 .
The main purpose of Chapter V is to derive a discrete expansion for Euclidean Green functions and Wightman functions which corresponds to a vacuum operator product expansion in the terminology of ref. [ S4] (i.e. an expansion of the vector distribution <р (хг )	(x^) |o> ). The derivation
is based on the above results on conformal partial wave analysis and on our previous study of the Clebsch-Gordan expansion for the pseudo-orthogonal group. This approach always involves a conjecture about the analyticity (and the asymptotic behavior) of conformal partial waves, which is partly justified by the analysis of the skeleton diagram expansion [M7 J . The identities among Clebsch-Gordan kernels at exceptional integer points in the representation space, derived in Chapter IV,are crucial for cancelling fake singularities coming from kinematical factors (Sec. 15.C). As a by-product we verify a positivity condition for the 4-point Wightman function, which was established in a different manner in ref. [М3].
We also present the complete conformal parti-.al wave decomposition of an arbitrary Euclidean n-point vertex function (Sec. 13 E).
9
We attempt to make the exposition reasonably self-contained and review a number of results of ref. [М2] . This introductory material also contains some new points: one example is the discussion of the (cp’tp)1" -model in Secs. 12D and 14A. We mention also the explicit expression for a general "basic field" which enters the operator product expansion of a pair of free fields (Sec. 14.C). The derivation of the vacuum expansion is presented in Sec. 15 (and the related Appendix G). We would like to stress the role of the relations between partially equivalent representations of the Euclidean conformal group established in Part One. Sec. 16 contains a discussion - but no ultimate solution - of the rather difficult problem of incorporating crossing symmetry in the present scheme. It closes by a summary of results (Sec. 16.C).
Each part starts with a brief synopsis.
10
PART ONE
TYPE I REPRESENTATIONS AND INTERTWINING OPERATORS FOR (/(211-1-1,1) SYNOPSIS
We present global realizations of unitary type I (symmetric tensor) representations of the generalized Lorentz group O(2h + 1,1), which are suitable for analytic computations needed in Euclidean conformal invariant quantum field theory in 2h dimensions (2h =2,3,4,...) . Whenever possible the results are extracted from Harish-Chandra's general theory of harmonic analysis on noncompact semi-simple Lie groups, supplemented by the use of the Knapp-Stein and Schiffmann intertwining operators. We also introduce intertwining differential operators, which connect elementary representations at partially equivalent integer points. The core of this part consists of a study of the properties and interrelations of all intertwining operators . We exploit the fact that the nilpotent factor N in the Iwasawa decomposition of G = S0't(2h + 1, 1) is commutative (it consists of 2h dimensional special conformal transformations). The commutativity of N makes elementary the harmonic analysis on the so called parabolic subgroup NMA (M»S0(2h), A = one dimensional subgroup of dilatations). Among other things, the harmonic expansion of the intertwining kernel allows one to single out the unitary representations of G. It also provides a simple criterion for positivity of the 2-point Wightman functions in a conformal covariant quantum field theory in 2h-dimensional Minkowski space.
11
I. ELEMENTARY REPRESENTATIONS OF THE PSEUDO-ORTHOGONAL GROUP
1. Group Structure. Preliminaries
1.	A The group O(2h + 1, 1) and its Lie algebra
We shall use the notation N = 2h + 1. Thus h is integer for odd N and half odd integer for even N (h plays the role of half sum of the restricted positive roots). We consider h >1 in the main body of the paper, since the lowest dimensional cases (h - 1 and h = | ) are degenerate (in some sense) and the representation theory of 0(3,1) and 0(2,1) is well known (see, however, Appendix В where these special cases are briefly discussed)-
The group O(N, 1) is defined as the set of linear transformations in the real (N + l)-dimensional vector space which leave invariant the quadratic form
% = ••11NN= ‘^OO = 1’ T’AB= 0forA^B •	<L1>
It consists of real (N + 1) x (N + 1) matrices g satisfying
fgng = n	d- 2)
(tg being the transposed of g). Its identity component G = SO^(N, 1) comprises those matrices g, satisfying (1. 2), for which
12
detg = 1, g°0 > 1.	(1. 3)
We shall mostly deal in what follows with the extended group
Gex = Of(2h + 1, 1) = {gCO (2h + 1, 1) | g°0 > 1 }	(1. 4)
which includes space reflections.
The Lie algebra CJ. of G consists of real (N + 1) x (N + 1) matrices X such that
+ nX = 0 .	(1. 5)
We can choose a basis X = -X (A, В = 0, 1, . . . , N) in 01 AB BA	о
satisfying the commutation relations
[ XAB’ XCdJ = ^AC^BD +t|BDXAC '*1ADXBC'’1 BCXAD (
(А, В, C, D = 0, . . . , N = 2h + 1).
(Note, that in the physical literature one uses more often the
"physical generators" Зд-ц related to the "mathematical generators"
X . „ by J , „ - iX ,	) A matrix realization of X . _ is given by
AB AB AB	AB
^ab’d = 11 ad 5 в 'Wa .	7)
1. В Subgroups and decompositions
The group G possesses several important subgroups which
are listed in the following table:
13
Subgroup
К = SO (2h + 1)
A = SO (1,1)
M = SO (2h)
N
N
Characteristic property
maximal compact subgroup
1-dimensional, non-compact (" dilatations")
centralizer of A in К (mam-'; a) ("Euclidean Lorentz group")
nilpotent, abelian (" special conformal transformations")
nilpotent, abelian ("translations")
H = H ~ AH noncompact ([ h] + 1)-dimensional abelian group of diagonalizable matrices ("Cartan subgroup")
Generators
X , a, b = 1, . . . , 2h + 1 ab
^2h+ 10 = D
X p , P - 1, . . . , 2h
И v
C = X -X ,, p	pop 2h+l
T = X +X , p	pop 2h+l
(p = 1, . . . , 2h)
D’X1 2’’ ’ ' ,X2[h] -1 2[h]
TABLE 1: Remarkable Subgroups of the Connected Pseudo-Qrthogonal Group G = SO*(2h + 1, 1) ([h] denotes the integer part of h)
According to (1. 6) the generators D, C
and T И
appearing in the
last column of the Table satisfy the following (non-trivial)
commutation relations:
[ D, C ] = -C , [D, T]=T,i[T,C]=D5	- X ;	(1. 6')
L PJ p L p J p L p Vs p v p v
[X. T ] = 5 T - 5 T [X, C ] = 5 C - 5 C L Xp , V W p p V X, L Xp , PJ \V p p V X.
The physical Euclidean space momentum operator is given by a
hermitian representation of iT .
14
Let M' be the normalizer of A in K, -i. e. , the set of those elements m'cK for which m' am' cA for all acA. It is easily Seen that M' is isomorphic to O(2h) and that M is an invariant subgroup of M' . The finite group
W = W(G, A) = M'/M	(1. 8)
is called the Weyl group for the pair (G, A) (or the restricted Weyl group). It has two elements W = { 1, w} . The nilpotent subgroups N and N in Table 1 are conjugate under the Weyl transformation
wNw = N.	(1. 9)
Whenever needed, we shall choose the following representative of w in M' :
w = exp brX2h 2h +1)	(1. 10)
i. e. rotation by tr in the (2h, 2h + l)-plane. In what follows we shall denote the elements of K, N, A, M, N by k, n, a, m, n, respectively.
The Iwasawa decomposition (see, e. g. , [V/3,W2]). Every element g of G may be presented (uniquely) in the factorized form
g = kna,	(1. 11)
or (equivalently)
g = nak	(1. 12)
(the factors к and a in (1. 11) and (1. 12) being, in general, different ). The dimension of +be abelian factor A in 4he J^asa^ja decomposition 6 = /<ЛГА of a Semi- simple Lie fjroup G Is called SpM rank
15
The order of the factors in both versions of the Iwasawa decomposition is a matter of convenience. Our choice of the first factor in both cases is related to the imbedding of the elementary induced representations of G in the left regular representation of this group (see Section 2). In this and the following chapter we shall use the form (1. 11). The form (1. 12) will be used in Chapter III in the study of the discrete series of unitary representations of Of(4,1).
The (GeV fand-Naimark-) Bruhat decomposition (see, e. g. , [G3, КЗ, K4.W3 W2j )• Almost every element of G (more precisely, every gcG which does not belong to the lower dimensional manifold wNAM) can be written in a unique way as a product
g = nnam.	(1.13)
on Finally we comment the lastline of Table 1. There is an
important difference between the groups SO*(N, 1) for odd and even N. For odd N (integer h) there is only one Cartan subgroup, up to conjugacy within the group. It is non-compact and is reproduced in Table 1. For even N (half integer h) there is, in addition, a compact Cartan subgroup, generated by ^34 . . . , X^ It follows that the set of elliptic elements (matrices with eigenvalues of modulus 1) has the same dimension aS G for even N,
16
but not for odd N. This difference has a bearing on the structure of the representations of G- only for even N there exists a discrete series of unitary representations of G (see Sections 7 and 8.B, below).
l.C The compactified Euclidean space as a homogeneous space of G  The "parabolic subgroup" NAM (which appears in the Bruhat decomposition) plays a privileged role in the Harish-Chandra construction of induced representations of G. It is notable that the (group theoretically) distinguished homogeneous space
G /VAM « К/М = S	-	(]
(isomorphic to the unit sphere S2h in 2h + 1 dimensions) is also relevant to physics (see e.g.fAl] ). It can be identified with the • »	•	zR
conformal compactification of the 2h-dimensional Euclidean space X = IR -the carrier space of Euclidean (quantum) fields. (S2'1 is obtained from X by adding a single point, sayoo; in the group theoretical language that is the equivalence class of the non-trivial element (1.10) of the restriced Weyl group.)
The group G acts in a natural way (by left translation) on the homogeneous space (1.14). In particular, there is a well defined local action of G on the vector space X which can be identified with the manifold of right cosets nNMA (n«N). Because of uniqueness of the Bruhat decomposition (1.13) for ge GX wNAM (see , e.g. , Theorem 1.2.1.2. of [W3 ] ) there is a one to one correspondence between the cosets xeX and the elements n € N such that x = nNMA. We shall use the notation n for x
17
the (uniquely determined) element of N corresponding to x (e X ) We endow X with the structure of a real vector space in such a way that
n n = n	(1 . 15a)
xi xi	x, + xi
That gives us an additive parametrization of the abelian subgroup N of G. The parabolic subgroup NMA of G plays the role of automorphism group of X . In particular we have
n у = x + у	(1.15b)
x
The stability subgroup of the zero vector x = 0 (corresponding to the unit element of N ) is the parabolic subgroup NAM of G.
The homogeneous space (1.14) is isomorphic to the set of
isotropic rays in R^ + (with metric given by (1.7)):
G/WAM » IK . / IR
1	+	4'	(1.14')
where IR+ is the multiplicative group of positive reals and
ik^. = ik*1,+ m (« g/mm ) =	r2R+2 i !?o > 0 .	-
= *>  - °}
[See remark in Section 2. В below, where the components of 1, are identified as linear combinations of the matrix elements of g (e6).J The components ofxeX are expressed in terms of the \ 's by
К 2h + 1	°
(J .J 6)
The left translation by g on. the homogeneous space (1.141) generate the natural action g: -» of G on , which gives rise to a well defined transformation law on the x's .We
18
deduce that the automorphism group of X is the parabolic subgroup NAM (conjugate to NAM). The action of the translations from N is already exhibited in (the second equation) (1. 15). The dilatations aeA and the rotations me M act as homogeneous transformations on X:
ax = | a | x, | a | > 0	(1. 17a)
(mx) = m x (p. = 1, . . . 2h)	(1. 17b)
p.	V
(summation is to be carried out over repeated Greek indices from
1 to 2h). For special conformal transformations ne N we can
only Speak about their infinitesimal actional X, since for every xkX there exists an ne N which would carry it to oo. For n = n^ in the neighborhood of the identity of N (e = ee being
1 2h
an infinitesimal 2h-vector) , we have the infinitesimal law
n : x — x* (e ), x1 = x + (x^5	- 2 x x ) e + O(e ^) 	(1. 18)
e	p. p.	|ip p. v v
We shall derive a global form of this law in the next subsection starting from an explicit realization of the Bruhat decomposition.
1. D Matrix realization of various subgroups of G. Construction
of the Bruhat decomposition
In what follows it will be useful to have an explicit realization
of the different subgroups in Table 1 which appear in the decompositions (1. 11-13), in terms of (2h + 2) x (2h + 2) matrices.
19
The matrices n and n will be parametrized by the corresponding
2h-dimensional vectors x(tX) and b, respectively. Using the matrix
realization (1. 7) of the generators we obtain the following
_ 2h
expressions for the matrices xT = Sx T and ЪС :
H	P=1 V V
( xT)X = 0	( xT)2h + 1 = (xT)° = x
P	PPP
-(xT)\.	= (xT)\ =xX(=x), X, Ц , = 1,.. . , 2h,
in + 1	U	К
(xT)An = 0 for A, В = 2h + 1, 0 В
or, in a more compact notation (in which the О-th row and column
appear in 2h + 2-nd place),
x
t ' x
xT
(1.19a)
x
x
°2
where O, stands for а к x к matrix with zero к
elements,
x = (x^,.
, x^) is a row vector, and Sc is the corresponding column
vector (cf. [ K5 ] ); similarly,
(1. 19b)
20
Using further the nilpotent character of the matrices (1. 19) (which
3	.	3
obey (xT) = (ЪС) = 0), we obtain
(1. 20a)
(12h
being the 2h-dimensional unit matrix),
(1. 20b)
Analogously,
we find that the matrix expression for a
dilatation a is
a -	I cha
\ 0 sha
0
2|a|
|a|2-l
2|a|
(1. 21)
|a | = e“ ( > 0 )
21
In the same notation
, (m )eSO(2h) ,
(1. 22)
Multiplying the matrices (1. 20)
1c	к
2h+lP 2h+12h+l
0	0
(k )eSO(2h + 1).
• ab
(1- 23)
- (1. 22) we obtain the following
expression for a group element g of the form (1. 13)
-x
g = n n am = x b x
\ x
, i 2
1-2 X
-2 X
mK p | a | ЪИ
|а|ЪИ
|a|2(l+b2)-l
[a|Z(l-b2)-l 2lal
| a|2(l+b2)+l
2 I a | >
тИ +2#b nf V a V
er , o’ . 2
x m +b m (x -1) er у er V
cr , er . 2
x m +b m (x +1) k о- у er v
|a|b^+(|a|b2--L- )хИ
laKxb)^2^^-1-"-’11)
2|a|
|а|1^ +(|а|Ъ2+-Ь)хИ lal
|a| (xb) LZ+(hfb2+l)(x2-l)
|а| (хЪ)|1а12+(Ы2Ь2+1)(х2+1)
(1. 24)
22
The condition that amatrix gc G can be written in the form (1. 24) is
JZ 4 _ i , 2h+l	2h+l 0	, 0 .
d(g) - 2 (g 2h+1 - g 0 - g 2h+1 + g 0)	° 	(1. 2 )
(The use of upper and lower indices is only necessary when the
II
index 0 is involved; for p, P - 1, . . . 2h we have m = m , f p f
u.
Ъ = etc' ) We note that d(g) is always nonnegative (for ge G)
and vanishes for g of the form wnam.
Using (1. 24) we find
(1. 26a)
Xй =хИ (g) _LJ
1	/р-	р. '
2d(g) \g ° ‘ g 2h+l
(1. 26b)
2ЪИ
0 2h+l p . 2h+l 0 p
(g o'g 0 )g 2h+l + (g 2h+rg 2h+l g 0 ’ (L 26c)
тЛ I =m^ (g) ] = g^ V\ BP I V
, p p ..0 2h+l , (g 0~g 2h+l g p~g P1
2d(g)
(1. 26d)
(The subscript В is to remind us that the quantities I a I g» в’ etc'
are determined from the Bruhat decomposition. )
We define the transformation x A x' from
gnx = n , n(g, x)a(g, x)m(g, x).
(1. 27)
23
This leads to the transformation law (1. 15) (1. 17) under the
automorphism group NAM of X and to the law
2 x + x b
X' =-----------YT~	(1- 28)
1 + 2bx + b x
under special conformal transformations (which agrees with the
infinitesimal rule (1.18) for b = e—0). We notice that the denominator
in (1. 28) coincides with the function d(n^n^) defined by (1. 25).
It is useful to observe that the special conformal transformation
can be expressed in terms of a translation n and the reflection R w olepntol in
of the first 2h + 1 axes (R is equal to the Weyl transformational. 10)
followed by a reflection § . ——£	i = 1, . . . , 2h-l; for odd 2h R
is a proper G transformation and could be taken as a representative
of the non-trivial element of W instead of (1. 10)). Using the decomposition (1. 27) of Rn^ we deduce the following transformation law under conformal inversion:
Rx = - *	.	= 6x	(1. 29)
x2 J	x2
It is easy to verify that = Rn ^R \ S is rejection of ike 2h-Hieixis.
24
Let us return to Eq. (1.27). It will be convenient for applications in part II of this book to introduce some special notation and state a few identities. We define p(x,g)e MAN by
g 1 nx = nx,p(x,g) 1 x' = g ’.	(J.27a)
Thus p(x,g 1 = ri(g5x)a(g,x)m(g,x) in the notation of (1.27).
From (1.27a) we deduce the cocycle condition
P(x,g]g2) = p(x,gj) p(gi'x,g2)
(I.27b)
The following special cases are immediate from the definition
p (x,n) = I for new ; p(x,ma) = ma for ma e MA	(1.27c)
For
Px в p (x ,w)	(1.27d)
Eq. (I.24) yields the explicit formula
Ъ = mxQixr'9x	mx - (-1L+22^')Is '	(1.27e)
We used matrix notation; x resp. x*" denote the column resp. row vector with entries x' ... x^. The following identities are immediate consequences
л , лп = tn m : a a x+y wy x wx+wy x+y wy
a a
x wx +wy
(1.27f)
and
p(x,n_Ry) = hx hwx_t^ nRu = nR<y x) kx kwx_wy where h = m a etc. and u = у + w(wx-wy)
(1.27g)
25
1. E Relationship between the Bruhat and the Iwasawa decomposition.
The Haar measure
The functions k(g), bj(g) and |a|j.(g) corresponding to the Iwasawa decomposition (1. 11) can be found in a similar way as the expressions (1. 26). In order to exhibit the relation between
the representations (1. 11) and (1. 13) it sufficies to write down the
We have
2x^
(n ) = x	(1. 30b)
I alT(n ) =_L_ = |a(x)| .	(1.30c)
,2 1+x
To prove (1. 30) it is sufficient to verify that
к n a(x) = n , XX	X
(1. 31)
which is a straightforward exercise in matrix multiplication.
In order to write down the Bruhat decomposition of k, we
first note that each kcK can be written in the form
к = к m with x^ x
1'k2h+l 2h+l
1+k
2h+l 2h+l
(1. 32)
26
where is given by (1. 30a) and m£M. Indeed, is nothing but a rotation in the (x, 2h + 1) plane by an angle
1-x2 в = arccos--------	(1. 33)
1+x2
and (1. 32) is a standard decomposition for SO(2h+l). Since the Bruhat decomposition of m is trivial (m^ m) the decomposition of an arbitrary ksK is reduced to the decomposition of k^, which is given by (1. 13) with
x(k ) = x, m (kJ =1,	(1. 34)
X	г) X
2 x
|a|-R(kJ = 1 + x , b (kJ = ---------- .	(1. 35)
B x	В x 1 + x2
Finally, we shall write down the invariant measure on G in terms of the invariant measures of its subgroups in the Iwasawa and the Bruhat decomposition. Since the group G is unimodular, its Haar measure is both left and right invariant; the measures of the non-unimodular factors will be chosen to be left invariant.
Let dk be the Haar measure on К normalized by
f dk = 1,	(1. 36)
К
Set further
da = d Ia I (=da), dn, = db . . . db , = db.	(1. 37)
ii	bl	Zn
a
27
Tht left invariant measure on NA can be written as
2 h d(na) = | a I dn da. ь
(1. 38)
In order to check (left) dilatation invariance of (1. 38), we note that
an = n । . a b b/|a|
(1. 39)
Then the Haar measure on G has the form
d^=dKcUda	(i.4o)
( for g. given by (l.Il) ). To see that we note that cLc^ has to be both left and right invariant (since the group G is unimodular) and the expression (l.40) is fixed (up to overall normalization) by the requirement that it is left invariant under К and right invariant under NA.
In order to express d^ in terms of the factors of the Bruhat decomposition (I.I3), we first note that the right invariant measure on the parabolic subgroup NAM is
(n&am)= dbdadm
where (Ln is the normalized Haar measure on M. That follows from the semidirect product structure of NAM = N$(A®M) (N being an invariant subgroup). Using again that is both left and right invariant, we obtain
dtdbdadm	(i.4i)
where dx = dx^ ...dx^^ ( = dn^ ).
It is also worth noting that the invariant measure on K/^ obtained from the decomposition (lo32) is
I LZk) f/y
28
2. Induced representations. Definition and various realizations
2. A Synopsis on the irreducible representations of the orthogonal group [ 29, 30 ]
Here we recall some basic facts (and fix notation conventions) on the irreducible representations of the compact orthogonal group O(n) to which we shall refer in the rest of this section. Some additional information on these representations is assembled for later use in Appendix A.
The finite dimensional irreducible representations (IR) of a compact Lie group are completely characterized by their highest weight (see [ 30] ). One has to distinguish the type of highest weights of SO (n) for the cases of even and odd dimensions.
The highest weight for the orthogonal group of rank V (that is, for SO(2V) or SO(2v + 1)) is I - (1^, . . ., I ) where
<*<•••<•*	for SO(2P)	(2.1a)
1 2 (^>1)
0<lj<	for SO(2P + 1) .	(2.1b)
The number s 1 . are all integers for single valued IR of SO(n) and all half integers for the double valued IR (i. e. for the faithful IR of the double covering Spin (n) of SO(n)). The dimension of the IR (f , . . . A r , ) of SO(2h) is given by
1 L hJ
29
d„ (f) = Zn
dar...,/h)
1< i< j<h
ay) =1 for
h = 1;
h-t
d,, (f) = Zn
d«i.....4-1
Г X. 3 for h = —
2
5
T’
(n.+n,) (n,-n,)
1 J J 1
(2h-i-j)( j-i)
2n
2(h-i)
for h=2, 3, . . .
(2. 2а)
(2. 2b)
(2h-i-j)(j-i)
J J
= 1
where
. + h - i
[h]+ 1
- j = 1 [ h] +
1 -j
(2. 3)
1,..., [h].
(The vector (h-[ h] , . . . , h-1) added to the highest weight in
(2, 3) is half the sum of positive roots of SO(2h). ) The (second order)
Casimir operator is
П(^
•^[h]’ гХу.1^ S C n[h]+l-i '	1 "S *[h] +l-i(/[h] +l-i+2h'2i)‘
(2.4)
(We are using here the
symbol X^
for the representation f of the
generators X, introduced in Section 1. A. )
The highest weight characterization of the IRs of SO(n) is particularly useful in studying their SO(n-l) content [G5]  A (unitary) IR f' = (/'j, . . . ,f1	) of SO(2P-1) is contained in the IR
30
i = се ... ,iv) of so(zp) iff
(2.5)
The IR {' = (f ’ , ...» St’y) of SO (2P) is contained in the IR i = (f ,... .f^) of SO(2p+1) iff
-S<	< *1 <	<2’6>
Each IR of SO(n-l) is contained in a given IR of SO(n) at most once. Let now O(n) be the full orthogonal group, including space reflections tr such that
2 t
tr = trtr = 1, det tr = -1.	(2. 7)
I
Let D (Л) be an IR of SO(n) acting in the (complex) vector space . For a given reflection ire O(n) (satisfying (2. 7) we define the mirror image of D as the representation
(Л) = D^trAtr'1)	(2. 8)
acting in the same vector space V (we note that ггДгг ^CSO(n) for any AcSO(n), so that the right hand side of (2. 8) is well defined. ). The following facts about mirror images are well known (see e. g. [u/4-] or Section 114 of ref. [Z2] ).
(i) The mirror image of an IR f = (f^, . . . ,1 J) of SO(2v) is equivalent to trf = (-f^, z’ ’ '
31
(ii) The mirror image of any IR 1 of SO(2f + 1) is equivalent to 1.
Thus any IR of SO(2P + 1) can be extended to an IR of
O(2P + 1), while an IR t = (f . . . t of SO(2p) can be extended to
an IR of O(2p) if and only iff = 0.
Having in mind the applications envisaged in [M2T>4 , we shall
be particularly interested in the symmetric traceless tensor IR (which will be also called type one representations) with highest weight
f = (0, . . . , 0, f ).	(2. 9)
The representation of this type can be realized in the vector space IT of all homogeneous polynomials of degree f
f (3r) ~	(2.10)
on the complex light cone
(2Л1>
(cf. [T6,Of 21,Ъ2]). Each such function allows a unique harmonic (homogeneous polynomial) extension to the whole n-dimensional complex space <C П (see [B2 ] and Appendix A). The group action on vectors of the type (2. 10) is given by
[ D (A)f] (£) = f (A J).	(2. 12)
It follows from (2. 5) (2. 6) and (2. 9) that the type I representations of SO(n) are decomposed into only type I representations of SO(n-l). Concerning 4he expression for +he convolution of 7wo -Sensors in the homogeneous -polynomial -formalism see Appendix A.5--
32
2. В Covariant vector valued functions on G. Definition of the induced representations
In this and the following subsections we will give three alternative realizations of the induced representations of G, (ex)
(induced by the parabolic subgroup MNA). We start with the most abstract and general one.
Let be the (finite dimensional) Hilbert space in which the I
unitary IR D of M = SO(2h) is realized. Let further c be an arbitrary complex number and write
X = [f ,c] = (f , -h-c)	(2.13)
(as it will become clear in the sequel the set of numbers in the parentheses corresponds to the generalized highest weight of an
t
elementary representation of SO (2h + 1, 1)). Consider the space
of infinitely differentiable functions 1 on G with values in , satisfying the covariance condition
^(gnam) = |a|h+CD^(m) ^(g).	(2.14)
The representation 7 (g) = Tx (g) of G, induced by the finite
- h- c S.
dimensional representation^ D (m) of MNA, is defined by

g. g'«G,
(2. 15)
33
(It differs from the left regular representation of G by the domain °Pera*"ors which is characterized by the covariance condition (2.14).) The representations so constructed are called elementary (induced) representations of G.
Remark: Let us exhibit the relation between this standard mathematical construction and the so called manifestly covariant formalism of ref. |M7,'T5j. In order to simplify the discussion we shall restrict ourselves to the case of a scalar representation (f=0).
It follows from the covariance property (2. 14) that a function is completely fixed by its values on the homogeneous space (1. 14). The points of this space can be identified with isotropic ("light-like") rays in R ^parametrized by the (homogeneous) coordinates
g A = |fgA - gA V A = 1,. . 2h+1, 0.	(2.16)
*	0	6 2h+l;
It is a simple consequence of the orthogonality condition (1. 2) that
t 2 = t Л A = g АпатЛ Б = 0.	(2.17)
Furthermore, using (1. 21) (1. 20Ъ), (1. 22), and (2.16), we obtain
£ (ga) = _JL i (g), g (gn) = g (gm) = g (g).	(2.18)
Ы
Thus the isotropic vectors g (2. 16) are in one-to-one correspondence with the equivalence classes gMNe G/ MN and can be identified
34
(2.19)
with them. On the other hand, a function F(g), satisfying (2. 14) with I - 0, only depends on these equivalence classes, so that (g) = у (gMN) =
Moreover, because of the first equation (2. 18) and (2. 14), £ (£ ) is a homogeneous function of degree -h-.c of £ . The representation (2.15) assumes the form
[сг[°’с] (g)^] (g ) =/(g‘^ )•	(2.20)
The vector (2.16) satisfies all properties of the isotropic vector £ which appears in (1. 16) and can therefore be used as an intermediate variable in the passage to the noncompact picture.
2, C The compact picture. К-content of the elementary representation#
By the Iwasawa decomposition every element g of G can be
written in the form (1. 11). The covariance property (2. 14) then implies that each (vector valued) function J- (g)d^ can be written in
<f X the form
^(kna) = ]a|h+c^(k)	(2.21)
and is thus completely determined by its values on K. Conversely, every smooth function^(k) on К satisfying the covariance condition (km) = (m \(k)	(2.22)
35
/5
can be extended via (2. 21) to an element of (_? (considered as a X
function on G). Hence, we may identify eachQ with a standard
spaceC(K,|/ )of covariant functions on K. The space C(K, If ) is independent of c. In fact due to (2. 22) ^(k) is completely determined by its values on the unit sphere (1. tf). Explicitly, according to
(1. 32) each kcK is decomposed uniquely in the form
/ 5й -________ । -g 0
/ v i+l г:
к = к л М with кл ={ - -	-	- ч - -	--- (2. 23)
ё £
\	!’2h+l О
\ о 0	1
„ , 2h+l л л
t = Е ёа = 1;
а-1
because of (2. 22)^(к) is fixed (for a given ) by its value for к = кл .
Let к^ and a = a(k, g) be determined from the Iwasawa decomposition of g ^k-
g ^k = k^na.
«x
Then we define the compact picture realization J of the elementary
representation x by
36
[^g)^J(k) = |a|h+y<kg)	(2.24)
We can define a К-invariant scalar product in CfK,^) (and, therefore, in eachu ), by setting X
(AA’ {dk7iw’A(k”	,2'25’
where < , > is the M-invariant scalar product on / , and dk
is the (normalized) Haar measure on K. The representation (2. 24) is continuous with respect to the topology defined by this scalar product. There exists also another natural (Frechet space) topology under which the space C(K,3^ ), regarded as a space of infinitely differentiable vector valued functions on the unit sphere (2.23), is complete. The representation^ so defined is obviously unitary, when restricted to the compact subgroup К of G, and the space C(K,2^ )(or its Hilbert space completion) can be decomposed into a direct sum of (unitary) IR spaces of K.
It turns out that each IR T of К appears at most once in a given elementary representation x of G. This is a consequence of the following (Frobenius type)
Reciprocity Theorem (see Corollary 5. 3. 3. 6 of Theorem 5. 3. 3. 5). The elementary representation x = [/, c] of G contains a given IR T of К exactly as many times as т contains the IR f of M.
37
We have already recalled in Section 2. A that each IR I of
SO(2h)(=M) is contained at most once in a given IR 7 of
(SO(2h + 1)(=K). Hence we have the
Corollary 2.1. Each elementary representation у = [/, c] of
G = SO (2h + 1, 1) contains a given IR т of К = SO(2h + 1) at most once.
The representation 7 is contained in x if the IR f of Mis contained in 7,
The reduction of an elementary representation x of G with respect to a chain of imbedded compact subgroups of the form SO(2h + 1) = KDM1^. . 3M2h'2 where
M1 = M = SO(2h) MJ = SO(2h + 1-j)
Q'-f provides a canonical basis in C(K, у ), because orthogonal groups are multiplicity free. If t \= t),. . . ,t are the multi-indices of the IR of M2(=M), . . . , M2h 2 then the basic vectors of the
. 7
canonical basis are given by a ,	. Such a basic vector
(1)	( —'ll “ 1)
transforms according to the IR f of j = 1, . . . 2h-l. The matrix elements of (g) in the canonical basis are called generalized spherical functions. Some of the functions will be displayed for the type I representations in Appendix A. 4.
2. D The noncompact picture: x-space realization
Most of our explicit calculations are performed in the x-space realization of the elementary representations based on the Bruhat
decomposition (1. 13) (1. 15-18) of G.
38
We start again with the general covariant realization of
Section 2. B. Using (1.13) and (2.14) we see that^(g) is completely-
fixed by its values
f (x) = У (nx)	(2. 26)
on the subgroup N K R . The space of all (vector-valued)
functions of this form, where/varies in will be denoted by C .
/	X	X
Defining x , a = a(g, x) and m =m(g,x) from g
g 1 n = n n 'a 'm 1	(2. 27)
X g
,n
(cf. (1. 27)) we see that the tranformation law (2. 15) for Zis equivalent
<7
to the following transformation law for f:
[TX(g)f] (x) = |a|‘h‘C D*(m)f(xg)	(2.28)
Defining D^'C(man) = |a| D^ (m) and using (2. 27), we can rewrite (2. 28) as
[ TX(g)f] (x) = D1’ С(п'\п ) f(x ) ,	(2.28')
g g
The expression for Xg for the various subgroups of G is obtained from (1. 15) (1.17) (1. 28) (1. 29) by going to the inverse transformation. We shall write down the explicit form of T^ for generic (Euclidean) conformal transformation in the special case of symmetric tensor representations (of type (2.9)j of M.
39
In this case the space C consists of functions f(x, £ ) on
R2^ x К	being the complex isotropic cone(2. 11)),
2h \ 2h	I
which
have the following properties: f is a homogeneous polynomial of
degree I in J , infinitely differentiable in x and admits for x—» oo an asymptotic expansion of the form
f(x,Z) ~ о x—co
co
_____i____ 23 я, (Rx, r(x)^>.
, 24h + c k=0
(x )
(2. 29)
Here H^is a homogeneous polynomial of degree к in the first argument and of degree I in the second, R is the conformal inversion (1. 29) and r(x) = m(R, x) is the O(2h) transformation, associated according to (2. 27) with the reflection R of the first 2h + 1 axes. We have
Rn = n n a r (x) with I a I = I a(R,x) I = * ,	(2. 30a)
x Rx x	1	2
x and x^ x
г(х)И = x2VK (Rx) = 2 ---------- бИр	(2.30b)
v	v	x
2
r (x) = r(Rx)r(x) =1, r (ax) = r(x) (for aeA), r (x)x = x, (2.30c)
The space is complete with respect to a Frechet space topology defined in terms of the Lie algebra generators by a countable set of (semi)norms (cf. [G2]| where such a topology is introduced
*) For the functional analysis terminology, which is used without explanation, see e. g. [r-i] ! Frechet spaces are defined in Chapter V of that reference.
40
for the case of the ordinary Lorentz group; see also the remark at the end of this subsection. ) The representation T^(g), x = [f > c] , is defined in the following way for the various subgroups of G ;
(a)	Euclidean transformations: (x'= (x!, . . . , x’ ), —----------------------------- д	Zh
me O(2h))
[ T*(nxpi)f ] (x,£-) = f (m ^x-x1), m J.)
(2. 31a)
(the formula is valid also for h = 1; in that case, for f>0, the elementary representation becomes reducible when restricted to the connected subgroup G of G );
(b)	dilatations:
[ T^alf (x,X)] = ------!--f (_L— ,Z), |a| > 0	(2.31b)
|a|h+C ia|
(c)	conformal inversion: у	1
[TK (R)f] (x,z>	--------- f(Rx, r(x)J).	(2.31c)
, 2 h+c (x )
The special conformal transformations (1. 28) are simply expressed in terms of the conformal inversion and a translation by b, so that it does not need to be considered separately.
We note that the assumed asymptotic behavior (2. 29) of f(x,^,) guarantees the smoothness of the right hand side of (2. 31c)
for x-» 0.
41
The equivalence between the compact and the noncompact pictures is displayed by the following relations:
(2. 32)
(1+x )	0
(2. 33)
vhere к is given by (1. 30a) while x(k) and m(k) are determined
from
(1. 32):
х(к)и
кИ 2h+l |
-------, m(k)
, , 2h+l 1+k , , 2h+l
ки
V
u 2h+l
k 2h+ ? V
, , 2h+l 1+k	,, ,
2h+l
(2. 34)
The variables ga
~ "k 2h+l
(a = 1, . . . , 2h +1) (see (2. 23)) and x^
V
are related to each other by a stereographic projection.
The above isomophism permits us to introduce a locally
convex (Frechet space) topology on C as the one induced by the X oo 2h
natural topology on C (S ).
3.	Further properties of the elementary representations
3.	A Equivalence, irreducibility, completeness
We shall formulate here (without proofs) some known general properties of elementary representations. They can be, roughly,
41
The equivalence between the compact and the noncompact
pictures is displayed by the following relations:
=-4^/<кх»’	(2’32>
(1+x )	0
z	\ h + c
/(k) =(----D^m'fk)) f(x(k)),	(2.33)
V + k 2h+l /
vhere к is given by (1. 30a) while x(k) and m(k) are determined
from (1. 32):
х(к)и
кИ 2h+l ,|i
-------, m(k) у
, , 2h+l 1+k 2h+l
кИ
v
u 2h+l
k 2h+ ?	"
, , 2h+l
1+k 2h+l
(2.34)
The variables ga = -k& (a = 1, . . . , 2h +1) (see (2. 23)) and хИ
are related to each other by a stereographic projection.
The above isomophism permits us to introduce a locally
convex (Frechet space) topology on C as the one induced by the X oo 2h
natural topology on C (S ).
3. Further properties of the elementary representations
3, A Equivalence, irreducibility, completeness
We shall formulate here (without proofs) some known general properties of elementary representations. They can be, roughly,
42
summarized as follows: Almost all elementary representations are irreducible. The exceptional--reducible--representations x form a denumerable set with infinitesimal characters related to the finite dimensional representations. Any irreducible representation of G is equivalent to some subrepresentation of an elementary representation.
In order to state these results more precisely we shall need some of the general notions of the representation theory of locally compact groups (see Chapter 4 of ref. |Wj]).
Let T and T1 be two continuous representations of G acting in the CY* C£>
Frechet spaces _X and A. , respectively. We say, that the representations T and T1 are partially equivalent if there exists a continuous linear map A: such that
AT(g) = T' (g)A for all g« G.	(3.1)
A map A with this property is called an intertwining operator for the representations T and T1 . If A in (3. 1 ) has a continuous inverse (that is, if A is bijective and bi continuous), then the representations T and T1 are said to be equivalent. Two Hilbert space representations T and T1 are called unitarily equivalent if the bijection A. is isometric (then if T is unitary, so is T1 ). If there exists an intertwining map A with a non-trivial GT	CY Г
kernel ker A (= the set of points of JL mapped onto the zero vector of Л ) then her A is an invariant subspace of 9C . More generally, if the representations T and T1 are partially equivalent, but not equivalent, then
at least one of them is reducible.
43
Most of the results concerning equivalence and irreducibility of elementary representations are stated for continuous representations on Banach spaces (Banach representations,, for short). In order to comply with this general framework we will replace each of the Frechet spaces of Sec. 2 by its Hilbert space completion	with respect to the scalar product (2. 25).
According to Corollary 2. 1 of the Reciprocity Theorem (Sec. 2B) each elementary representation X of G (in either	or C ) is simply
redicible (i. e. , multiplicity free) with respect to the maximal compact subgroup К of G. therefore the elementary representations of G pertain to the class of К-finite representations defined as follows. Let T be any continuous representation of К on	(in particular, it can be a restriction of a
representation of G on	). Let К be the set of all equivalence classes of
finite dimensional IR1 s of К. For each т e K, let £ denote the character T
of T ^T(k) = tr T(k)Jl, and d(T) be its dimension. According to Schur1 s orthogonality relations the operator
П T = d(T) T(£t) = d (T) JeT(k)T(k)dk ,	(3.2)
where dk is the normalized Haar measure on К , is a (contihuous) projection.
A (continuous) representation T of G is said to be К-finite if the projection | | r has a finite rank (i. e. , if the space Тт.Пт% is finite dimensional)
for every T«K. In a simpler language a representation T of G is called
К-finite if each irreducible representation of К appears in T with a finite
(or zero) multiplicity.
44
If T is a К-finite Banach representation of G then the (dense) set of
К-finite vectors
К
;f=£f.,f.eX
being the subspaces
introduced above) has the following
remarkable properties: Each vector
, is analytic. In other words, К
the mapping G3 gi— T(g) defines an analytic function on the group G (considered as an analytic manifold) with values in the Banach space
(see Lemma 4. 5. 5. 1 of ref.	This fact allows one to define a representa-
tion Tj^. опЗ^к of the Lie algebra (and thus of the universal enveloping
algebra Ot (Л (CJ^) of i*-s complexification 0^,	).
Now we are prepared to complete our list of equivalence concepts with a
more subtle notion, which will be used in the formulation of the subrepresentation
theorem below.
The representations T and T1 of on -X and X are К	К	0. К	К
called algebraically equivalent, if there is a nonsingular operator
К
on A).
X such that T1 = AT A (no continuity requirement being imposed К	К К.
CV* 5У'/
The К-finite Banach representations T and T1 of G (on X and A ,
respectively) are said to be infinitesimally or Naimark equivalent if the corresponding representations T and T’ are algebraically equivalent.
For infinite dimensional representations there are several nonequivalent notions of irreducibility. Every elementary representation of G on % operator irreducible (in the sense of Schur1 s lemma): each bounded operator in JC , which commutes with the representation operators T^ (g) , is a multiple of the identity (this follows from the results of |К5 ,K4,Z4,2_5j ; the statement
45
f
is true for all elementary representations of. SO (2h + 1, 1); for G = Spin(2n,l) (n = 1, 2, . . . ), it is only valid for c # 0 ). Nevertheless, as we shall see, some of the elementary representations are topologically reducible in the sense that there exist nontrivial (closed) invariant subspaces in some of the . We shall present a heuristic discussion of the question: which elementary representations should be expected to be reducible?
First of all, one would expect that representations, whose label (f ,	-h-c) coincides with the highest weight of a finite dimensional
1 LhJ representation of G might contain a finite dimensional invariant subspace.
The type I representations satisfying this condition can be written in the form
X J = (0, ... 0,f , f + v - 1) = [f , 1 -h - I - p] ,L =0,1, . . . 1/ = 1, 2,. . . (3. 3a)
It will be shown in Sec. 6A that these representations are indeed reducible. Other elementary representations which may be suspected of being reducible are those, obtained from x^ v by a chain of intertwining maps. Since the Casimir operators of G are multiples of the identity for each elementary representation (this is true for any semi-simple Lie group since the Casimir invariants are polynomials of the labels c of the character of A) they have to have the same values for every pair of representations x and x' which can be related by (a chain of) intertwining maps. There are four type I representations (including (3. 3a)) with this property; they are X^p given by (3. 3a) and = [f , h + f + 1/ -1]	(3.3b)
x'/p = [f +P, 1-h- t ]	(3.3c)
46
х^+ = [/ + v> h + t -1] ’ w =°u, • •.; v = i, 2 ...).	(3. 3d)
This is easy to verify for the second order Casimir operator (/>	.
As we shall see in Sec. 6E below, the eigenvalues (x ) can be obtained from the expression (2.4) for the group SO(2h+2) via analytic continuation in c. They are:
G2(x) = 2 Хдв X =.2^ * [h]+i-i^[h]+l-i+2h’21) + c "h (3’4a)
- i (f + 2h-2) + (v +f -1) {v+t + 2h-1) for the representations (3.3) (3.4b).
If we go beyond the class of type I representations we will find exactly 2[h]+2 elementary representations with the same Casimir operators (or with the same infinitesimal characters in the mathematical terminology, cf.[Z4,Z5j /.For example, for h = 2, these are the representations
X = [I p * 2; C1 given by
Xfp = t °’* :± (*+и+1)1	(3- 5a)
= [0,1+ P;±(f+1)]	(3.5b)
x'^ = [±U +1), ! + »; 0].	(3.5c)
It appears that all these representations are indeed related by a chain of intertwining operators [Z5,G"l]. All but the last pair of representations
turn out to be reducible. (The representations (3. 5c) belong to the principal
47
series of unitary IR, studied in Sec. 3D below. )
All nonexceptional type I representations (different from (3. 3)) are topologically irreducible. This is most easily demonstrated by infinitesimal methods (see Hirai, ref. [H/ ] ); ). The irreducible components of elementary representations (including the irreducible elementary representations) are call ed sub repre s entati ons.
The importance of the class of elementary representations stems from the fact that every IR of G (in a certain sense, to be specified) is equivalent to some sub representation of an elementary representation.
The following statement is a consequence of a recent result of Casselman [d j and of the fundamental subquotient theorem of Ha rish-Chand га [Н'1 , Wjj t (In the context of elementary representations of SQ (N,l) a subquotient is either a subrepresentation or a factor representation in an elementary representation. )
SubrepresentationTheorem. A representation tr of G (or (^) which is Naimark equivalent to an algebraically irreducible К-finite representation of Oj. is also
Naimark equivalent to some subrepresentation of an elementary representation of G. In pa rticular, every unitary IR is equivalent to some such representation.
48
3. В Characters of elementary representations
Let x = [f , c] be an elementary representation of G in the covariant realization of Sec. ZB. For infinitely differentiable functions <P (g) of compact support on G, the operator
tfX(<P) = f <P (g)^ (g) dg	(3.6)
is known (see Theorem 8. 7. 4 of ref. [;V2] to be trace class. Therefore, one can define its trace
©x (<p) = Tr 0ГХ (<p);	(3.7a)
we shall also use the heuristic distribution theoretic notation
©x (g) = Tr ГХ (g),	©x(<?)= /©x (g) <P (g) dg .	(3.7b)
©x is called the character of the elementary representation x • We will write down an integral representation for ©x . To do this we first introduce two auxiliary functions.
Let, as usual, m«M , aeA ; define h /*	“1
F^ (ma) = |a | J V (kmnak ) dk dn	(3. 8)
KxN
and, for к^, k^ eK.
v	I h- cl	— 1
F^ (^1 ’ ^2^ = /lal	(m) (kj mn a ^2 ) dm	(3. 9)
t
where D (m) is the representation I of M = SC(2h), defined in Sec. 2A.
49
F<P
(ma) is a scalar function on M while
у
F* (k^, k^) is a matrix
valued function on Kx K. It follows from the definition that the functions
F satisfy the covariance conditions
F	(m' ma m1 ”^)= F (m a) (m cM , m«M, ae A) (3.10)
F* (k.m к m ) = D;(m )-1 F* (к к ) D* (m ).	(3.11)
Y 1	1 t,	-L	t,x4>	£
The proof of (3.11) as well as the verification of (3.10) for m'eM is trivial. The only non-trivial point is the validity of (3.10) for m' = w (where w is given by (1. 10) ) It is an immediate consequence of the following
Lemma 3. 1.(cf. jW2J Lemma 7. 7.11 ). Consider the coset space G/A with elements g = к n A and measure dg = dkdn. The function F^ can be written in the form
F^ (h) = A (h) J" d g cp (ghg ^), for h = mac MA ,	(3.12)
where A(h) is independent of and invariant under the restricted Weyl group.
Proof:	Using the Iwasawa decomposition (1.11) for g we see that
ghg 1 only depends on the coset g (since A is in the center of МАЭ h) and that consequently the integral in (3.12) is well defined and equal to
f dkdn <P(knb h n^ k’1 ) .	(3.13)
Assume for the moment that a ^1 ; then the map p. : N—N defined by
50
, -1 -1 -1 р. : n — n’ = m nman a
is bijective. Indeed, noting that
-1	-1	-1	-1	t'l ui
m nb m = n л ,аПъ a = n b/}, ,	(3.14)
mb
we find that the Jacobian of the transformation p. is (the matrix under the sign of the determinant is 2h x 2h):
| det (m’1 - (al'1) |=| det[exp{-(^ XE+<P2 X34+ ’ ‘ ‘+<P[h] X2[h]-1, 2[h] )} 'H’1
(3.15a)
[h]
= |l-!a|	[h] )|~1(i+|a| ^-г|а| ^сов^.)^ |1-|a| 1|^1> 0 .	(3.15b)
(In writing down (3. 15a) we have used the invariance of the determinant
under similarity transformations. ) Thus the integral (3.13) is equal to:
| det(m 1 - | a| *)| 1 Jdk dn’ <p(kmn’ak 1 ).
Then (3.12) coincides with (3.8), if we set
h -1-1 rr-r 1 2(h-rh])[2t]	!
A(h) = |a| |det(m - | al ) | = | V |a -	— |	| | (|a| + — - 2cos <P . ).
i=i	iai
(3.16)
The result will remain true (by continuity) for a=l , for which Д(Ь) =
Д (m) may vanish. It implies, in particular, that
F<p
0 = Д (m) for half-integer h .
(3.17)
51
To complete the proof of the lemma we only have to note the w-invariance of Д (h) (indeed, each factor in the right-hand side of (3. 16) is invariant under the transformation w; a—a ) .
Now we are ready to state the following result of Harish-Chandra [HI] .
Theorem 3. 2 . The characters Q (<p) of the elementary ----------------- ------------------- x representations are given by:
& (<P) = L. tr FX (k, k) dk = f lai [trl/(m)] F (ma) dm da	(3.18)
x K	MA	?
where the trace tr in both cases is taken in the finite dimensional vector space 1/^ .
The proof of this theorem (see Sec. 8.8.2 of ref. [_W2j ) is so simple and straightforward that we shall sketch it here.
Let G ; then « X
[	] (kj) = J<P(g)^(g Ь^) dg = fo(g)D* (g H^mjdg
for any m . Performing the change of variables g ^m — g and using the invariance of dg and its expression (1.40) in terms of the Iwasawa factors, we obtain
[(Гх (<?)/ ] (k ) = ftp (k mg’1) D* (m)^(g) dg a 1	1	q
- J<p(k^ma ^n ^k ^)D* (m) (kna) dk dn da
52
Using again the covariance condition (2.14), performing the change of variables n* = a n a (|a| dn = dn’ ) a' = a and integrating the
result with respect to meM
we find ( f dm = 1),
[ITX (kj) = /FX (kp k)/(k) dk.	(3.19)
Thus, we presented J (^) as a covariant integral operator with kernel (3. 9). To compute the trace of such an operator one writes down its matrix elements in the canonical basis (defined at the end of Sec. 2. C).
The result of this calculation (see p. 246) is
Tr (<P) = f tr F^ (k, k) dk .
This proves the first equation (3.18). The second one then follows from the definitions (3. 8), (3.9 ) .
Corollary 3. 3. Set for any x - [f > c]
X = [f , -c]	(3.20)
where i (= tr I ) is the representation of M obtained from i. by space reflection (see (2, 8) ) . Then the elementary representations x and X have the same character:
®x W =	•
(3.21)
Proof:	if w is the non-trivial element (1.10) of the restricted
Weyl group it acts on M as a space reflection, and therefore,
53
D* (wmw ) = D (m) .	(3.22)
In order to obtain (3. 21), we apply a Weyl transformation to (3.18) and use (3. 22), the transformation law waw = a for dilatations, and the w-invariance of F (ma) .
3. C The spherical trace function. The character of a subquotient of an elementary representation
A useful expression for the character 0	, also applicable
when U is a subrepresentation (or a factor) of an elementary representation is given in terms of the so-called spherical trace function defined below.
Let U be a subquotient of an elementary representation of
G which contains the IR T of К . Let further П T be the projection operator on the corresponding subspace • The spherical trace function
U
t (g) is defined by:
t^(g) = Tr (Пт u (g) ПT > •	(3.23)
Proposition 3. 4. Let U be equivalent to a subquotient of the
elementary representation x of G s: SO (2h + l, 1) (or Spin (2h + l, 1) )
which contains the representation T of К : then tU (g) = tX (g) T	"T •
54
subrepresentation acting
С /l , but not in both.
X X
proposition follows.
Corollary 3. 5.
Proof: Assume that the representation space C of an
X elementary representation x contains an invariant subspace I (possibly trivial). Then byCorollary 2.1 of the reciprocity theorem, if an IR T of К is contained in x , then it is contained either in the
in I or in the factor representation acting in Thus nTU(g)nr = HTTX(g)H Tand the
Let U be any IR of G^ contained in a given
elementary representation X • Then the (distribution valued) character
of U is given by
©„(g) = S t* (g)	(3.7c)
TeU
This formula is useful for the study of the discrete series (see Section 7A).
3. D The principal series of unitary representations
So far we have not discussed the question which of the elementary (sub)representations of G are unitary. This problem is closely related to the study of invariant bilinear forms on pairs of spaces C^ . Here we shall start such a study by showing that the representations x = [f,c] are unitary for c pure imaginary.
55
To do that we shall use the noncompact (x.-space) realization of the elementary representations (see Section 2 D).
First of all we shall construct an invariant bilinear form on a pair of dual representations
X=[^c], x = [f,-c]	(3.24)
for arbitary (complex) c.
Proposition 3.6. If f, eCL> fe C , then the bilinear form --------------- - 1 X 2 X -------------------------
B(f,f,) = f<f.(x), f_(x)> dx	(3.25)
14 J 1	4
(where <f^(x), f.,(x)> is the M-invariant scalar product in the finite n t
dimensional vector space if —ci. (2.25)) is invariant under the representation TX® T X of G.
Proof.
Using the transformation law (2.28) for f
and f^
we obtain
B(TX(g)f TX(g)f ) = f|a(g, x) I ’2h <f(x ), f (x )> dx, (3.26) 1	z,	1 g 4 g
where x and a(g,x) are defined from the Bruhat decomposition (2.27) of
-U, g	Dx
g n . Noting that the Jacobian_____is nontrivial for dilatations and
X	Dx
special conformal transofrmations of the type (1. 28e) only, and using the relation
^b
Dx
= [ 1 - 2bx + Ъ^х2] 2^ for
x-x b
l-2bx+b2x2
56
we conclude that
|a(g,x)| dx = dXg .	(3.27)
Inserting (3.27) in the right-hand side of (3.26) we reduce it to the original form (3.25). This completes the proof of the proposition.
Corollary 3,7. Let f^.f^e with X = [ ^> c] , c pure imaginary (c = io~). Then the sesquilinear scalar product
(f ,f_) = Г <f (x), f (x)> dx	(3.28)
1 L	J 1 L
is invariant under the action of the representation T of G. --------------------------------------------------- X ------
To reduce this statement to Proposition 3. 6, it is sufficient to note that if f(x) e C r . , (<r real), then the complex conjugate
L r, i<rj
?WeC[ I,-io-] *
The Hilbert space completion of C^ with respect to the norm topology defined in terms of the scalar product (3.28) will be denoted by
We shall identify the representation T (for x = [ 1, i<r] with its extension to a unitary representation of G in (ft . The family of unitary representations so constructed is called the (unitary) principal series.
The irreducibility of these representations is guaranteed by a theorem by Hirai [Hj] (see also [Wj] vol I p. 463). It, however, fails for the case of the two-fold covering Spin(2h + 1, 1) if 2h + 1 is even and c = o.
57
We mention finally that for unitary representations of the principal series the scalar product (3.28) coincides (up to a constant factor) with the scalar product (2.25) in the compact picture, so that our present notation for the Hilbert space completion of does not conflict with the notation used in Section ЗА.
3.E Infinitesimal generators and Casimir operators of the elementary representations
The emphasis in this work is on the global realization of the elementary representations. It is nevertheless useful to write down the expressions for the infinitesimal operators of these representations. We shall collect here the corresponding formulas in the noncompact picture. ^They will be used, for example, in Section 6A, to establish intertwining properties of some differential operators for exceptional elementary representations of type (3.5).)
We shall restrict ourselves to type I representations (i. e. , symmetric tensor inducing representations of M,--see (2.9)]. In this case all infinitesimal generators appear as first order differential operators in x and We shall list them in the same order in which the
-v 2h corresponding global G transformations onN«R were described
in Section l.C. (and the noncompact realization of the type I elementary representations was presented in Section 2 D). We shall deal throughout
58
with the so-called mathematical generators (see Section 1 A and Table
1 of Section l.B), which are antihermitian for the unitary representations of G.
(i)	Translations. The generators T , defined by
P
T f(x) = —— f(x-x')| । have the form p	x'=0,
т =-V	( =	, X = l,...,2h.	(3.29)
И P ^P
(They are related to the Euclidean momentum operator P by P - iT . ) PPP
(ii)	Rotations (or M = SO(2h) transformations):
X =-(xaV+JA^) =x V - x V +S <? - 5-c)	(3.30)
|1 V	0 'цу V p. p. V QV p, up, V
We note that the "spinorial part" of X
s	= . £ <) -X J , ц, у =1,	2h,	(3. 31)
|1V 0V p. (f|i if
are interior differentiation$on the cone К (2.11), since -------------------------- n
РДМ] T2_0 =°	<3-32)
for any choice of f(^), and therefore, we do not need to introduce independent coordinates among the J.’s (cf. [ 33] ).
(iii)	Dilatations. Differentiating (2. 31b) with respect to |a|
and setting a = 1, we obtain
59
D V-
(3. 33)
(iv)	Special conformal transformations. They act non-trivially
on both x and
2 X
C (=X „ -X ) = 2(h+c)x + 2x (xV) - x V + 2x s. , p pfl p2h+l	p p	p	Xp
(3.34)
where s is given by (3.31). X p
The Casimir
operator
xBA
ABX
2	2
|X + D + 2hD + С T
“ pp	p p
(3.34)
turns out to be a constant. A straightforward computation of the
right-hand sideof (3.34) gives
2(X) = Ш + 2h - 2) + c2 - h2
(3. 35)
This expression coincides with the analytic continuation (3.4) of
the eigenvalues (2.4) of the Casimir operator for the finite dimensional
representations of О (2h + 1, 1) to arbitrary values of c. We note that,
in agreement with (3.22), the Casimir operator (3. 35) does not
depend on the sign of c:
(3.36)
60
II. INTERTWINING DISTRIBUTIONS AND THEIR FOURIER TRANSFORM
4. Intertwining operators: x-space realization
4. A Group theoretical definition of the intertwining operators
By a well known theorem (see [wjlvol. I, Corollary 4. 5. 8. 3
to Theorem 4. 5. 8.1 and subsequent remark on p. 343) a character
determines every К-finite unitary representation of G (in particular,
every unitary IR) uniquely (up to equivalence). Since, according to (3.22), the unitary principal series representations x = [ 1, c] and X = [ f, -c] have the same character, they must be equivalent. (We recall
that according to (3. 20) (2. 8), if L =(lj. • • , f [ h]	then
1 = f(-l) (h t h] t . . . , t r )Y Therefore, there should be an
4	1 L	[ hj '
(and i/^__) X
2’h]
intertwining operator A^ (see definition (3.1)) mapping onto (Z ) аПС* COmmu^ng the representation operators:
AX^X =TX\ (x = H,c], x = [7,-c] ).
(4.1)
Let^(g) belong to the space ~
of functions on G satisfying
the covariance condition (2.14) with X replaced by X' Then the intertwining
operator A
X
can be defined by
61
[ Av /] (g) = / skwnJdx =	w)db,
X '	NZ	N °
(4.2)
where w is again the non-trivial element (1.10) of the restricted Weyl
group. To verify this statement we have first to show that satisfies the covariance properties for an element of indeed,
C- / (gwmWa 'n ) dx1 N f	b' + x?
= |a|2h f |(gwn mWa"l) dx = |a|h+ C (mW) J /(gwn ) dx
n’ x	nz x
= |a|h+c D^m’1) (A^)(g).	(4.3)
In deriving (4. 3) we used the following (notation and) identities:
w w = wn, w 1 = rf , b1 = (b , . . . , b_, ., -b ),	(4. 4a)
b	b	,	1	Zh-1 Zh
b1
w 'aw = waw ' = a ' ,	(4. 4b)
_ 1	..J	Tf?	£	-yj	£
w mw = wmw = m , D (m ) = D (m),	(4. 4c)
a *n a = n	(4. 4d)
X	f ।
I a| x
(cf. (1.10) and Section 3. B). The covariance property (4.1) is
verified by a straightforward application of (2.15).
62
Using the Iwasawa decomposition (1. 31) of "n^ and the covariance property (2.14) for^e^ , we can rewrite (4.2) in the form X
(A /)(g) = fj(gwk )---------- .	(4.5)
x/ N4 X (l+x2)h-c
Since k* varies in the compact manifold К we see that;
A^ is well defined and analytic in c for Re c < 0 ;	(4. 6a)
similarly we deduce that:
A„:^	is well defined and analytic for Re c > 0.	(4. 6h)
XXX
Both A and A^- can be extended to meromorphic functions in the entire X X
complex plane c(see [к 3,K4,S1]).
From now on we shall consider the special case, when the representation i of M is equivalent to its mirror image (see Section 2A) and can therefore be extended to a representation of G = O^(2h+1, 1), ex
including the space reflection
I (x, x_ ) = (-x, x,,)	(4. 7a)
s — Zh — Zn
and the (Euclidean) ’time inversion”
® (x, x ) = (x, -x ), _x = (x , . . . ,x ). Zh	1	£П-1
(4. 7b)
We note that I is either a proper rotation (for 2h odd) or related to 0 by the proper rotation I 0 (for 2h even).
63
Using (2. 8) and the fact that the transformation
I 0: x-»-x commutes with M = SO(2h) we find s
i	1 w i -If	-I t f t -1
D (m) 3 t> (m ) = D (0m0 ) = DX(I ml ) = D (I )D (m)D (I ) . (4. 8) s s	s	s
Hence, we can define an equivalence map
r(Is):^r7	-by
s [ i, -c]	[f,-c] y
[Tds)/] (g) =	as)/(g)-	(4.9)
(Note that we use distinct notation f and f for the mirror image representations even if they are equivalent, since the matrices
T i
D and D , appearing in the equivalence relation (4. 8), are different except for the trivial representation 1 = 0. )
We define the normalized intertwining operators G^ :
^X = I ,-c] “*1 f, c] =^X and G~ X -*4. Ъу
G = у A T(I ), G— = y. T(I )A_	(4.10)
X X X s X X s X
where y^ is a normalization factor to be determined in such a way that the operators G satisfy the normalization condition
G G.
= 1 = G_G X
X X
(4.11)
X
64
for c pure imaginary. Condition (4.11) can indeed be satisfied, since
G G___is a unitary map	onto itself which commutes with all
XX	X
T (g) (g e G ). Schur’s lemma (asserting that G G_ is a multiple X	ex	XX
of the identity in^) can be applied--according to Section 3. A. --to all elementary representations.
The operator G^ (considered as an integral operator in the
noncompact picture) appears as analytic continuation (in c) of the
Euclidean 2-point function in a conformal covariant quantum field
theory	15»M21, (This accounts, in particular, for our choice of
notation. )
4. В The intertwining distribution in the noncompact picture
Now we shall find the action of the operator G on functions z	9	* X
f (x)e C„ (related to £ (g) by (2. 26)). Accor ding to (4.10) (4.2) we have
CGxfl
r i J> ~	„
у ID (I ) *(n wn ) dx = X J s о x. x
= у fl) (I m(x,w) ^xtx*) (x )C ^dx ,	(4.12)
X J 8
In deriving the last equality we have used the relation
wn = n rij. a(x, w) m(x, w),	|a(x,w)| =
x' s	x
(4.13a)
65
where
x -x	1
x' = I Rx = (	,	) , m = m(x,w) =1 r (x), m =r(x) I (4.13b)
s	2	----5---- л
X x2
and r(x) is given by (2.30). Performing two consecutive changes of
Z 1	2 - 2h
variables x -’x1 : x = I Rx' (x = —— , dx = (x’ ) dx*) and
s	2
x'
x? x2: x' = x^ - x^ in (4.12) and using the identity
I r (I Rx’)I = r(x')	(4.14)
s s s
(which is implied by (2. 30)), we obtain
(G f ) (x ) = /Ъ (x -x_) f (x ) dx_,	(4.15)
X 1 J X 1 2	2	2
where G (x _) is the (matrix) kernel
X
\ f
Gx(x12) =——-------- D (Hx^)), x12=x1-x2,	(4.16)
/ 2 \k+c
(For fixed x G is a (matrix) operator in the finite dimensional 1^ X
representation space У .)
In the special case of type I (symmetric tensor) representations t
of O(2h) (see (2.9)) we can replace D by the corresponding generating homogeneous polynomial
Jl Щг(х ))^2	=( r (x)J-2) >forJ,1,J2 eKa (4.17)
66
(it suffices to assume that at least one
2 satisfies the light-cone condition -
of the variables^ or 0).
Expressing у

in terms of another normalization constant
n(x).
/
_ и(х )W 2 h+c X -------------
(2tr)h
(4.18)
in order to conform to the notation of refs. [4,5], we obtain
n(v)	2	h+c	l	n
□ (х;^Л)=--------— (________)	(-J-r(x)^) (jeK2h,	(4.19)
(2^)h	x2
The expression for the Green function (without using the^, £ t
formalism) was first obtained from О (2h + 1, 1) invariance consideration (see, e. g.	The derivation of (4.16) presented here
which follows the mathematicians’ work[K3 ,К4,М2|51з due to Koller [ K5] • It makes transparent the role of the Weyl symmetry w and the conformal inversion R.
5.	Momentum space expansion of the intertwining distribution and positivity
5.	A Fourier transform of G
The intertiwining kernels are used, among other things, to construct scalar products for the complementary (and some exceptional "discrete") series of unitary representations of G.
66
(it suffices to assume that at least one
. .	.	.	2
satisfies the light-cone condition =
of the variables^ or 0).
Expressing у

in terms of another normalization constant
n(x).
/
_ и(х )W 2 h+c X ------------
(2tr)h
(4.18)
in order to conform to the notation of refs. [4,5], we obtain
n(v)	2	h+c	l	n
□ (х;^Л)=---------— (________)	(-J-r(x)^) (jeK2h,	(4.19)
(2tr)h	x2
The expression for the Green function (without using the^, £ t
formalism) was first obtained from О (2h + 1, 1) invariance consideration (see, e. g. ,&иЦМ'10,Р5[| The derivation of (4.16) presented here
which follows the mathematicians’ work[K3 ,К4,М2|51з due to Koller [ K5] • It makes transparent the role of the Weyl symmetry w and the conformal inversion R.
5.	Momentum space expansion of the intertwining distribution and positivity
5.	A Fourier transform of G
The intertiwining kernels are used, among other things, to construct scalar products for the complementary (and some exceptional "discrete") series of unitary representations of G.
67
It is, therefore, important to investigate their positivity properties.
It turns out that this is most conveniently done by carrying out a harmonic analysis of G on the parabolic subgroup NAM.
The group NAM is a semi-direct product of an abelian subgroup N and a reductive group AM which is the direct product of a compact semisimple part M and of a one-dimensional dilatation subgroup A. The harmonic analysis on such groups is elementary and proceeds in two steps. First, one performs the (ordinary) Fourier ~ 2h transform on № Й. , defining
GX(P;W2> = /Gx(x:Jl’^2)e‘1PXdx-	(5Л)
Using the integration formula
F(h+c)	i °o	12.
________ f /_2 .Ы с -ipx 1 Г ,	h+c-1 г	-ipx J (——- J	e dx =_ J da a	J dx e
,, Ji xZ	, h 0
(2ir)	(2ir)
00
= f da aC 1 exp(- J______ p2) = Г (-,c) (|p )C	(5. 2)
0	2a
valid for -1 <Rec < 0, and extending it by analytic continuation as a distribution theoretic identity to all noninteger c, we obtain
f
л c (2-rr)	k=0	r	x
68
п(х)(гР2)С
F(c+h+f )F(c+h-l)
Л 1	WW , *‘к к
go(£)r(/-k-c)r(h+k+c-l) [--------L.]	(^Г =
i 2 гР
П(Х)Г(-С) (PJnXPJ-o) l ? - l„ I Ъ 7\ (.1Л- --------------[ A1-!2.]^ )c рс-л h'2) (-)
F(c+h+f)	! 2
гР
(5. 3)
where
2, P ^1?2>
ш = cos в = 1 -______
(5.4)
[ в being the angle between the (2h-l) vectors;, andz in the ’’rest frame"
P = (0> IPI)
|p| =(pM
(5.5)
of p ] . Here we have used the following expansion formula for the Jacobi polynomials (see Eq. 8.962.1 of ref. [G7]):
t
(c-f, h-2)_ (-1) Pf (ш) —~
£ A) rd-l-c)rh+k+c-l)
k=0	Г(-с)Г(с+Ь-1)
(5.6)
For real c, if f(x)eC^, then also f (x)c (and we could have
у considered the restriction of C to real f as a real T covariant vector
X
space if we wished to). For such c, G^ defines a hermitian sesquilinear
form on C~ x C~ :
X X
69
(f , G f ) = f f<f (x ), G (x _) f_(x_)> dx dx =
1 x *'*'11	% 14 & 4	1 z
= J<f/p)’ Gy(p) f2 (p)> -Ah •	(5-7)
1 X L (2ir)Zh
(Following the physicist tradition we distinguish between functions and their Fourier transforms only by their arguments. ) Thus, we can expect to find complementary series of unitary representations of
G, , for some real values of c only, (ex)
5. В Harmonic expansion of G^ (p)
Since the Fourier transform (5. 3) of G^ (x; jp^) automatically homogeneous in p (and therefore irreducible with respect to A) it remains as a second step to carry out the harmonic analysis with respect to the stability group
U = SO(2h-l)	(5. 8)
P
of p. In order to perform this step we shall use the frame (5. 5) (which is possible by O(2h) covariance) and will set
2
X.. = (z., i), z. - real, z. = 1 j = 1, 2; z, z _ = cos 0 .	(5. 9)
dj “J “J	“J	-1-2
(That is no restriction on generality, because G is a homogeneous polynomial in J and 5-_. )
70
We consider the Jacobi polynomial P jc	as a
function of for fixed _z ) and carry out its harmonic analysis. Since Lt	1
it depends only on the scalar product z z , it is invariant under the 1 Lt
3
•7
(z z ) of SO(2h-l)
stability subgroup SO(2h-2) of_z , so that one is lead to the decomposition -2.1
(c-i, h-2)	1	h-‘
of P. ’ (z,z_') in zonal spherical functions C f —1—2	H	s
(defined in Appendix A. 2):
p(c"-f> h-2)	.	. у*
S-U
2h+2s-3
(f-s).’
(c+h-1) s
(c-h-f+2)
(2h‘3)f+s+l
^2 'f-sCs
3
2 (w),	(5.10)
where we use the notation
(a) = Г(а+к) = a (a + 1) . . . (a + k-1). E(a)
(Let us caution the reader that in deriving (5.10) we had to correct
(5.11)
some
misprints in Eq. 7. 391.10 of ref. [ G?] . )
By construction, the expansion (5.10) corresponds to the
decomposition of the Green function G (p) (regarded as an operator in
the finite dimensional space = ^(2h))	®®^h-l) projection operators
f S	Л A* f	TA'S
П (p), which map V onto the subspace J . of SO(2h-l) symmetric (2h)	(2h-l)
traceless tensors of rank s^ f. [ The easiest way to see that is to note
that due to Eqs. (A. 18) and (A. 19) of Appendix A the zonal spherical h.2
function C 3 belongs to the eigensubspace of the Casimir operator s
71
П [ 2h-l] of SO(2h-l), corresponding to eigenvalue s(s+2h-3).] These projection operators have the following characteristic properties:
9 e f q 1	9 q
П S(p)lTS (p) = 5ss,n S(p) ,	(5.12)
Tj П ls (p) =1,	(5.13)
s=0
S. s	Р-й $. s	.
<p).p	...p
= n'^plp®'	far 0 < s< I.	(5.14)
The completeness relation (5-13) follows from the remark at the end of
Section 2. A, which says that type I representations of SO(2h) contain
only type I representations of SO(2h-l). Eq. (5.14) follows from the
argument given below.
Consider the operator Q: I/”
(2h)
(2h)’ definedbV
f = (f
Qf = (P
It is obviously U “SO(2h-l)—

covariant and satisfies
1	(f-1)	is
QDX(u) = D* '(u)Q, 0П
f —1S П Q.
Applying the second equality f-s+1 times we obtain
72
_f-S+l is s-ls i- s+1
Q П = П Q = 0
is
(since П =0 for i < s). This proves (5.14).
In terms of the generating functions
n4i-h’ ^i1-•	..p v.... Wz -• • %
the completeness relation (5.13) assumes the form
9g	9 Г	#1
„Ьп ^5-1’^z’ =	H”-1’]5	<5-16>
the last equality (in brackets) hold for the special choice (5. 5) (5.9) of
the vectors p,^, an^^"2' ^°^ows from the preceding discussion that

f ч	s
П	=Afs(‘1)
i 2 гР
1 h i
C 2 (ш). s
(5.17)
Here A i s
is a normalization constant determined (in principle) from
(5.12). In order to evaluate it, we insert (5.17) in the completeness
relation (5.16); the result is
E (-1)S AC2 (ш)
s=0	ls S
(5.18)
Using the orthogonality property for Gegenbauer polynomials and their normalization (see (A. 20)) as well as the integration formula
73
(l-u/+h‘2(l+u)h‘2 C 2 Mdu = (1)Г(Ь-1)Г(МЬ-1)—-------------(-2-h-t.^21_
s	-	(2h-4)! (2h+l+s-3)!
(see [ G7] Eq. 7 311. 3), we obtain
r	(h-1)
A = (2h + 2s - 3)-------------- ---------=--- .	(5.19)
(<-s)!	(2h-3)^s+1
Noting that .	v	1
lim £— C (Cos в) = — cos s0
p-о Lv	s ,	s
(see [ G7] Eq. 8. 934. 4) we see that in the special case when 2h = 3 and
Eq. (5.19) becomes meaningless, the projection operators (5.17) can
still be defined and are equal to
П

f g
(2h=3) ^I'h’ = ('1)
S 2 1! (i)f Ц-s)! Ц+s)!
cos sS, (5.20)
i 2 zP
where cos d (=w) is given by (5.4). In this special case П for s > 0 is a projection on a two-dimensional space which is reducible with respect to the proper rotation group SO(2), but becomes irreducible, if we
consider the group 0(2) including reflections. The decomposition of t s
П into one dimensional SO(2)-irreducible projectors corresponds
i is0
to the decomposition of cos s0 into e . (We note that in this case sin 0 = (p^PJ-^) Vp2
representations will then vary between -i and i.
(px^A^), )	s	SO(2)-irreducible
74
The preceding discussion does not apply to the case h = 1
♦ (corresponding to the ordinary Lorentz group О (3,l))which is exceptional
from our present point of view. That case is treated in Appendix B, f s
where it is shown, in particular, that the operators П vanish for s > 1 (h=l).
In order to be able to write down in a simple way the scalar
product of two symmetric tensors in terms of the corresponding
homogeneous polynomials f, it is convenient to extend f(j)toa
homogeneous harmonic polynomial F(t,) (~f jj( ?>)) t, e <C2b (see Appendix A). f g
The factorization property of the projection operator П
exhibited in (5.17), persists for its harmonic extension. It is related
to the decomposition of an arbitrary SO(2h)-transformation into a rotation
in the plane (2h-l, 2h) sandwiched between two
SO(2h-l) transformations (which leave the axis 2h invariant). Its group theoretical derivation (given in Appendix A. 4) also uses the factorization and orthogonality properties of generalized spherical functions. The results for the harmonic extension of (5.17) is
n'S(P:4^ =bfs Lfs(P’ nSS (P;4’ Lfs (p^2>	(5-21a)
2h
where are arbitrary complex 2h-vectors (t,j, f, 2 e ®
,	\	f ~s
?‘S V-s'^fs^^^ 2
p = (p ) p ,
a 2
£ = u ) c
h+s-1
Cf-s(n>
(5. 21b)
75
=(p?^'s for£-2 = 0
• n s/2 h-i
nSS (р;СД2) = (-2)S J—Ihlk (к, ) С (Ш),	(5.21c)
1 2	(2h-3),	11 22
Lt S
= ——i^L
’s/lrlllr22
.	(s+h-l),
b = 2*‘S ------------— (*) ,	(5.21d)
0 q	a
(2s+2h-2)^
and tr.. are the elementary "projection variables"
11	(p4)(pSJ
ir . . =П (p; t,., t,. ) = £,.£,. ----, i, j = 1, 2.	(5.22)
ij	1 J 1 J p2
The simplest way to prove (5. 21) consists in verifying that the right hand
side is harmonic in both £ and and goes into (5.17) for (5.19) to
2
t,j = J-- (^. - 0)- We shall apply this factorized form for the projection operators to the derivation of some differential identities among the intertwining operators at exceptional integer points in Section 6. D.
5. C Normalization and positivity for non-exceptional representations. Complementary series of unitary IR' s
Putting together Eqs. (5.3)(5.10) (5.17) and (5.19), we obtain the following harmonic (SO(2h-l)p) expansion for the momentum space intertwining kernel:
76
G (p) = n(x )Г(-С) (Ь > Б к (c) I?s(p),	(5.23a)
X (h+i+c-l)r(h+c-l)	s=0 S
where
,,„.<.3 ГЦ...,-1|ГЫ.1-| e lhtS-C-11,-.	(5.23b|
S	r(h+c+f-l)r(c+2-h-f)	(h+s+c-1)
The normalization condition (4.11) does not fix the factor n(x) uniquely and there seems to be no universal choice of n(x ) equally suited for all purposes. Knapp and Stein [KJ] [Section 13 (Lemma 36)] require (in addition to (4.11)) that n(x ) is a meromorphic function of c with all its zeros in the (closed) right half plane and all its poles in the left half plane and that it is real for real c. The simplest choice of this type which satisfies the above conditions for both integer and half integer h, and has only simple poles and zeros, is given by
(n(x)=)n (X) = (h+f+c-1) r<h+c-1> .	(5.24)
Г(-с)
The normalization property (4. 11) follows from the identity
Kfs(c)Kfs(-C) =1
for the coefficients (5.23b).
We shall also use three other choices of n(x ) [ all compatible with (4.11)] each appropriate for a different kind of problem: The
normalization
77
n (х ) = Г(Ь+1+с)Г(Ь-с-1) t	(5.25)
° r(h+f-с-1)Г(-с)
adopted in refs. [M2,B4]is particularly convenient in writing down the differential identities between hermitian forms for exceptional representations (see Section 6. D below). The corresponding intertwining operator becomes infinite for the exceptional representations X (3- За) and x’^* (3. 3d).
In the derivation of operator product expansion in the framework of quantum field theory pursued in[M2,B41 (see also Chapter V) Where one is concerned in particular with the Wightman positivity condition for composite fields (see Section 5. D, below), one should demand that n(x ) has neither poles nor zeros in the right half plane and is positive for c > 0 (h > 1). If we assume in addition that n(x ) is polynomially bounded at infinity (in any direction different from the negative c-axis) we end up with the following (unique) choice (cf. [ K5] ):
n (X) = (h+f+c-1) —t-^c-l).. .	(5.26)
С	Г(с)
A third choice is appropriate in the study of exceptional points for which h+c-2 is a negative integer. In that case we shall use
n (X) = nr(X> = (h+f-c-1) 
C	Г(-с)
(5.26* )
78
We shall use the notation G , G, G and G for the intertwining operator with normalization (5.24), (5. 25), (5. 26) and (5.26’ ), respectively.
We remark that the product n(x ) n(x ) is the same for all four different choices of n(x ) (in fact, it remains the same for any n(x ) consistent with (4.11)). Knapp and Stein [KJ] have shown that this product is proportional to the Plancherel measure for G (see Section 8. A. , below).
We are now prepared to study the positivity properties of the p-space intertwining kernel ,	- f	.	- f (h+s-c-1).	.
G (P) = (lp )C S к (c)I?S(p) = (|р2)С Г -----------------I?S(p) (5.27)
X	s=0	s-U
ts
(where П are the projection operators (5.21) satisfying conditions (5.12)-(5.14)) and to single out the complementary series of unitary type I
f
representation^ of О (2h+l, 1). Since the projection operators are positive semidefinite (having eigenvalues 0 and 1), it is enough to control the sign of
2 c the coefficients к and of the distribution (p ) on the space C~. (of the Fourier transforms of functions of C„ ).
X
Let us start with the case f = 0. In this case, , 00	\
к00(с)=1(=П (p)j	(5.23c)
79
and the right hand side of (5. 27) looks superficially positive for all 2 c
real c. However, the distribution (p ) requires regularization which destroys positivity for c < -h. On the other hand, the positivity of G+ (p) guarantees the positivity of the scalar product
f ff(x) G+(x - x_) f (x_) dx, dx_ for f(x) eC J J 1 X 1	4	X
only if the Fourier transform of f(x) exists and is square integrable with weight G^ (p) and Eq. (5. 7) makes sense. Now f(x) is infinitely differentiable so that f(p) falls off fast for p -* ». The only trouble may arise from the behavior of f(x) at infinity (or of f(p) at the origin). According to (2. 29) the asymptotic behavior of a function f(x)tC„ (for X = [ 0, c] ) is given by д f(x)~---------------- for x -* oo .
, 2 h-c (x )
2 -c	2
This implies that f(p) behaves at worst as (p ) (for c > 0, p -* 0).
Then the positivity condition is ensured by the existence of the integral
JG (p)|f(p)|2d p = A J(p2)C(p ) d2hp = A J -------jJP--	(<oo)	(5.28)
*	0 (p2)C-h+l
2_2	2	2
p <A	p <A
which is satisfied for c < h. Thus, the representation X = [ 0, c] is unitary for -h < c < h. (Note that this was the first place where the special
80
choice of normalization (5.24) did matter; for the choice n = n (5.26), C
G^would have poles for positive integers c’s including the points c = 1, 2, . . . , h-1 in the middle of the complementary series, and would change sign in each such point. )
Similarly one proves that for t > 0, h > 1, the sesquilinear form
(5.7) on x defined by G+ is positive defininite if 1-h < c < h-1. XX	X
The integer points (3. 5), in particular the boundary points for the complementary series, require a special investigation which will be postponed to Chapter III.
The results of this section can be summarized by the
following
Theorem 5.1. The sesquilinear form (5. 7) on C^, x
with kernel G given by (4.19)(5.24) (with Fourier transform (5.17))
is positive definite for
i = 0
-h < c < h
(h> 1)
(5.29a)
f = 1, 2, . . .
1-h < c < h - 1 (h > 1) .
(5.29b)
The unitary representations of G, , Y (ex)
and 7 in the domain (5. 29)
(which differ only in the sign of c) are equivalent.
The unitary IR x in the range (5.29) with c / 0 are called the
complementary series of type I unitary representations of O+(2h + 1, 1).
81
5. D Wightman positivity
It turns cut that the harmonic expansion (5.23) is also suited for the study of Wightman (Minkowski space) positivity of the 2-point function in a (weakly) conformal covariant quantum field theory.
The problem has also a purely mathematical formulation--and justification (without explicit reference to quantum field theory) and it would be in the spirit of the present Work to emphasize that aspect of the matter.
It was shown recently (see [Li] ) that for some range of real c the elementary representations of SO (2h + 1, 1) can be "continued" (in a peculiar way) to a special class of unitary representations of the universal covering G(2h, 2) of the conformal group SO0(2h, 2). The maximal compact subgroup SO(2h)®SO(2) of this group has a continuous center--the subgroup SO(2). The unitary representations of G(2h, 2), obtained via analytic continuation of the elementary representations of G, can be characterized by the requirement that the (hermitian) generator H of this distinguished SO(2) subgroup (which plays the role of a "conformal Hamiltonian, " see [ S5]) is non-negative. This is precisely the interpolated holomorphic discrete series of unitary IR of G (2h, 2), studied for instance in [ R2, GTO, 02], The subsequent analysis of Wightman positivity could therefore be regarded as determination of the range of x for which the
82
corresponding representations of G(2h, 2) are unitary. It also appears to be equivalent to studying the Osterwalder-Schrader positivity (see [Mj5] ).
Here we start with the "Euclidean 2-point Green function"
(5.23) with normalization (5.26) and continue to Minkowski space
momenta p^ = ip^ (p^ - real)
The Wightman 2-point function
for composite fields in momentum space w (p)
is obtained from
(5.23) (5.26) in two steps. First, we define the Minkowski space
2 c	2 c 2	2 c
т-function by replacing (p ) in (5. 23) by (p -iO) = (p -	- iO) :
T(p)	(b2- i0)C S к, (с) n (p).	(5.30)
(Here and in what follows we use the notation and generalized functions’ techniques of Gel’ fand et. al. [	. ) Then we determine w(p) in
terms of the discontinuity of 7 :
W (p) = - i o (p- ) [ 7 (p) - 7 (p)] = u X X
=	-2 sin ^(c-fjLk^ 0(p H-jp2)^ £ K (c)(jp2/ I?S(p) =
Г(с)	+	S=° is
(5. 31)
83
Tn deriving the last equality, we used the distribution theoretic identity
(Q + iO)X - (Q - iO)X = 2i sin trX (-Q)X	(5. 32)
where t^ = 0(t)tX (see [ G4] ). It can be verified that the distribution w (p) (5.31) is the Fourier transform of the x-space Wightman function, defined, as usual, as the boundary value
w (x) = lim G (x,- s - ixn).
C e*0 X
In order to find the range of positivity of the right-hand side
of (5. 31), we first notice that the operator . .,s ,i 2. t is . .
(-1) (jp ) П (p)
is Minkowski space positive, since, according to (5.17) (5.19), for
3
(-1)S(aP2) П S(p) = IP |2^ Vo x S	s
(5. 33)
for Ш = 1-p2	> 1.
M2
The last inequality (w > 1) is fulfilled, because we have
_	_ о	о
>0	'°,=J >
(5. 34)
and p = p - p^ (because of the 0 (-p ) factor). Now it is quite straightforward to establish the following positivity condition for the
composite field Wightman functions
84
w (р) >0 if с > 0, f = 0, or с > h + f - 2, f = 1, 2, .. . .	(5. 35)
The limit case c = h + I - 2 (f = 1, 2, . . . ) corresponds to the conserved 11 canonical currents. ” The composite character of the fields is reflected in the restriction c > 0. If we wish to incorporate elementary fields it is appropriate to use the normalization
n (x) = 2C (h+f+c-1) Г (h+c-1),	(5.36)
which violates condition (4.11). (The essential distinction from (5.26) is, of course, the absence of the factor [ Г(с) ] \ ) With this choice the Wightman function assumes the form:
2 c-f V' (c+2-h-f)	2 f fs
w(p) = 2ir--------(-P )C S -----------------— (-D (p )VS(p),	(5.37)
Г(с+1)	(c+h+s-1)
I - s
which is positive in the wider range c > -1 for f = 0. The limit case f = 0, c = -1 reproduces the two-point function for a free^spinless, zero mass field and (5. 36) is adapted to the canonical normalization of such fields.
The positivity restriction on the "anomalous dimension" h + c of (weakly) conformal covariant quantized fields was also obtained by other methods in refs. [R2] , [ F2] .
85
III. PROPERTIES OF ELEMENTARY REPRESENTATIONS AT EXCEPTIONAL INTEGER POINTS
6. Nondecomposable representations and intertwining differential operators
6. A Subrepresentations of exceptional elementary representations
The present chapter will be devoted to the study of the four families of exceptional type I representations, x^ = [	±(h+f+P-l)] ,
= [f + v, ±(h+f-l)] . We shall use the shorthand notation
C . instead of C ± or C >. iV
As was pointed out in Section 3 A the space C contains a finite dimensional invariant subspace E^ which can be described in the (x, J, (-picture as follows. It consists of all polynomials P(x,J)eC^ 2h 2
(x£R ,	= 0) satisfying the P-th order differential equation displayed
below:
E^= £p(x. J-) £C^ v P - polynomial; (j.V)PP(x, J.) = 0 for V =•	(6.1)
This and similar statements in the rest of this subsection follow from the intertwining properties of the corresponding differential operators which are established in Section 6 B. , below. The general solution of (6.1) can be written as a finite expansion of the type (2. 29):
p-1
p (x’<P = So (x }V' ' hk(x: x})
(6. 2a)
86
where { x} stands for the 4-tuple
2
{j ,x} = (^, x^, xAj., 2(x^) x - x J>).	(6. 2b)
(хл J) = x 5- - Z, x .
<7 p,p	Zp. v
and h, к
is a harmonic polynomial of degree к in the first argument
and
a homogeneous polynomial of degree f in the last four variables:
Axhk(x; {J-, x'}) = o, -f)hk (x; {^, x}) =0.	(6.2c)
Note that the four linear forms (6. 2b) of satisfy the first order differential equation
(jV) {y. x} =0.	(6 3)
They are in one-to-one correspondence with the generators of G
displayed in Section 3. D if we identifyjwith the (infinite dimensional)
generator -V of translation (see (3.29)), then x^.would correspond to 2 dilatations, xAj would represent SO(2h) rotations and 2x(xj) - x z, would correspond to special conformal transformations.
The space contains an infinite dimensional invariant
subspace
(Д^ f(x,£) = 0); 3g(x,£)ec’*:
f(x, t,) = (^V)" g(x, t,)}	(6.4)
87
(The notation for the subspaces E and F^ is taken from ref. [61] , where the integer points in the space of SL(2,C) representations are studied. )
The subspaces E^ and F^ are orthogonal with respect
to the invariant bilinear form (3.25) on С , x C, . Indeed, if tv tv
PeE, and f eF , then tv	tv
B(P,f)= —— fp(x, S )f(x, ?,)dx = —— Jp(x, <} )((? V) Pg(x, £)dx = £ f	=	I f	=	=
= (-I)" — /[(} v/p^i)] g (x, t,) dx = 0.	(6.5)
ц t	=	=
(We note that this calculation is only legitimate if both P and g are harmonic in their second argument. )
The space C' contains an infinite dimensional invariant t v
subspace
F'iv= {ieCiv :3geCL: f (x.p = (5V)P g(x,jj}.	(6.6)
Its dual, C,+ contains another infinite dimensional invariant subspace tv	r
(for h > 1):
= {f(x, £)e cy* : ( ^V)P f (x, ^) = 0 }.	(6.7)
The subspaces and F^ are orthogonal with respect
to the invariant bilinear form (3.25). If f(x, t,) is the harmonic
88
extension of (jV) Pg(x,j^ )e F1^ , andf'	then
В (f, f' ) = _J_ f f (x, () ) f * (x, t,) dx = 0	(6. 5’ )
(f + р)!
The proof is the same as in (6. 5).
Conversely, if for fixed / ’ in C^+ Eq. (6. 5^ ’) holds for
any Ре E (geC ) then feF (f‘ tD ). Thus, one can use (6.5) tv tv	tv tv
and (6.5) as alternative definitions of F^ and D^. Similarly,
assuming that F and D are defined by (6.4) and (6. 7), we can use
(6. 5) and (6.5) to define E and F', tv tv
Remark ; In the special case 2h = 3, the exceptional
representation X^p=[^+P>l+^1 °f SO (4,1) is still reducible when '+
restricted to the subspace D^CZC^ . This is most easily seen by
exhibiting the К content of . According to Corollary 2.1 of the
reciprocity theorem of Section 2. C an IR of SO(4) with highest weight
' +
т = (Tp T^) is contained in x^ p iff 7^ and 7^ are integers and
|	| < t + v < 7 . On the other hand, according to (6. 4) (6.7)
the factor space	is isomorphic to F^ (С C ) and hence has the
same К -content which is given by the representations 7 = (Tp 7^) such that I 7 I < I,	7 > i+ v. Thus the K-content of D splits
1 1.' —	2 —	tv r
in two disjoint sets:
D [ SOf(4,1)] = D* x D.‘ , I v L ’ J tv tv ’
89
where
. v co
D^iggf+i’ Mp+k)’
v 00
D‘ =®©(-e-i, •e+i'+k)	(6.8)
i=l k=(J
t
There are no elements in the Lie algebra of SO (4, 1) which would
connect D, and D . f V tv
However, according to the results quoted in
+
Sec. 2A D. are mirror images of one another. Therefore, the direct tv
sum space D is irreducible with respect to the extended group
f
О (4,1) (which includes space reflections).
We note that in the case of the Lorentz group (i.e.for h=l)
= ап<^ Df v = °	1 (see -Appendix B). For h>l all the
(<)± subspaces I are non-trivial for any t.
6. В Intertwining differential operators. Partial equivalence among
the representations X p
Proposition 6. 1. The differential operators
dP=(Ff: C'tv-*C\v	(6-9a)
and C'+-*c+	(6.9b)
L,	1V Lv
are intertwining operators for the pairs of representations
(X£„, X^) and (x^, x/„), respectively.
90
dPT =T d"	(6.10a)
Vp xlv
d* PT 1+ =T . d' V.	(6.10b)
X SLv x£v
(The operator d|P is always assumed to act on harmonic functions of t,, so that we can set Э2 = 0. )
The simplest way to prove this proposition is to show p	V
that the operators d and d1 have the right commutation relations
with the generators of the elementary representations exhibited in
Sec. 3D. Verification of the infinitesimal form of (6.10) is only nontrivial for dilatations and special conformal transformations.
For the dilatations (3. 33) it is sufficient to observe that
(qV)P(-e+>'-l-xV) = (f-l-xV)(qV)\ q = £or 5	(6.11)
To verify (6.10) for the special conformal transformations (3.34)
(with c taken from (3. 5)) one uses identities of the type
[ (qV)^, x ] - pq (qV)^ \ [ (qV)\ x2] = 2p(xq)(qV)P И H
[ (qV)\ X (x\7)] = x^(qV) + (y-l)q^ + q^xV)] (qV)P !-
2
In deriving the second equality we have used that q =0 for both
q = J. and q = 3^.
For exceptional x> the dual representations x and X
are no longer equivalent. In that case the intertwining operators
91
I
, v ,1 ZJ+v+h-l^v (ZhH +s+ v- 3)! (v+^-s)! .	.
G^(p) = %\(p)--(*p ’	^о(2ьш+,-3)!7;--------(-n	n (p)
and
0 J. у , _+	, , ,1 2.f+h-l	(р-l)! (2h+f+s-3)! (	f + ps, .
(P)=GX’+(P) = (|P }	s=^+l (s-f~-l)! (2h+2f+p-3)!-----П (p)’
(6.12)
(6.13)
establish only partial equivalence between the dual representations,
while the operators G (i are not well defined on C , since Xfp	(,
n (x) (5.24) has a pole for x = X/l7
(They can be defined, though,
as hermitian forms on the corresponding invariant subspace, see
Sec. 4C, below). On the other hand, the operators G defined by
(' )•
(4.19) (5.23) with normalization (5. 26' ) do make sense for у = x.
(' )+ - lV
(but develop poles for x = X« )• In particular, G = G" _ is lv	tv v
a distribution supported at the origin in momentum space:

- (2 , )h (2O^2h-2)	t-lVp)1'-1 [	<6.14.)
= (2f + P+2h-2)! (f +P+2h-3)!
'	e1i0r(l-p+e)
2 h+f + p-l-ey(-l)S I?S(p;,y, ij) p2	„(f-s + p)!(2h+.f+s+p-3)!
r	s = 0
(6.14b)
(' )±	(* Я1
The operators G^ annihilate the invariant subspaces of
while their images
(' )±
are the invariant subspaces of •
(We notice
that for integer h G^ is, according to (6.12), a differential operator if (l<)p<h-l, then the same is true for G, +. ~	f v
92
The properties of the intertwining operators G^ and
their relation
to the operators d
can be formulated in a compact
way, if we
use the language of exact sequences. We recall that a
dire cted
sequence of homomorphisms of vector spaces is called exact
if the
image of each homomorphism of the sequence coincides with
the
kernel (i. e. the set of vectors mapped into zero) of the next
one.
Proposition 6. 2. Let i' denote the imbedding map of
the corresponding invariant subspace into C'	. Then each directed
sequence of homomorphisms in the "quartet diagram" displayed on
Fig. 1 is exact.
Fig. I-
Quartet diagram of intertwining operators at exceptional points
93
We have, in particular, the identities
dPG' = (zv/g" (х;?Л ) = ° = G + G"	,c .
h 4	0	i. v Iv	(6.15a)
d'	(6.15Ы
с; V clt <p:t.K>(ijp^ - о - o;*,c;;.	<C15=>
tv	v
G" d*" = C’ (p;j,^)(i^ p)P = 0 = G" Gt .	(6.15d)
tv	tv r /’ t, t,	tv tv
Proof. The intertwining kernel
g" (x;7, £) = n-(xfp } X2 v+t~1 l-7TM^	(6.16)
tv * ?
is a polynomial in x of the type (6.2) and is, therefore, annihilated v	+
by d . On the other hand G^^fp) is a homogeneous function of p of degree 2(f + P+h-l), which annihilates Fourier transforms of polynomials P e [since? Pfp,^) = P (i\7 ,^)6 (p) and the degree of P(x,j) does not exceed 2{v+t-l), ci. (6.14a] . This takes care of (6.15a).
To prove the first equation (6.15b) we use (6.13) and
(5.14) which can be written as
(р^)"пМЧр;^) = 0 fors=f+l..............t+v	(6.17)
The last equation (6.15b) is a consequence of the orthogonality property of projection operators, since
94
I?	g I? g
°;;w "-^<^-3)1	l(',1’.”(2ht,tp.l-,>. <«_is>
S =0
Eqs. (6.15c) and (6.15d) now follow because of the
symmetry of the kernels G^ (.	, t,^) with respect to the last two
arguments.
Proposition 6. 2 implies a number of isomorphisms
(and equivalences of the corresponding IR1 s of G ) among the
factor spaces
X' I
for each exceptional (integer)x
and the invariant
subspaces of the partially equivalent representations:
F * C /	=*F* — С+/	. E ^C+ /	. D »C* / . .	(6.19)
tv tv' E, tv tv' D ' tv tv' F, ’ tv tv F' ' tv	tv	tv	tv
We remark that the exact sequences of the quartet
diagram of Fig. 1 are parts of longer exact sequences relating
by partial equivalences all exceptional elementary representations
(3.5) with the same infinitesimal character [Z5,GH] (and not just the type I representations, considered here). We also note that similar
diagrams of intertwining mappings are known for the complex semi-simple Lie groups (see [G2] [ Z$] ).
6.C Hermitian forms on invariant subspaces. Exceptional series
of unitary representations
As was observed in the preceding subsection the operators G+ (with normalization factor n (5.24)) are not defined on the spaces
X	+
95
Ctv
Similarly, the operators G
(with normalization n (6. 14))
become infinite on the spaces C , r	tv
We shall prove here that
these singular operators can nevertheless be defined as quadratic
forms on invariant subspaces.
We start with a heuristic discussion of the problem,
-f-referring, for the sake of definiteness, to G . The
kernel G
X
(x)
differs from the kernel G (x) (which is well defined X
left half plane) by the numerical factor
for all c
in the
П+(х) _ (h+f+с-1)Г (h+c-1) . n (x ) (h+f-с-1) Г (h-c-1)
(6. 20)
subspaces of C
subspaces 1^
(I )-(. Let us use the uniform notation for tv
(‘ )±
, . The operator G vanishes on the invariant fp	x
(')_	(i )_
°f	(for x = X^V ), while the numerical factor
the invariant
(6. 20) becomes
infinite for exceptional representations with negative
c.
We have to show that their product G defines a
non-vanishing
(and finite) bilinear
form on I x I ,	. The most
tv tv
natural way to
do that is to use a
limiting procedure (reminiscent to
analytic
renormalization).
Let, for
example, X£ = X^p (e) = [ f ,1-k-f-p+e] ;
consider the hermitian form
£
96
which is well defined for positive £. In order to find the limit of
(6.21)	for e-»0 we expand the kernel G _ in powers of e: Xe
I 1/ I 2/ 7}V+Jl~1	2	t
<-(x; * >= тйь ШrU^:i)[7-^* °(£)1 e
crucial point is that the singular term in
because of (6.5) (since (x )p+^ (J-r (x)t,/б E^
Thus we can go to the limit UO, obtaining
(fl’B^f2)s^TBf,P,s (fl<f2) = M fl(xi)’BL (xrX2)f2(x2) dxi dx2 ’	(6‘23)
The
(6. 21) vanishes
and f _e F. ).
where
p	2	1	2	2
Bf p(x’7’?,) = (2тт)Ь(/+1/)'. F(h+f + P-l)	l°g^ [ (xj)(xt,)-—(^)] .	(6.24)
The kernel B, (x,^;z..t,) is not covariant under ,	f v 12 6ъ
т
TX tv of G, .. Under dilatations and special (ex)
the representation
conformal trans-
formations it acquires non-homogeneous terms (because of the log).
For instance, under the conformal inversion (1. 29) (2. 30) (2. 31c)
B^p transforms as
BL (X12’^1^2) ”(X12 x2)MP'1BL(Rxl’-Rx2;r(xl^l’r(x2)^2) =
= Bf>i2’71’^2) +
(-D^l P(1°g Xl + 1°g X2} ,4 ,.-lr ,	„ ,, i 2	.,f
+ (2ir)h (f+p)!F(h+f+ p-1)	2 5	X12^1 X12^2 " **12 %^2	'	( •	>
% note that (/<,- [£	(6.25’)
(see (6.21)), because	L"
= As	I lo^ - HlfM) -	] (6.2V)
where V'(i)- ’inFfe). The last three terms in the square brackets do not contribute to the right-hand side of (6.2J1) for the same reason as the A term above.
97
However, it is not difficult to see that the hermitian form (6. 23)
remains invariant, since the inhomogeneous terms can be split (as exhibited by (6. 25)) into a sum of polynomials of in one of the variables (multiplied by more complicated functions of the
other variable) and therefore have vanishing expectation values
(6. 23) for f^ f2 e F^ .
Let I be the invariant subspace of C for any of the X	X
(type I) exceptional representations (3. 3). In general, we define the
hermitian form B, on I x I by \ XX
(fl’ BI 9 =	ff < fl(x? •	(x12} f2 (x2}> dx! dX2 ’
X	£	(6.26)
1 2 X
Ь (' )	(' ) c )+	{’ )	{> )	{> )
where X£ = [•£	, ±(c	- e)] for x = [ f , ±c ] (c >0),
One finds by inspection that they are all finite (non-vanishing) and
T
invariant under the corresponding representation of О (2h+l, 1).
The explicit forms of their kernels look simpler in the x-space
picture for c>0 and in the p-space picture for c<0; in particular,
we have,	2
T5i+ i r\	ylog(2	,*2 J-lr. / .„nf+V
Bb	~ (2тг)И f!r(h+f-l) 2 }	[^Г<ХН]	.	(6.27)
. . 2, h+f-1 i s
= f ! (2h+f-3)! (2h+2f+p-2)l	<2h+*+s" 3)! f1" sn (p) +
4
/i 2 . p . .4 (2h+^+s-3)! s '
+(log72)ski (-1) —(ZiT-i)T n (p)j-	(6-28)
98
The definition of
Proposition 6. 3,
(' )±
В implies the following
The hermitian form B^. on 1^, ----------------------I v tv
(1Ц.	(Mi
is related to the hermitian form (F , G, ’ F ) on c; 1 by ----------------------------------- Г tv 2 ----------- tv —
(f,, B „
Г tv
f ) = (□'. ’ F , B\ ’
2 tv 1 tv
G . F ) = (F , G\ ' F,) tv 2	1 tv 2
(6.29)
where f.
C )±
= G\ ' F., i = 1, 2; tv i
F _ can be regarded as cosets in the 1 f Ct		
factor space
C" / jf-HOF) f v
(6. 30)
(or as arbitrary representatives of these cosets).
The second equation
(6. 29) can be written in the symbolic form
□' ' B' ' □' '	= G ,
tv tv tv tv
(6.29' )
where both sides should be regarded as kernels of
a hermitian form on C. (or on the product of-the factor ---------------------- l v J------------------------------
spaces (6. 30) )
According to this proposition and to the isomorphism
(6.19) between the invariant subspaces I and the dual factor spaces, the study of the positivity of the (singular) hermitian forms (6. 26)
is equivalent to the simpler problem of studying the positivity proper-
ties of the (regular) intertwining operators □'	, defined in
Sec. 6b.
There are three classes of subrepresentations to
be
studied: those realized in the subspaces E, , Fl =« F, and r	tv tv tv
Dfy-
99
The only finite dimensional unitary representation of
О (2h+l, 1) is the trivial one (realized
in the one dimensional space
E^). It corresponds to the end point
(f-Ю , c = -h) of the complemen-
tary series of unitary representations
(5.29a).
In order to single out the
infinite dimensional unitary
representations in the spaces F and D it is necessary to study
the
positivity properties of the p-space kernels (6.12) and (6.13).
We
see that G, !.v
is only positive for I - 0, while G^ (p) is positve
for
all f and;p.
The unitarity of the representations in F follows
immediately; the
proof of positivity of the scalar product in
requires an additional argument, which is presented in Appendix C.
Thus we have
Proposition 6. 4. The only unitary irreducible subrep-
resentations of the exceptional type I elementary representations
(' ) ± f
X. of О (2h+l,l) are those realized in E„,, F„  kF! and D fp —	--------------------------- 01 Op Op ---------------- fp
We remark that the elementary representations Xq >
containing the IR' s F^ are the limit points f = v, c = h-1 of the
complementary series (5. 29b). For 2h = 3 the representations in
formthe discrete series of sqiare integrable unitary representations
of 0^(4,!) which will be studied in Sec. 7 below. For integer
h> 2 the representation in is known [Z 5	] to be equivalent
100
to the irreducible principal series representation [ (1, 1, . . . , 1, i. +1, f + p),0] (which reduces to one of the representations (3. 5c) for h = 2).
The unitary representations in F„ (^F' ) and in D Op Op	fp
for all 2h>5 are called exceptional series of unitary representations of 0^(211+1,1).
A last remark concerns Thieleker assertion [ 13 ] about the existence of yet another set of exceptional unitary (typel)
representations. These are the representations listed in case
П.В.1) of Theorem 3 of [13 1 with A = n. , j>2 (in Thieleker* s
notation). In the special case of type I IR* s (j = 2) they would be contained in the elementary representations x^ = [^,h+f-l], f>0 (in the notation of the present paper). It appears that this assertion is not correct. Indeed, it follows from the results of Hirai [H3 ] and Knapp and Stein[K3 >K^that the representations x are topologically irreducible and that the only invariant hermitian form in C is Xf the one generated by the intertwining operator G-x. considered here.
Xf
On the other hand, Eq. (5.23) above shows that the kernel G-%, (p) Xf is not positive definite. Therefore, we maintain that the representa-
tions
are (irreducible and) non-unitary.
101
6J3 Differential identities between hermitian forms for exceptional representations
The relation (6.29) between B, and G, is not the only lv	!.v	’
one between the hermitian forms on invariant subspaces and equivalent
factor spaces of exceptional elementary representations. The objective of this subsection is to establish two other sets of identities
у	v
of that type, which involve the differential operators d and d'
In order to write these new identities in a simple and
symmetric form, we first notice that the singular bilinear forms
B* and B^ are proportional to the appropriate limits of the
intertwining operator
G normalized according to
(5. 25) for X~*X,.
X i
and X“* X^ > respectively. Indeed this follows from (6. 26) and from the proportionality of the intertwining operators G and G° (for non-exceptional x )• It appears more convenient for our present purposes to use G° rather than В-ъ .
Proposition 6.5. In the limit points x and X*jp (in which G° becomes infinite) the products (pj-^)PG°-
v о	v
and (рЭ„ ) G , + (p; t,, Эу) (p^,) remain finite, and the following identities
take place
, .v_o	i , .v (2h+f-2)p	0 ,
(p/? Ct v- (р:Л ’^2 (p?2 " (h+f-l)p Gfp-(p:?'r^2)
(6. 31)
102
(Zh+f-Z)
te-.l (P9/Gk°+ (p^9?)(p/	G^+ (P:^}	(6'32)
,	r(’)° r°, '
where v± = c (* )± • v
Proof. Eqs. (6.31) and (6. 3Z) follow simply from
similar identities for the projection operators:
,	. v f
<PJ1> П
v _ U-s+1) (Zh+f+s-Z)^ (f+1) (h+f-1) v	v
Z v t+v s
(6.33)

»!?>.».•	v (f-s+1) (Zh+f+s-Z) z v
Ur)!1 (p9t,) n	= (/+1) (h+z-D	(г~) n
V V	(6. 34)
for s = 0, . . . , I (v = 1, 2, . . . ). The first of this equalities is a
direct consequence of Eqs. (5. 17) (5. 19). The derivation of (6. 34),
on the other hand, uses the explicit form (5. Zl) of the harmonic £ s extension of П , and requires some work.
First, we shall establish the following auxiliary relation
for the functions & (p, t,), defined in (5. Zlb):
(P\)bf s Lf s (p’= *bf-ls Lf-ls (p’f°r *-S+1, P =v4“P'	(6‘ 35)
P
To prove (6.34) we notice that according to (5.21)
is 3	3
bfsLfs(p’^ (h+SAJ ’^2 ~ ^V^pV)3^
and we use a known identity (Eq. 8. 961. 3 of ref. [ 40] ) for
103
Jacobi polynomials:
Applying (6.35) v times, we find
(рЭ )%, L, (рД) =-^7F^b. L (p.t,) (for s<f)
H t, f+vs f+PS	f! fs fs
v	(f-s+1) (2h+f+s-2)
=	----7^------------LL (р,Э )	(6'36)
'	2P(h+f-l)	S '
v
Inserting (6. 36) in (5. 21a) we obtain (6. 34). This implies (6. 32)
since Eq. (6.17) guarantees the vanishing of the singular contribution
2
(proportional to log —) in the counterpart of (6. 28).
P
7. Discrete series of unitary representations
7.A Definition and general properties of the discrete series of SOT(2n, 1)
Let g-»U(g) be a unitary IR of a semisimple Lie group acting in a Hilbert spaced. It is said to belong to the discrete series if there is a non zero vector ФеХ for which
J |(L,U(g)L)|2dg<co	(7.1)
where dg is (suitably normalized) invariant (Haar) measure on G.
It is known (|W$] vol.I 4.5.9)that (7. 1) implies the existence of a
103
Jacobi polynomials:
Applying (6. 35) v times, we find
(рЭ )%, L, (рД) =-^T^b. L (p,t,) (for s<f)
H t, f+vs f+PS	f! fs fs
v	(f-s+1) (2h+f+s-2)
~-------------"Lfs(p’95>	(6‘36)
'	2P(h+f-l)	S '
v
Inserting (6. 36) in (5. 21a) we obtain (6. 34). This implies (6. 32)
since Eq. (6.17) guarantees the vanishing of the singular contribution
2
(proportional to log —) in the counterpart of (6. 28).
P
7. Discrete series of unitary representations
7.A Definition and general properties of the discrete series of SOT(2n, 1)
Let g-»U(g) be a unitary IR of a semisimple Lie group acting in a Hilbert spaced. It is said to belong to the discrete series if there is a non zero vector ФеХ for which
J |(L,U(g)L)|2dg<co	(7.1)
where dg is (suitably normalized) invariant (Haar) measure on G.
It is known (|W$] vol.I 4.5.9)that (7. 1) implies the existence of a
104
positive number dy, called the formal dimension of U, such that
J l(«’,u(g)'₽)Pdg = dy1 Jl<pll2 ПФП2 ;	(7.2)
(the value of dу depends on the normalization of the Haar meafu^re dg).
The question of existence of a discrete series is answered by the following theorem of Harish-Chandra [H2](Part П, Theorem 13).
Theorem 7 1, A semisimple Lie group G with finite center has a discrete series (of unitary IR) if and only if it admits a compact Cartan subalgebra (i.e.iff rank G = rank K).
For a proof the reader may also consult [WJ] Vol. II, Theorem 10.2.1.2. (K is, as usual, the maximal compact subgroup of G).
Corollary 7. 2. The groups SO^(k,l) and Spin (M, 1) (N>2) have a discrete series iff N is even (N = 2n).
This Corollary also follows from ifeorem 9 of Г'1'l'lJ and. from the observation that the restricted rfeyl group of SO(N,'l) contains the matrix	(see ('1.'!)).
A convenient characteristic of the discrete series representations is given in terms of the spherical trace function t^. (g) = Tr (nTU(g) 11^)	(t: IR of К = SO(2n) in 1/.; П^: projection on If?)
introduced in Sec. 3.C.
be the character of a discrete
series
Let %
representation . Let further U be a subquotient of the elementary
representation x- Then according to Corollary 3.5 of Sec. 3. C.
у
0y is expressed in terms of the spherical trace function t .
105
®U	= Ьт ~ Ьт iOT T£ U’	(7- 3>
It can be shown that t^ (g) is a spherical trace function of some discrete series representation iff it is square integrable on G.
There is a conjecture (Blattner1 s formula) which purports to give the К-content of the discrete series representations for an arbitrary semi-simple Lie group. This conjecture was recently proven to be true for the groups SO(2n, 1) by one of the authors [ M4] . Here we shall restrict ourselves to the discrete series representations contained as subrepresentations (or quotients) of type I elementary representations of G. In this case the situation is simpler and our analysis can be based on the following
Theorem 7. 3. The unitary irreducible discrete series representations Ц of SO (2n, 1) and Spin (2n, 1) are not unitarily equivalent to their mirror image U (defined in Sec. 2A) Both U and U appear as subrepresentations (more generally subquotients) of the same elementary representation.
The proof is based on the Harish-Chandra classification of discrete series representations for arbitrary semisimnle Lie groups (see [H2,W$1 and [M4] ) and will be omitted.
Corollary 7  4. Discrete series representations of
106
G—SO (2n, 1) (or Spin (2n, 1) ) never contain a completely symmetric tensor representation of К (for n>2).
Corollary 7. 5. For n> 3 no discrete series representations of G (—SO (2n, 1) ) occur as a subquotient of an elementary type I representation у = [ f,c] of G.
To prove 7.4 , we note that completely symmetric tensor IR1 s of К - SO(2n) are equivalent to their mirror image for n>2. Therefore if such an IR T of К occurs in a discrete series representation U, it will also occur in its (inequivalent) mirror image U. Theorem 7. 3 then implies that 7 will have to be contained twice in the elementary representation X which contains both U and U as subquotients. But that contradicts Corollary 2.1. of the (Frobenius) reciprocity theorem (see Sec. 2G).
To obtain 7. 5 we recall that a representation
T = (t , . . . , T ) °f К = SO(2n) n> 3 contains the type I representation 1	n
(f) = (0, .. . 0, t , = t) of M = SO(2n-l) only if т = . .. = т = 0 n-1	1	n—
(see Sec. 2A). Such representations of К are equivalent to their mirror image and, therefore, we can apply to them the argument, which led us to 7. 4.
Since the representation theory of SO (2,1) is well known (see [ВИ] [G2] and Appendix B. 4, below) Corollary 7. 5 only leaves us with the group SO (4,1) (and its covering Spin(4, 1) which, however,
107
contains no
(unitary) IR
additional type I representations). In this case every
I of M=!SO(3) is of type I. According to the analysis
of Sec.
6A
Theorem 7. 3 implies
that the only candidates for the
unitary
discrete series IR'
s are
the subrepresentations in the sub-
spaces
D (6. 8) and their I v
duals
in . In the following subsection
we shall demonstrate that
they indeed
belong
to the discrete series
of SO'(4,1).
We shall close
this general
review
by quoting a result
of G. J. Zuckerman[Z4,Z5](specialized
to our
case).
Let U. be the discrete I v
series representation of О (4,1)
(acting in D Г C ). Let further lv~ lv
E, denote the finite
I v
dimensional
representation contained in v . (as lv
well as
the space in
which it
acts). Then the character 6. of U. can I v tv
be expressed
in terms of
the character ®
sentations (see Sec.
of E. and characters I v
3B) as follows:
of elementary repre-
® . = ® , .
lv X» f v
-® + +e^
xf v Etv
(7.4)
7.B Unitarily induced representations on G/К
To identify the discrete series representations, we use a method which is fairly standard and has been applied to arbitrary semisimple Lie groups [ -H7,S2’] . It singles out the discrete series IR1 s as subrepresentations of suitable unitarily induced representations of G [ =SOf(2n, 1)] on G/K.
108
Let т = т (к) be a unitary IR of К on a finite dimensional vector space ]A. Consider the space X = X^ (dg, IT.) of functions f on G with values in IT which satisfy the covariance property
^(gk) = T(k-1)/(g)	(7.5)
and the square integrability condition
(^) = < °° (7- 6)
Here < , > is the hermitian (sesquilinear) scalar product in/^ and dg is (as usual) the Haar measure on G. We define a (highly reducible) unitary representation U^_ of G on^ by
[UT(g)|] (g1) =^(g_1g')•	(7.7)
U can be identified with a subrepresentation of the left regular representation of G (^^ is isomorphic to a closed subspace of the corresponding representation space). Therefore, it admits a decomposition into irreducibles in which only principal series and discrete series representations of G appear (see Sec. 8 below).
The procedure of picking a single discrete .series representation out of U^_ is based on the following two observations.
(i) A standard reciprocity theorem (similar to the one quoted in Sec. 2C) says that U^_ contains the discrete series representation U of G iff the restriction of U to К contains T.
(ii) For every IR of G the Casimir operator must be a multiple of the identity:	= xZ The eigenfunctions
109
of (^2 belong tocC^ iff the corresponding eigenvalue X is discrete.
y> 2
The representation U in the corresponding eigensubspace TX	7л
of
is square integrable, t Consider now those groups SO (2n, 1), for which К is
t semisimple (this excludes SO (2,1)). According to Corollary 7.5
(Sec. 7A) only the group G = SO (4,1) may have discrete series of
type I unitary representations. On the other hand Eq. (6.8) gives
us the K-content of the possible candidates U for the discrete series representations in this case. A fixed eigenvalue X of may cor
respond, in general, to a finite number
representations U . (there are two such
of inequivalent discrete series
representations for each
eigenvalue X in the case of the group G
under
consideration). For
every given U = U we select a unitary X10
in U, , , but does not appear in U, . for
X iQ	Xi
that the subspace	which consists of all
IR T
of К which is contained
This guarantees functions
i r i
T 0
satisfying the differential equation	= 0
carries an irreducible
discrete series representation, equivalent to U.
It follows from (6.8) and from the explicit expression (3.35)
for the eigenvalues of that a suitable choice for T belonging to the 2
representation U, .which acts in D. , is
r	f v ’	tv
110
т+ = Г s, slC U,+ , т = Г -s, s] C U, for s = f+p. s L J fp s L J f v
It coincides with the choice made by Takahashi [ТИ] , but it is not unique. Another choice is made in ref. [ H?] , + +
We shall consider the case т = T and will write U_ = U . s	т s
The case T = Tg is quite similar (and can be obtained from the first one by taking the mirror image). + ,
7.C Realization of the unitary representation U in the space (NA)
The covariance property (7.5) implies that the functions ^(g)	are determined from their values on the homogeneous
space G/ К which is isomorphic as a manifold to the subgroup NA
T
of G (cf. the Iwasawa decomposition (1.12)). For G = SO (4,1)
that is the 4-dimensional manifold
В s G/K = SO1'(4,1)/SO(4)^NA - Ж? x .	(7.9)
It can be parametrized by 4-vectors
x = (x, y), x = (XpX 2,x3> e Ж. ,	y>0,	(7.10)
or by the corresponding matrices b = na in NA:
b—> = n a = x x у
1 -х/ у х/ у \
2 2 , -»2 , I . I	I ->2	2, 2
х у -х +1 х -1 I , (у =|.а |, х =х +у ).
2у 2у I	У
2 2	-2., /
\ х У ~х ~1 х +1 /
\ 2у 2 у /
(7.11)
111
Here we have used the realization (1. 20a) (1. 21) of n and a . As x у n 2
already stated each function J can be specified by its values on B: ф(х) = J-( b_») .	(7.12)
x
The correspondencey_> ф given by (7. 12) generates a mapping of ^f2
oa the Hilbert space of vector valued functions on В with scalar product т
( Ф ,,Ф ? ) = f ( Ф , ( 3?) , ф ( 3?)> ^5	•	13)
i’z j ' i	z zy
Here <> > - <>>s is the SO(4) invariant Scalar product in У (we use that the representations (7.8) of SO(h) remain irreducible when restricted to
S0(3) and can therefore be realized on the same space); y’4dx* is the (left) invariant measure on the homogeneous space В (7.9). The space a^_can be considered as the norm closure of the space of real analytic functions on В with finite norm:
ii 11 2
I I Ф I | = (Ф.Ф) < 00 .	(7.14)
Considering the infinitesimal operators of the representation U in T r
we shall always assume that they act in the dense subspaceof Ж
The vectors Ф can be regarded as (scalar) functions ф(х;^) on
В x which are homogeneous polynomials in of degree s. The action of the operators U (g) on these functions is defined as follows.
112
Let xg and k(g-1,x)
be determined from the Iwasawa decomposition
(1.12) of g-1b_ : x
g-1b— = b—* к (g 1, aT ) ; x x g
(7.15)
then
[Ut ( g) ф J (3q
> = *	k( g'\x) ]^,)
g	b
(7.16)
where к—Л[к] is the 3-vector representation (1,1) of К =SO(4). We shall
write down the explicit form of x* =x*CT, k( g ^x*) and Л[ к] for a complete set О
of subgroups of G (such that each element g of G can be factored in a
product of elements of the subgroups).
a)	Translations and dilatations:
n .a e NA x
(n , a) x
-1
X
x-x1 |a|
(7.17a)
b
(
lal ’
thus

(7.17b)
b)	Rotations m e M = SO( 3):
m h_» = b -1 m , x ( m x, y)
(7.18a)
so that
Xм (=x* ) = (m JX, y), k( m \x*) = m \ A[m = m \	(7.18b)
m
113
c)	Special compact transformations (rotations in the (3, 4)-plane):
Let
t
1
0
0
I 0
\	0
\
0
1
0
0
0
0	0
о	0
1-t2	2t
1+t2	1+t2
2
2t	1-t
1+t2	1+t2
о 0
(7.19)
then
k^b—> = b k(-t; x), t x o?(t)
(7. 20a)
where
2	_k?
•	1+‘2	•	1?	•	(1’t,x3+t(x -1»	,	1+t2'	,7?0Ы
x! = ------ x., i = 1, 2, x’= ----------------, у =-------- у ,	(7. 20b)
i cr i	3	o'	cr
2 ->2
o' = cr(x,t) = 1 + 2tx^ + t x ,
114
k(-t;i?) =
2t	t
6.. --x.x. 1	-2— (l+tx,)x.
ij <r	i j i <r	3i
L _________
t ,,	!	t2 ,-»2	2,
2- (l+tx3)x. j 1 - 2 —(x - x3)
I 2	। tv
-2t v1 ; -2fd+^3)
s °	;	°
- 24-yxi °\
2^(l+tx3) 0
1 - Zyy2 0
0	1 /
J
(7. 20c)
= m к m , t = t,(t,x) =	, A = (1 + 2tx + t^x^)2
1	** L. 1	1	1	A	j
i, j = 1, 2;	(7.20d)
finally,
if we use the same notation m
for the non-trivial 3x3
sub-
1
matrix of the above

(7. 21a)
where к
‘1
is given by (7. 19) and
115
(7. 21b)
We notice that the
conformal inversion R (1. 29) is a proper
SO'(4, 1) transformation (corresponding to simultaneous
rotation in tt in the
(1,2) and (3,4) planes) such that
X
k(R,x)
Rx = ( - ^, x
—*	1	2	2
(Л[к(г,х)] )^ = ~2 [2(x^-ye^K xK)+(y - x ) 6^] , p, v = 1, 2, 3
(7. 22b)
7.D К  invariants. Solution of the eigenvalue problem for the Casimir operator. The discrete series
The infinitesimal generators of the representation U^_ constructed in the previous subsection can be written down as first order differential
116
operators in x and in a similar way as the generators of the elementary
representations displayed in Sec. 3D. In particular, the generators of
3
(IB.-) translations and M = SO(3) rotations are given by exactly the same
formulas (3.29-31) (with 2h=3). The generators of dilatations and special conformal transformations include in the present case the variable y;
D(= X ) = - x V = - xV -yV	(7.23)
40	x у
_. —>
С (= X . - X ) = 2x (x V) - X v + 2 X s - ye s	(7.24)
p pU p4 'P	p <r <гр p<rX <r X
(H = 1,2,3),
where s is the spinorial part of the rotation generators, given by
(3.31). The Casimir operator (3.34) is a second order differential
operator:
^,(u J = lx XBA = У2 V2 - 2y v + s(s+D +2y (s V + S V + s V ), l, T	Ad	у	c, j 1 Di 6	1£ j
(7.25)
Here we have used that
Is s^^ = s(s+l)	(7.26)
p\
for the representation = (s, s) of К = SO(4) (which reduces to the representation (s) of SO(3) when restricted to M).
In order to find the discrete spectrum of it is sufficient to study some special class of solutions of the eigenvalue equation
117
Ф = [s(s+l) + (t -1) (f + 2)]Ф .
(7.27)
[In writing the eigenvalues of in such a special form, we have
anticipated the result, given by Eq. (3. 35) with f replaced here by s,
3
c = h + f -1, h = — .
However, we need not assume for the moment
that I is a non-negative integer (or even that it is real). This will
come out as a result of the solution of the eigenvalue problem. ]
We shall look for solutions in which the Z-dependence is factored
out in the following simple form:
ф(х;^) = 0s; (x) ( ^Q+ (x,y+1)J-0)S ,	(7.28)
where
(|Q+(x,y+l)^) =	C^(x,y+1)^	(7.29)
will be determined by the requirement of K-covariance, ф i® a fixed 2
vector of (i. e. ф = 0, and different ф correspond to different
solutions ф of Eq. (7. 27)). (The reason for the peculiar way of writing
the second argument of Q will become clear in the next subsection.)
We
shall further assume that
Ф„,(х) is К invariant si
(which does not impose
any restriction on the eigenvalues). Under these assumptions the vector
(7.28) belongs to the K-irreducible subspace of the representation
T = (s, s) of SO(4). s
It belongs to the class of К - finite ve ctor s
118
which plays an important role in the algebraic approach to representation theory. The above choice will not only allow us to solve easily the eigenvalue problem, but will also prepare the ground for finding the G-invariant 2-point Green function associated with Eq. (7. 27) (see Sec.
7E below).
We start by writing down the most general К-invariant function
ofx. There is just one algebraic SO(4) invariant, which will be determined
as follows. We first observe that every G = SO(4, 1) invariant of the two points and X£ is a function of
- . 4yly2	(7.30)
u = u (x ; x ) =	--------------------
(Xi-x2) + (y1+ У2)
[One way to establish that is to take a homogeneous rational function 2	—
of УрУ2 and (x^-x^ > which is automatically NAM invariant, and to impose invariance under the conformal inversion R (7.22a).] Then, we notice, that the point x = (0,1) is К-invariant (since b^ = 1 ), and deduce that
u = u(xj 0, 1) =	~z— 	(0 < u < 1)	(7. 31)
x + (y+D
is a basic 1-point К-invariant (every К-invariant of x is a function of u).
In order to find Q we demand that
ho+(0’y++1)?2 = ^ih
(7.32)
119
for
= U (1 + /1 - U) 2 > 1
(0 <u < 1).
(7. 33)
Then
^QU.y + 1)^
is obtained by applying the (compact)
"boost"
x mx ^-t(i?) mx
(0,y+)'	=(x,y)
(7. 34a)
where к is given by (7.19) with
= Г x2+(y+l)2] [x2+(y-l)2] + 1 - X2- y2 , 4y x2+(y+l)2
(7. 34b)
The non trivial 3x3 part of m is given by
(7. 34c)
Using further the explicit form (7.21) of the representation Л [k] of
SO(4) we find
+	u	2
( ^Q(x,y + 1)^) = (Л [k->]^ , ^ф) =	— [ (xF)(x ^) + (y+1)	+
+ (y+l)(xjA ^ф)] - ^ф~-
2(х5)(х£ф)+[(у+1)2’
-	?	?	(7« 35)
X + (y+ 1Г
120
Inserting (7.28) (with Ф = 0g^(u)) *n (7-27) and taking into account
the differentiation properties
Эи
Эх
И
2
и	Эи „	у+1
- Ту V =и(1	2
(у	- 2у V )и = -2и ;	(7. Зба)
{y2V2-2yV + 2у (V.s + V,s + V s )} Q+ = 2и Q+	(7. 36b)
У	£ Сл j	Сл jJ. j Let
we obtain the following ordinary differential equation for ф^(и):
? d2	d
{ (u-l)u ~ + 2u ~ + [(f-l)(f+2) - s(s+l)u] } ф .(u) = 0.	(7.37)
_ £	au	si.
du
The solution of this equation is
«+2
Ф ,(u) = cu F(f + s + 2, f + 1 - s; 2f + 2; u)	(7. 38)
si
(see [КИ]). It is a polynomial in и (square integrable with respect to the
- 4	,
scalar product (7.13) with volume element у dx ) if t is a non negative
integer smaller than s:
s - t = v (=1,2,...), I =0, 1, 2, ...	(7.39)
We see that the eigenfunctions (7.28) (7.35) (7.36) are (real)
analytic in the half space y> 0 (in other words they belong to ^3 ). In
-f - 2	—»	00
addition, the functions у	ф(х; ^-) have a C limit for y—>0 (which
will be shown in the next subsection to belong to
Df,>-
The set of all real
analytic solutions of (7. 27) which belong to
and have the above
T
121
properties will be denoted by Я v
Thus we define the (Hilbert) subspace	°f %-r
as the set of(square integrable) solutions of (7.27) with s and I satisfying (7.39), and identify the discrete series representation with the restriction of U on ,+ . Its mirror image U, can be
T lv	° lv
defined as the IR obtained in the same way from U_ , where
= [-s, s] s = I + v.
(7.40)
As it was mentioned before the representation
U, = U, t v lv
+ Ц,
t v
(7.41)
A
is irreducible with respect to the extended group О (4,1). Similarly, we shall write for the (dense) subspaces of real analytic functions
A, -A,®	C
The functions (7. 28)(7. 35)(7. 38) are finite linear combinations of
the vectors of the canonical basis [defined in Sec. 2C(see also Appendix A)] .
If we act on them by (some of) the non-compact generators [say D(7.23)]
we will obtain the basis vectors for "higher" representations of K. In this
way one can verify that the representations have the same K-content
(6. 8) as the elementary subrepresentations acting in
Df,
122
7,E Two-point Green function. Equivalence of with the sub-t +	+
representation of p acting in
The functions ф(х) of vt (which behave like 2 f(x) for
у —► 0) are in one-to-one correspondence with the elements f(x) of +	-jf 4"	4"
Df V  The maPPinS L: t p Df p is given by
f(x) = [L0] (x) = lim у 2 Ф(х,у). у 4-0
(7.42)
Let T , be the restriction on t v
D.
t v
у
of the operators T of the elementary
representation Y , . r	t
It is not difficult to verify that L plays the role
of an intertwining operator between the representation U (on
and T+ : tv
L U.+ = T* L .	(7.43)
tv t v
In order to establish the equivalence between U, and T , we have n	t v	tv
to construct the inverse operator L . This is done in a standard fashion
in terms of the Green function & of the operator ^2 ”
(X = (v+t )(v+f + l) + (t-l)(f + 2)).
The G-invariant solution of the equation
((^, -X)<0+ B[y2v2-2yV + 2y(s V, + s V,+ s,,V_) -U-DU+2)]. “ Z	у Z3 1	31 Z 1Z 3
•o#+(x,£;x',^') = У4 6(x-x'Xj-J!.1/+p	(7.44)
is
рЭ + (x, ; x',^.') = N^pu^+2 F^p<u) [ Q+(x-x’> y+y') J’1
(7.45)
123
where u - u(x,i? ) is the invariant variable (7. 30), Q is given by
(7. 35) and F^ is the singular solution of the hypergeometric equation
d2	d
{u(l-u)	+ 2 [f + 1 - (f + 2)u] — + (p-l)(2f + p+2)} F (u) = 0
_ Z	du	£ V
du
(7. 46a)
(given by Eq. 9. 153. 3 of ref. [G?] )• We will only need to know the
behavior of F , tv
in the neighborhood of the singular points u = 0 and
u = 1; it is	.	-1
(zt + v + 1\ fZt +v\
F (u) и - \	j I J
ИУ	L V '	' p - 1
-Zl -1
u	for u —► 0	(7. 46b)
P-1
Fo(u)«(-1)
(v-1)'.
(2f+2) p
for U —> 1.
(7.46c)
1
1 - u
The singularity for u -» 1 is responsible for the 6-function in the right-hand side of (7.44). The overall normalization is determined from the Green formula (for ф satisfying (7.27))
>= (TTji^ &	ф (*'* 5-' > -
3
- cZ)+(x, Э ) [i^'0(x'	' )] } d *dy
°	У
U+pX	.Э^,)] ф(х’ ,^ ' ) -^	, ф&,%' )}-y,
4 where B-> is chosen as a small ball in HR. with center in x S is its sur-x
face (with surface element d<r), the prime on	and -т—у indicates that
Z	dn
124
the differential operators act on the primed argument and the normal n'
points inside B—>. The result is
N =	-------
2	(p-1)1.
(4n)
(7.47)
Applying the same Green formula to the entire half space Ж. x we express ф(х) in terms of its boundary value f(x) (7.42). This gives us an explicit construction of the operator L . Only the term (7.46b) of F , enters the final formula; we have tv
Ф(х,^, ) = [ L-1f] (x,^-) =	fs+tv(^’i ;x',9^)f(x',^')d3x'
(7.48)
where f e and x v
S;p(x,rx^,) = - firn
(-I)"'1 (4n)2
v'(2t + 1) '. Г 4y (2f + p)! [Jx-x' )Z +yZ
t + v
Q (х-х',у)Э^, ]
(7.49)
A similar expression is obtained for
Ф (x ) еД iv with
f e D .
X v
and Q replaced by Q where
125
2 2
2х х + (у -х ) 6	± ух е.
~± ,	. ц. Р______________ ц Р \ л-м-р	. , ,
Q (х,у) = —е------------z----, ц,р =1,2,3 .
РР	2	2
г	х + у
(7. 50)
Because of the orthogonality of the two spaces
1 t* —	~	3	4"
7ГГЙ! / sfP(x’^;x'	>%">d x' =0 forfe Dfp (7-51)
where S is given by the counterpart of (7.49) with Q replaced by Q .
Thus we established the equivalence between the representations and T* . It implies the proportionality of the scalar products
(7.13) and (6.26-27) in the corresponding representation space. It seems
rather intricate to derive this proportionality directly from the definition
of the two scalar products. In order to show what is involved, we shall
verify the above statement in the simplest case I = 0 v = 1. (In
Appendix B.4 we present a complete discussion of this point for the
analytic discrete series of SO^(2,1) ~ SL(2,HR.) /Z^. )
Proposition 7. 6.
The elements of are 3-vector functions of

which satisfy the identity
f(x) = V A (- Д)’ 2 f(x) = curl f f(x' ; , x •> ,	, .2
(x-x )
dx1 (2т)2
(7.52)
F or
functions in Dqi the hermitian form (f , В'
f^) (6.26) (6.27) can be
written in the following way:
126
'+	1 rr ^/X1^2 ^X2^
(fl’ B01f2)= ,^~2 ff 2 dxl dx2 (flf2B fl|i f2|J /2п	x12
(7.53)
Proof. According to the remark following Eq. (5.20) the
projection operator D(p) (= П1] (p)) can be split into two (covariant) one
dimensional projectors (for 2h = 3):
where
П+(р) +
П (p)-
(7. 54a)
П(р) = 1(6	- p p ± ip, e ,
P-v p. v HX pX^
2” 2
(p = (p ) p) .
(7. 54b)
(7.52)
has zero divergence and is orthogonal to I I :
I I projects on the invariant subspace	of . Eq.
the fact expres ses^that any
(Vf) (x) = 0
(V ft Dq1)
(7. 55a)
( ri’f) (x) = 0
(feDoi
(7.55b)
f
In order to rewrite the scalar product (6.26) (6.27) in the simple
form
(7. 53), we use the
identities
1
2 ^2
lip	’ Vlp
^1A
2 X12
2(x ) (x.?)
12 |i 12 V 4
X 12
6	,
+ rV. V, tn x
2	lu. 2v 12
X 12
127
and integrate by parts taking into account (7. 55a).
Now we shall demonstrate that the scalar product (7. 13) for functions in ,A can also be reduced to the form (7. 53).
According to (7.48-51) the function
f(x, у) = у 2 Ф(х,у), (0сЛо1 >
(7. 56)
is expressed in terms of its boundary value f(x, 0) = f(x) by
„	1 „	. Г f(x' )dx' _ У ftx'ldx’
f(x,y) = -v V A J -------------^2 -2 - r, 2 'Sr
(2 л)	(x-x')+y "JLU-MOJ
тут • (7.57)
The scalar product (7.13) is expressed in the following way in terms of the functions f(x' ) e D* (using the second equality (7.57)):
(01>02)= f ^(х»У) f2 (x, у) dx dy =
1 (2*)2
f2^x2^
2
X 12
dxldx2 = 2? (frB« f2>-
(7.58)
Л
128
8. The Plancherel theorem. Concluding remarks
4-
8.	A Harmonic analysis of the left regular representation of SO (2h+l , 1) for integer h
Here we shall apply our knowledge of the IR1 s to the standard problem of harmonic analysis of functions on G .
Let D = D (G) be the space of all infinitely differentiable scalar functions <p (g) on G of compact support. Let G be the set of all (equivalence classes of) unitary IR1 s of G and let к eG be realized by operators T (g) acting in a Hilbert space $ . Let © (<p) be the К	к	к
character (3. 7) of the representation к . The Plancherel problem is to find a positive measure dK on G such that
<p(l) = f ©K (<p) dK , for all (fleD .	(8.1)
G
where 1 is the unit element of G .
For (fie D define <p D in terms of the left regular action of the group:
<Pg(g') = [L-4 <P] (g1 ) = <P(gg').	(8.2)
Then Eq. (8.1) implies that
<p(g) = f Tr T (<p ) dK = f Tr(T*(g) T (<p) ) dK	(8.3)
*	К £	*	n	rl
G	G
(we have used that T is unitary, so that T (g ') = T* (g) ) . К	к	к
129
Multiplying both sides of Eq. (8. 3) by <p (g) and integrating over ge G , we obtain
J|<p(g)|2dg = f Tt (TK(cpf Тк(<р) dK .	(8.4)
Using continuity of both sides with respect to the Hilbert space topology in (G) we can extend the validity of Eq. (8. 4) to all square integrable functions <p on G .
In order to solve the Plancherel problem one has first to know which unitary IR1 s enter into the integral in the right hand side of (8.1) . A partial solution of the problem for the generalized Lorentz group is given by
Theorem 8. l.(|N2] , [H5] ) Let G - SOT(2h+l , 1) (or Spin(2h+1,1) ) with integer h . Then only the principal series of unitary IR1 s
X =[(f),c( = l<r)] enter into the Plancherel formula (8.1) (8.4) and
co
fdK =	f p(£ ,c) -^4-	(8.5)
J	*	*	1
(!)tM c=-i co
with the Plancherel measure p expressed most easily in terms of the
variables n. , defined in (2.3): ---------- J ---------------------
p(f , с) = C | | (n2 -n.2) П (p2 - c2) , l<i<j<h J 1 k=l
n . = I . + j -1 (j -1, . . . , h) , c = i <r
(8.6)
130
( C being a constant, depending on the normalization of the Haar measure on G ) ; the sum in (8. 5) is carried over the set M of (equivalence classes of unitary) IR1 s of M .
We see that in the special case of type I representations the weight p(f , i<r) is proportional to the absolute value square of the normalization factor n(x) [ (5. 24) or (5. 26)] of the intertwining operator:
2
p(f,i<r) =Ajn(X)|2=	| r4h(it°)~"l [ (h+^ -I)2 + <r2] ,	(8.7)
П, . .... .	. (f + h -1)!
(j-i) (j + i -2) -.
l£i<j£h-l	’
It was proven by Knapp and Stein [Kj] (see also [W2] ) that such a relation is valid for all principal series representations; moreover, the analytic continuation of p for non-imaginary c is given by
P (i , c) = A^n(x) n(x) .
(8.7)
131
л
8J3 Harmonic analysis on SO (2n, 1). The role of the discrete series
For the group SO (2n, 1) [or Spin(2n, 1)] (n integer) the principal
series representations are no longer the only ones entering into the
Plancherel formula. We will explain the reason for this difference.
To do that we shall use the function
F (ma) (3. 8)
introduced in Sec. 3B.
A function e D(G) is called a cusp form if
F^(ma) = 0
for all mae MA, a 1.
(8.8)
For G = SO (2n, 1) (or, more generally, for any semi simple Lie group G,
which has a compact Cartan subalgebra) there exists a cusp form <p(g)
depending only on the conjugacy class of g, i. e. such that
<P (gg' g Ь =	), for all g,g' e G,
(8.9)
which cannot be expanded in principal series representations alone.
To see that, we shall introduce two more auxiliary notions: the notions
of a hyperbolic and of an elliptic set of G.
Consider the torus ( = abelian compact connected Lie group)
Tor^'(G) of elements of G of the form
t = exp{2D 0. X } (q = n, n-1).	(8.10)
, , K Zk-1 Zk
k=l
We call a t singular if 0. = 0 for j Ф к or 0, = 0 for some j (in particular J k	J
the identity, t - 1 is singular ), The singular elements of Tor^ forma
lower dimensional manifold. If t e Tor^ is not singular it is called
regular. The set of regular elements of Tor^ will be denoted by ' Tor^b
132
We shall also use the notation 'A = {ae A; a Ф 1} . We define the elliptic and hyperbolic sets, G , and G, , of G by ’ ef	hyp’	’
Cef ~ ( g € C’ gl€ C’ t€ ' Тог^П^: £ = ё11 2 * * * * * )	(8- lla>
G = { g e G; 3 g.e G, h = tae ' Tor ' A: g=g h g 1 } (8. 11b) nyp	i	11
The intersection G , C\ G, is empty. It is easily verified by considered hyp
ing a neighborhood of the identity that G has the same dimension
(as a manifold) as G. Harish-Chandra has proven that the set of regular
elements of G, defined by
1 G s G . U G , el hyp
(8.12)
covers G up to a lower dimensional submanifold.
-1	00
Since g G g = G there are C functions <p on G with support
on G , which satisfy (8.9). They will have automatically compact support
since G , el
has a compact closure (because it consists of matrices of
2	n4*l
uniformly bounded norm j| g j| = tr(g*g) <2	),
Proposition 8.2. Let <p(s.) eD and supp (pCG^. Then у is a cusp
form, so that F^(ma) vanishes 
The statement follows from Lemma 3. 1 and from the remark that
the intersection G , f) G, has Haar measure zero.
el hyp
133
It follows from the second equality (3. 18) that if supp <p > then
© (ср) = 0 for all elementary representations. Let (p be a cusp form satisfying (8. 9) . Then
T (<P) = T (g) T (Ф) T (g-1) ; К	К К	л
since the representation is unitary and irreducible we can apply Schur1 s lemma and deduce that
\(Ф) = U ,	(8.13)
where X = \^(<p) is a complex number. This allows us to rewrite the representation (8. 3) in the form
(g)= f QJg) dK • * л к
G
(8.14)
The contribution of the principal (as well as of the complementary) series to the right hand side of (8.14) vanishes for g in the interior of G^ , while the left hand side is assumed not to be identically zero on such points. Thus, an expansion of cp (g) in terms of principal (and complementary) series alone is not possible. However, such an expansion becomes possible, if we
also include the discrete series representations. This was exhibited for
К-induced representations in Sec. 7 . The measure p (c) in (7. 8)
is given again by (8.6), which for the case of 0^(4, 1) assumes the form
(8.15)
(2 s + 1)! I
2S + 7/2
2	2
[ ( s + 2 ) "c ]c tg tt c
tr
s!
134
The formal dimension d. of the representation U. (in D ) is tV	H	t V	tv’
given by the residue of p + lz (c) for c - t +
,-l	„	(21 + 2 V+ 1) ! ’
= z^^u.2
(f+|) (2f +P+U.
(t + v )!
(8.16)
In the general SO (2n,l) case, we have the following theorem due to
Hirai [H5] .
♦
Theorem 8. 3. For G = (S)O (2n,l) (or Sp'in(2n,l)) only the
principal and the discrete series enter into the Plancherel formula. Using
the notation
6
for the character of the discrete series representation
U6
with formal dimension
we obtain the following explicit form for

(8.1)	in this case:
<p(l)
7 йтp" c> +,<p *
(t )e M -1«5
(8. 17a)
with P and d given by
p(t , c) = C(f+i)...(f	+n-3/2)]“|	®.+ j-i)2- (f.+ i - j)2] c.
l<i<Kn-l J	1
n-1
» П [ (\+k-i)2-c2] tgirc;	d’1 = Resp(6) .	(8.17b)
135
8.C Synopsis on unitary type I representations. Summary of equivalence relations
In the preceding pages we presented a rather comprehensive discussion of the properties of elementary type I representations of G = О (2h+l, 1 ) (induced by symmetric traceless tensor representations of the parabolic subgroup MAN where M = SO(2h) ) . Here we shall summarize for the reader1 s convenience the main facts about such representations.
First, we give below a complete list of the (unequivalent) unitary type I irreducible representations of G * , thus summarizing the results of Secs. 3D, 5C, 6C and 7 C. We are using throughout the paper the notation x = [f , c] for the elementary representations, where f is the number of tensor indices, characterizing the IR of M = SO(2h), and -h-c is the (length) dimension fixing the representation (2. 31b) of the f dilatation subgroup A = SO(1,1). We recall that an elementary representation X is irreducible unless h+c is an integer of the type involved in Eq. (3. 5) and considered in Sec. 6A.
a)	Principal series (Sec. 3D): c-pure imaginary, (c = i<r), f -arbitrary. (These representations are still unitary for arbitrary unitary IR's t of M .)
b)	Two classes of type I complementary series (Sec. 5C); 1=0, -h < c < h (h > 1 ),	(5. 29a)
t = 1, 2,. . ., -h+1 < c < h -1 (h > 1) .	(5. 29b)
In both cases the point c = 0 is excluded from the complementary series since it belongs to the principal series.
136
c)	Two exceptional series of unitary representations (Sec. 6C):
(i) the subrepresentations (acting in the subspace F
of the representation
Xqv ~ v + h -1 ] ; (ii) the sub representations (acting in the subspace
D v ) of the elementary representations X'^y = [f + ff + h -1 ] . In
the special case of h= 3/2 these latter representations are reducible under t
the identity component G = SO ( 4, 1) of	and split into two discrete
series representations	of G (Sec. 7). For n > 2 , the discrete
series representations of SO (2n, 1) are not of type I.
Secondly, we shall summarize the results about equivalences among
elementary (sub) representations.
The elementary representations
X = U > c] and X = [ ?» -c]
(8.18]
(where £ is the mirror image of £ defined in Sec. 2A) are equivalent for non-exceptional c (i. e. , for h + c different from the integer values involved in Eq. (3.5) ). For type I representations £ = £ [ = (0 , . . . , 0 , £ ) in the notation of Sec. 2A ] . The intertwining mapping exhibiting this equivalence is given by (4. 16) and (for type I representations by) (4. 19). The (Fourier and) harmonic expansion of the intertwining kernel (4.19) is given by (5. 23). For the exceptional points (with integer h+c) the representations (8. 18) are, in general, only partially equivalent. There are additional intertwining differential operators in this case. The picture of
intertwining mappings for the integer points is summarized by the quartet
137
diagram of exact sequences of representation spaces' homomorphisms presented on Fig. 1 (Sec. 6B). The intertwining differential operators (defined in Sec. 6B) provide a link between invariant hermitian forms for subrepresentations and factor representations (Proposition 6. 5 of Sec. 6 D).
138
APPENDIX A.
Symmetric tensor representations of SO(n) and their decomposition in IR’S of SO(n -1)
A. 1 Harmonic extension of homogeneous polynomial functions on the light cone
We have collected in this Appendix some facts about the type I representations of the orthogonal group, used in Sec. 5.B (for n = 2h). The results quoted in this first section are taken from [ W1]	[B2] •
Proposition A. 1. Every homogeneous polynomial.
'<»>  f|11	5-^ •	<A-»
on the cone (2. 11) has a unique (homogeneous) harmonic extension
fTT(f,), f, С СП such that
>2	~)2	p. u
^H	Rl...
1	n
(A. 2)
fH(£) =f(pforFKn.	(A.3)
The homogeneous polynomial
is also determined by its values

f (t, ) on the real unit sphere H
S11'* 1 = {b ; I2 = 1} ,
which satisfy the equation
[ Д +f U + n-2)] f (i ) = 0 , D	A A
(A. 4)
(A. 5)
where Д is the Laplace-Beltrami operator on S [see (A. 12) below] . D		 I ,
The scalar product of two tensors
/1..Л н
is proportional to the scalar
139
product on
Ж(п> ,X2(s«-lb
('ih’ 'гн’у-!
f	fl...
Ш	2Ц

(A. 6)
where (d£) is the normalized surface element on the sphere.
Proof; The first statement is a consequence of the known fact that
every homogeneous polynomial P( £) (of degree t ) can be expanded in a unique
way as a sum of (homogeneous) harmonic polynomials
хгл 2 к
p(c) = z; & ) y. io.
0i2k^t	*
Ys (C)(see,e.g.p.l5 of [W^
(A.?a)
To prove that we set
where
Г° - At-2j
p(;>
(A.7b)
is the Casimir operator of SO(n) with eigenvalues
s(s^-2)	(A-9)
corresponding to the eigenfunctions	commutes with
shall, apply (A.7) to an arbitrary homogeneous extension
of j-(£) and shall identify with	• To see that
indeed harmonic, it suffices to note that the Laplacian A
We
W
is
can be written in the form
(Л.Ю)
and to use (A.9) with b-l . It remains tp show that ($) o-nly
depends on	and not on, the particular choice of its extension
in t . Let indeed	81314 ^^//0 tw0 sucb extensions;
then the difference	~	vanishes on the cone	’
140
and hence has the form
- 5‘ Pt-zk),
where	a homogeneous polynomial of degree
projection of this polynomial vanishes:
The harmonic
The
one to
one relation between f ft) audits restriction to the real H
unit sphere
(A. 4)
is established trivially. Because of the homogeneity
l
condition,
fH 1S
extended to arbitrary real £ by homogeneity
v2
*Hft) = ft2) f„ft) (for t,2 t о ) , M	H
(A. 11)
and then to any complex £ by analyticity. We note that П is related
to the Laplace-Beltrami operator A on the sphere (A.4) by о
° = -As .	(A. 12)
Finally, since the representation (t) of SO(n), realized on the space
„ L	T/M
of symmetric traceless tensors, or equivalently, on the space ub
of harmonic polynomials on the unit sphere, is irreducible and since both
scalar products ( ,	) . and ( ,	)	, . are invariant under the action
V X(n)
of this representation given by (2. 12) , Eq. (A. 6) follows from Schur's
lemma. We shall evaluate the constant a. in Sec. A. 3, below. I
i. 2
Example: The harmonic extension of the polynomial (b^ ) ( r. =0)
2 2 2
is found from covariance consideration to be of the form (b t, ) P(bt>)
- -	2 2-^
where b£ = (b t, )	(bt.) and P is a suitably normalized solution of
(A. 5) regarded as a function of t,
The result is:
141
£1 1
H . (b, t,) =	(~~b
2 2^ t, )
с/2 Лч) =
(A. 13)
t £
-- (4) F(-p
1- £
2 ’
, 2 2
. b_L_ (fat,)2
p
where and F(a
(3; у ; x) are the
Gegenbauer polynomial and the hyper-
2  z
geometric function, respectively, and the symbol (a)^ is defined by (5.11) .
A. 2 SO(n-l) expansion of homogeneous polynomials.
The zonal spherical functions
The existence and uniqueness of a decomposition of the type (A. 7)
implies that each polynomial (of degree £ ) on the unit sphere (A. 4) has
a unique expansion in harmonic polynomials
£
s
s = 0
P(^)
Ys (U
UsS11'1
(A. 14)
(where Yg
,	n-1
(and S
by
s s11'2
(t, ;P) satisfy (A. 9)).
we can use this result
expansion
giving to
Replaci ng
n by n-1
to obtain a
SO(n-l)
for an arbitrary function f(^) °f the type (A. 1). □ the special value:
Indeed.
U )
2 _
Z , 2	- 1
(A. 15)
(cf. (5.9)
), we express f( ) in terms of the (I -th degree) polynomial
n - 2
on S , and by (A. 14) obtain
f(z , i )
142
f(z , i)[ = (-i^n)
2?	Y(n'1)(z).
S ;0 S
(A. 16)
Consider now the special case, in which f (^) is a covariant function of three vectors, say p, b and (and therefore depending only on their
scalar products). By an appropriate choice of the coordinate system we can arrange that the vector p points along the n-th axis, while b lies in the (n -1, n ) -plane. Then the function f(^,)( = f(^,;b, p)) will be manifestly SO(n-Z) invariant [SO(n-Z) being the subgroup of SO(n) acting non-trivially in the subspace ЖП 2 of ЖП spanned by the first (n -2) axes] . In this case, the harmonic polynomials Y (z) n -1
in the expansion (A. 16) depend on the single scalar variable
i.
2 ‘ 2
w = (b ) sb (= cos z b )	(A. 17)
(cf. (5.4) ) . They are determined from the harmonicity property uniquely up to a constant factor. The normalized solution of
s	z	aw
de*)
(A. 18)
given by
Y<n -1) (u ) . (i<gn -1») 2 c 8^(U )= (Xg(n -1’ 2	Ps(l/2 *2 ’ n/2 '2) (w),
(A. 19) where
T (n -1) Js
n -3
f [ c8 (e 5)]2 (de) = ^—-
1	5 -z
~	2 2	2	?
J (1 -w ) [Cg (w)] dw =
r( 2 1 24~nTT r(n-3+s)
P /ri, -1 \ 1/2	Q Q 2
2 Л 8.0+^3)Щ^3)]
1 n -3	(n+3 -4),'
s ! 2s+n -3 (n -4)1
(A. 20)
143
is called (normalized) zonal spherical function.
[In deriving (A. 20), we have used the formula
V
P+1 2л 2 ,P+1 Г(—
(A. 21)
for the surface of the
unit sphere , Eq. 7.313. 2 of ref. [G7] for the
normalization integral of the Gegenbauer polynomial^and the doubling formula
2^P 1 Г (f) Г (P + 1/2) = tt^2 p (2 p) for Г -functions (see Eq. 8. 335. 1
of ref. [G7] ). ]
A. 3 Evaluation of the proportionality constant a^ between the scalar
products in
and in Л , ------- t
In order to find the proportionality coefficient a^ in (A. 6),
we shall evaluate the scalar products in both sides for the special case of
vectors of the type (A. 13). It follows from the definition of (b , t, ) that the usual square norm of the symmetric traceless tensor (bw -tr)
is
( b
b -tr) (Ъ b -tr) =
H (b > b ) n£
(A. 22)
Choosing for b a real unit vector
and
using Eq. 8. 937. 4 of
ref. [G7] we obtain
e) =
J-1 i! t (i)
(П -2)I _
24 Л ’
Г(2 4)(п+1 -3)1 rf| -i) (n-3)i
Г(а+Х) q Г(а) J
(A. 23)
^ ...
e
(Hnf(e’^’ Hnf
(e’ U) I = Hn£
ТГ nt
e ,
t

144
On the other hand, according to (A. 20)
g f	2 r .
<Hn£(e’^’Hn£(e'^>	(n)	=	T =
Xf 2	1
i1	n -2	(n+f -3)!
	2Z+n-2	(n -3)1
(A. 24)
From (A.b^fA. 23) and (A. 24), we find
1 I
(A. 25)
A. 4. Derivation of a factorized expression for the projection operators | | Here we shall use (for the first time) the canonical basis in
and shall write down some of the (generalized) spherical functions evaluated in Vilenkin [VI ] .
Type I representations of SO(n-i) are labelled by just one number B. ; therefore, we can denote the canonical basis vectors of the symmetric tensor representation (t ) of SO(n) by
= ~ c >	1 - si	- 3 ->	• • • - 3 ,	>	I s ? I	•
S’ 1	2	n-3	—	1 n- 2
(A. 26)
(Since the representation
is fixed, we shall omit the superscript (f ) on the
the unit sphere. To do that we define the rotation R in the plane
145
(Г е ) (Г, С Sn 1 , е is the unit vector of the nth axis) such that n	n
R,e = £	. trR = n-2 +2 (e , t,) = n-2 +2 cos <p(t, ) .	(A. 27)
t, n	t,	n
Let
l	t _ '
Dec, (A) =	, D (A) S
OO	О	D
(A. 28)
be the matrix elements of the representation D in the canonical basis. Then
(f ) *	1/2	t	-1
s' '(C) = d (£) d; (R 1 )	(A. 29)
S	n	Ob	t,
where d (f ) is the dimension of the representation (1 ) given by
(cf. (2. 2) ). To prove that, we first note that the matrix elements
I -1
D. (A ) only depend on the cosets Uo
ASO(n-l)e SO(n)/SO(n-l) ,
since
£	-1	4	£	-1
Dnc (u л > = Dnc<A ) if u e = e (u C SO(n-l)CSO (n) ); UD	Ub	n n
(A. 31)
this means that we can replace the rotation R
in the definition (A. 29) by
any A^ of the form A^
that the rotation R -1 A t,
= R u , u e SO(n - 1)
belongs to the same coset
Secondly,
as A R^
we observe
. Hence,
the
have the correct transformation law
146
ПЛ(Л)2 (£ ) = d 2(f ) D* (R^A) = 2 S , (i ) D* (A) s n os r, s' s' s s
(A. 32)
for basis vectors in the representation space
The normalization
factor is chosen in such a way that the canonical basis vectors are (ortho)
normalized:
1 / „-1 W> = 1>. о
(A. 33)
(This follows from known normalization properties of matrix elements of representations of compact group, -- see, e. g. , [30] , Sec. 27. )
Each vector f = f(t,)e	can be written in the form
f(i) = z (l)< \ , f > = d}(£) <D*(R ) S , f> .	(A. 34)
о	о	n	c, u
In deriving (A. 34) we have used the unitarity of the representation D (A) ,
which implies the identities:

Let A be any operator in
with kernel (i , ?) :
(Af)(t) =	f(C)(d4')-	(A. 35)
Then the kernel
is given by
A (i ,1') = dn(f) < Eo , dSr’1) AD^R,) E0/> .	(A.36)
147
Indeed, noting the connection between integration on the unit sphere S
and invariant integration on the group SO(n) and using (A. 34), (A. 32) and the orthogonality relations
r I	t	rl
JDSS (A)Ds s(A) dA=dn(n5ss 5ss	(A-37)
bl 2 b3 4	ЬГЗ b2^4 '
( [	] , Sec. 27, p. 73), we derive
,V) f(t') (dV) =d3/2U) / < A*D/(R)S0 , D*(A) E Q ><d\a)S Q, f^
= d^U) < D'(R^)S0, Af>
(Af) (t.) .
1—S -—A s
In order to apply (A. 36) to the projection operator [ |	= | | (en)
(which projects onto the invariant subspaces '}{, of SO(n-l) ) s	e
n
we shall first write down the rotation R^ (A. 27) in the form
R = u R(<p)u ,	(A. 38a)
Then
R {cp ) belongs
to the same coset as R^
and the kernel
of the projection operator can be written in the following
manifestly positive form:
148
JZl3 (L i') =dn(i) < So> DJ!(R‘1(<p))nJ!SDJ!(u‘1 u^,) D/(R(^,)Eo) =
= d U) D*-(n) (R*1^) ) Воо^’СиЛ ,) D^n)(R(<p') )	(A. 39)
n us	uu g g su
i (k)
Here D (к = n, n -1) stands for the representation matrix of the IR (t ) of SO(k), and our notation does not distinguish between an element и of SO(n-1) and its canonical imbedding in SO(n).
In order to evaluate explicitly the right-hand side of (A. 39) we set
t = ( z sin (p , cos (p )	g = (z1 sin (p' , cos <p' )	(A. 40a)
•l" - z' = 1 , z z' = w (= cos 6 ) .	(A. 40b)
The matrix elements appearing in (A. 39) are evaluated in [V9] . In particular, _s(n-l), -1	s! _n/2-3/2 ,	.
W	cs (u) «	(A-41)
is proportional to the zonal spherical function (A. 19) (it is denoted by
P^qq1 S(“) in Vilenkin [V1] ):
D^(n)(R4(<^)) = D^n)(R (</>)) = Os	st)
r-	-,72
_s, n ,,	f!(f-s)!	2s+n-3	. s _ „п/2+s-l ,
= 2 - -1	—H—— ~П— 1—7Г"	sin (PC. cos (p .
2 s s! s+n-3)	(n-2)	* t-s
__	1+1 t
(A. 42)
Combining together (A. 39) - (A. 42) (A. 30) and taking into account that .I , n. 2 ( —)	и и
S(L i) --	1 • -t 1 rYs I'"1. •. i'Vi <A- 43>
(because of (A. 6) (A. 25) ) and identifying cos (p with pXj , cos (p1 with p£ we end up with Eq. (5. 21).
149
A.5 Interior differentiation on the complex cone. Expression for the convolution of two tensors in terms of homogeneous polynomials
There are two equivalent ways to write down the convolution product of a rank v tensor f with a rank -f tensor g (v$t) in the language of homogeneous polynomials. One (used throughout Sec. 6) consists in taking the harmonic extension gft) of g<3) and then noting that
= —f-r-	(A.44)
We shall describe in what follows another technique which does not need a harmonic extension. It uses instead an interior differentiation
D„ on the cone (2.11) (see [B2 ] ). D,, is defined up to a factor by /*	Л*
demanding that it is the lowest order differential operator with the following three properties: (i) it is a lowering operator, - that is, it maps the space V? of homogeneous polynomials of the type (2.10) for f - 1,2,... into V? 1 ; (ii) it is an n-vector, in particular it has the same commutation relations as with the generators
of rotation (i.e. [ X ,D ] = S . T> - S jD -cf. (1.61);
(iii) it is an interior operator on the cone (2.11), - that is, for any polynomial (j) on IK n we have
ajjWi’11 f- j1-0	(A
Examples of interior operators on IK„ (which are, however, not
lowering operators) are given by the generators of rotation
У = s
(3.31) and dilatations
X=h--r+j9 (Э =	, fe, |-ri )
on the complex cone. There is no first order lowering interior
differentiation, since = 2^ (fO for J2 = 0). We shall show that
there is an unique up to a constant factor second order operator Э
satisfying (i) - (iii). We first study the uniqueness claim. The most
general second order operator satisfying (i) and (ii) is
+ &X^ + cJ/4 Д , (А- Э/+-+Э2) .
Applying (A.45) with f = 1 and f =	, we find
150
( a +	+ 2cA ) 2j^ * О
[ a + & (fit-i ) + 2 (hn") c ]	= о
hence, c = —b, a ~ °> 80 that
э, - Ихэ„ -IU)
Л \	/	2 9,u	(A. 47)
The operator so defind is indeed interior, since for every polynomial jf(j) one has the relation
^«Г37(3)] = зЧ^2^^	(A.48)
so that (A.45) is verified.
In what follows we shall choose the normalization constant
6 = <	(A.49)
With this choice the operators X..w ,
/“v
= V  Vn+( Гзл+ v	(A,50)
and Xo n = - i. X are the (mathematical) generators of the (real) Lie algebra of the conformal group S0o(n,2). That follows from the easily verifyable commutation relations
[	, x 1 » X S + Xuv	<A-51>
*•	» j у J	yxv /^V
With the operators (A.47) (A.49) we can replace Eq. (A.44) for the covolution product by
« I
4(v^C5') -	+	, 4*3)(3) •	(A<52)
We note that the vector operator D = (D,,...D„, ) satisfies the iden-1 Zn, tity
О	(A.53)
hence , we do not need the harmonic extension of either factor in the left -hand side of (A.52).
It is useful to have an explicit formula for the action of (-pD)'z on a homogeneous polynomial in 3 of degree k?v (where p is a fixed 2h-vector). We note that
(p'D)1' = TT [ (h+k-1-i. ) (т>Э) - l. рз Д ]	(A.54)
151
Obviously (A.54) is a polynomial in p , pj , p3 and A .
Accounting for the degrees of homogeneity in p and 3 we can write
the following general expression for (A.54):
(?D)V - E a(p, m,n) (ip1)n(p3)m'7p^)v"T,'n (-i^m	.
m, n	(,A.t>t>)
where the sum runs over nonnegative exponents (04n«miy-n) and the coefficients a(y, m,n) may also depend on к and h. Applying to both sides of (A.55) the operator pD we find the following recurrence relation for the a's:
a(v+1,m,n)= (m-n-n )(2 h 4 2Л -	- 4 - ПИ-n ) <a ( V , rn ,	) +
л <	(A-56)
/ii	л	xz	.	/	i	1 $ m < v
+ (л+к-и-2-»п*п)а(р, mln)+a(P,m-l,n)
k	4 ’	X ’	’	'	1 $ n $ min (v-m
which have to be supplemented by the boundary conditions
a (v+1 , m, о ) = (Ji+k-y-z-m )a(v,m,o) + Я(и,ш-1,о )	, 1 ± m £ v
d (v+1, m, m) - (h + k-v-2')[2a('p,ni1w-l) +a	, 1 < m S
a (v+f, п>Ли-") * (2m-y)(2h*2k-v - lm-3)a (v,m,P-tn)« a (>< m-1,y+1-m); £ +
2(h + k-v-2)a(p, 41,
a (v , о , о ) = ( h+ fe - v - <
a ( v, v , о ) = 1	(A.S7)
Since it is difficult to solve directly the resulting system,
we shall apply both sides of (A.55) to the simple homogeneous polynomial (-63 ) where b is some fixed 2h-vector. The right-hand side gives
To evalutate the left-hand side we use (A.13):
<pl»’(S3>b.[(₽i»4vJ,.,' ^7Г^>Г'р3,’">м(4)^л -
f()l+K-v-l)'n, П ( iJ - n - m ) ' (k- y- m i-n ) ! ( tn- n ) ’ n. !
152
Comparing (A.58) and (A.59) we obtain:
a (v, m, n ) =
v'. (А + k-V- 1 ) v-m
(v-m - n ) ! (m - n )! n !
(A.60)
It is easy to verify that the coefficients (A.60) satisfy the recurrence relations (A,56) and (A.57).
153
APPENDIX В.
The special cases h = 1 and h = | . Relation to the formalism of two by two matrices
t
В. 1 Splitting of the representations x of О (3,1) into elementary representa-
tions of SL(2, C)
T
It turns out that the elementary representations [I , c] of О (3,1)
f are reducible for I > 0 , when restricted to the proper Lorentz group SO (3,1).
The universal covering Spin (3,1) of SO (3,1) is isomorphic to the group
SL(2, C) of 2x2 complex unimodular matrices studied thoroughly by Gel' fand, et al. [G2] .
In order to exhibit the precise relation between the representations of f
О (3, 1) and of SL(2, C), we shall first recall the definition (and labelling) of
the elementary representations of SL(2, C) adopted in ref. [G2].
To each pair of complex numbers (n^, n^), whos e difference n^-n
2
is an integer, we make correspond a respresentation of SL(2, C), acting in
an appropriate space	n ) °^ infiniteiy smooth functions (p (z) of a com-
plex variable z = x^ + ix :
if g =кб/в6 ’	= 1 1

П -1 n -1	/ .
, .	,	. 1 ,___.2	_ { 5 z -P
(z) = (a-yz) (a-yz) (pl----------c-
l a - у z
(B.l)
(We have converted the right translations used in [G2] > into the left translations used throughout this paper. )
154
We shall show that each of the representations x = [1 >0, c] of О (3,1)
(see Sec. ДЛ) can be decomposed into two representations
<nl’ n2>
of the
type (В. 1):
[f,c] = (n , n )ф (n , n ) where n -n = 21 , n + n = - 2c .	(B. 2)
L &	L> L	L L>	L Lt
In order to prove this statement, we first notice that each symmetric traceless
tensor f	(1=1,2,...	) in two dimensions (u. . = 1, 2 )
has just two independent components, say,	and	^2 * ®'or each
f	(x, , x_) e C we define a pair of functions Ф (z)tD, , and
1	2 x	r	T+ (ni ’n2)
^(z)‘D(m,n.) by
^(Xi±ix2)=fi...n(Xi,X2)±ifi...i2(Xi,X2) .	(B.3)
Then, it is straightforward to verity that the transformations (2. 31 a, b) for f go into appropriate transformations of type (В. 1) for the (p1 s . For instance, dilatations correspond to matrices
/I |V2n \
/ |a|	0 \
a =(	-]/2 ) >	4a)
\ 0 ]a | J
so that
Tn n (a)<^+
П1П2 ±
-1- c
(z) = |a|	<Z’±(z/|a|),/c=-|(n1+n2) / ,
(B. 4b)
*
The notation (n , n ) used in this Appendix should not be confused with the symbol for the highest weight of an IR of SO(n) used in Sec. 2A.
155
while rotations (on angle 0 ) in the (1, 2) -plane are represented by
(B. 5a)
(B. 5b)
On the other hand the conformal inversion (2. 31c) goes into Rz = - 1/z
which is not an SL(2, C) transformation. It could be replaced by the proper
rotation RI , where I (x , x ) = (x , -x ) ; we have £	£ X £	X	£
g DT = ( I 1 TZ	I'/’x	(Z) = ---('“) •
Ш 2 ^-Ю/	(nl’n2> M2 *	(zz)1+c	± z
(B. 6)
The operator of space reflection T (I )
X L
in C X
mixes the two terms
in the direct sum decomposition (B. 2):
Tx(I2)</’± (Z)	= ?’:F(I2Z) ' X2Z = Z •	(B,7)
_ —
For 1=0 the representation [0 , c] is equivalent to the
representation (c , c) of SL(2, C) .
__1 s
B. 2 Vanishing of the projection operators | | for s > 1
We already observed that for h = 1 and any i > 0 the number of
independent components of f	is two. Therefore, we can have at
156
most two non-vanishing projection operators in the momentum space expansion
(5. 23) of the invariant two-point function. Actually, we shall show, that
(—.10	(—
for h = 1 , t > 0 the operators | | and ] j are 1-dimensional - £ g
projectors, while all other ] ] (with s > 1 ) vanish.
To see that, we notice that the variable w in (5.4) or (5. 21c) is
equal to -1 for h = 1 . Indeed, since w in (5.4) is a homogeneous function of and we can take (for non-collinear g.' s) = (1 , i) ,	= (1, -i)
so that
ш = 1----------2 = -1 	(B. 8)
|Pl+iP2l
Furthermore, we notice that
-1/2	s -1/2
C ' (-1) = (-1) c 4 (I) = 0 s	s
for s > 1 ;
(B. 9)
.—. ss therefore, according to (5. 21c) | |	(p) = 0 for s > 1 . Inserting in
(5. 21a) we find
n00(p)	= 1.
rf°(p)	= 2* 4^0(p)^0(p), t >1,
(B. 10)
ПП(Р) =	-1Ln(P) Ln(p) ПП (P) >
П* S(p) = 0 for S > 1 .
157
В. 3 The structure of exceptional representations for h = 1
First of all, we observe that the structure of the representations
changes for the pair x^
We shall show that for 1 > 1 the representa-
tions
x'*
are irreducible.
To see that, we notice that
C'~ = F.' for h = 1 , t > 0 . tv tv
(B.ll)
Indeed, the (sub) space
F^' can be defined as the kernels of the intertwining
operator G' (6. 13) . r	t v
On the other hand, it follows from (6. 13) and from
Section B. 2. that
G'^ e 0 for h = l , t > 1 ( v = 1, 2, . . . ) .	(B. 12)
This proves (B.ll).
It also shows that the subspace D. C c' + is t V t v
trivial:
D, = { 0} for h = l , t > 1 , tv	—
(B.13)
since
D. is the image of c' tv	tv
under the mapping
G' lv
On the other hand, the representations
x'*
are reducible (like in
the general case)	) being nontrivial invariant subspace of
c.'~(cH’ tv t v
This is not true, however, for the SL(2, C) representations (v, - v) and (тР , v) appearing in the direct sum decomposion
xot = C = [ p,° 1 = (p’ ‘p) ® (‘I/’ P) ’
(B. 14)
158
To see what happens we consider the special case f = 1 . In this case, the invariant subspace DQ C c'*	is defined as the set of vector functions
f (x) i C_, which satisfy the transversality condition
p. 01	1
} f (x) = 0 .	(B. 15)
И И
Using (В. 3), we see that (B. 15) is equivalent to
J
y— ([,>(z) + (p (z) = 0	(B. 16)
and, thus, mixes the two representations (1, -1) and (-1 , 1).
For h = l , t > 0 the first quartet diagram on Fig. 1 becomes
degenerate and can be replaced by the following simpler diagram of exact
(directed) sequences:
E ________s C~
1V	7 1 V
(^J)P
0 ________
Fig. 3
The quartet diagram for the Lorentz group (h = l) and t > 0
Finally we note that although the diagram on Fig.3 has a similar
structure as the diagram on Fig. 4 (Sec. 3. 3 of Chap. HI ) of ref. |G2],
it has a different content, since each representation
-iv
is again a sum
SL(2, C) , and to each arrow in our picture picture of Gel' fand, et al. For instance,
the homomorphism ( J . 3)p :	—C" is split as follows in SL(2, C) variables;
159
/J
D(-2£-l>, p)
9
D
(p, -21 -v)
-C^
(B. 17)
B. 4 Elementary representations of SO (2, 1). The analytic discrete series
The elementary representations of SO (2,1) are labelled by a single
number c , since M is trivial in this case. They correspond to the representation [ £ = 0 , s = - 2c] of the two-fold covering group SL(2,ffi_)
in the notation of ref. [G2] . (The representations of SL(2, Ж) with e = 1
also studied in [ВИ ,G2] are double valued representation of SO (2, 1 ). )
If we s et again
g =(“f ) e SL(2, Ж) y5
(a , (3 , у ,5 - real) then the
elementary representation T is given by its action on functions f(x) of a
single real variable x according to
I T (g) f] (x) = (a - у x) 1 2C f (д^уР ) 	(B. 18)
They are irreducible except for the half integer points
± с = I + h -l=f-|	1=1,2,...	(B. 19)
The negative type representations (with c = | -1) contain the finite dimensional invariant subspaces E^ of the polynomials of degree 21 -2 . The vectors of E^ are annihilated by the intertwining differential operator ?> -1
V , which maps the space	onto the invariant subspace D of
x	2 ~1	1
C of functions f(x) , satisfying
* ~Z
160
оо
J f(x) xk dx = О к = 0 , .. . , U -2 .	(B. 20)
-00
The representation T^ 2 of 0^(2 ,1) is irreducible on . However, it splits into two irreducible subrepresentations Tg+ and Tg when restricted to the identity component G = SO (2,1) of the three dimensional Lorentz group. The space Dg then splits into two invariant subspaces Dg and Dg which consists of boundary values of analytic functions in the upper and lower half planes z - x+ iy, respectively. The IR' s Tg on D* form the (analytic) discrete series of unitary representations of G . Since they are analogous to each other we shall restrict ourselves to a brief discussion of T+ f
In order to make more transparent the interrelation between the K-induced and NAM-induced pictures of the discrete series, considered in Sec. 7 , we shall outline the corresponding results in the simple case at hand.
It is convenient to use the SL(2, HL) notation as in (B. 18); then
G can be defined as the factor group SL(2, ffiL)/^^ •
Each factor in the Iwasawa decomposition (1. 12) is a one-parameter subgroup of G = SL(2, HL)
8 cos —
e
- sin—
the decomposition (1.12) assumes the form
(B. 21)
161
~ 1
g = n a к = —	.
B x у 9 уу О
9
COS —
. 9
-sin —
 ,	2
9 sin —
9 cos—
X
1
9	9	9	9
у COS — - X Sin— У Sin— + X COS-
~г
We consider the
. 9 - sin—
9 cos —
(В. 22)
space Xf
of square integrable functions on G ,
У
which satisfy the covariance condition
ue e
(B. 23)
These functions are defined by their values on the homogeneous space
G/K ~ NA , which is i somorphic to the upper half-plane C = {z =x+iy ;
The representation	acts on such function as a left (quasi)
regular representation [ see (7.7)] . Introducing the functions ф(х)
(7. 12) on the homogeneous space
C
ф(х) = 0(z) =^(^х ау) >
z = x + iy
(B. 24)
one derives the following transformation law for the <p's :

, 8 z -p .	a
<p(-----^) for g =
a -y z	у
(B. 25)
5
Eq. (В. 24) maps <<
onto the Hilbert space
of functions ф with
invariant scalar product
162
(фг , Ф2) = f Фг (х) Ф2(х)
(В. 26)
Using (В. 25) one easily finds the infinitesimal generators of the
repres entation

The second order Casimir operator is:
& = у 2 Д -2if yV (Д = V2 + V2) . 2	x	x у
(В. 27)
Its eigenvalues are I (I -1) , 1=1,2,.
the corresponding eigen-
functions of the canonical basis have the form
, n 1	• k
ф (z) = A u (-------------r) ( ——- )
к	fk ' z+1 ' z+i
u = —— , к = 0, 1, 2,... (В. 28) x +(y+l)
They are also eigenfunctions of the
compact generator
X12 = aWU = -{l(i+^-y2)Vx+xyVy + ify}
(B. 29)
(corresponding to eigenvalues -i(f+k) ), the vectors
are related
to the analytic functions f(z) (in ) with boundary values f(x) (in ) by
!
Ф (z) = у f(z)
(B.30)
In particular, the canonical basis functions (B. 28) go into
Ik
A i . f+k
z +1
/а
Ik
к
, +k
(z+i)
(B.31)
163
The operator
-I
L : 0(z) —f(x) = lim у 0(x+iy)
У ^0
intertwines the representations U^+ and T* . Its inverse is given by
-1	i
the Cauchy integral formual: L : f(x) — ф (x) = у f (z) where
f(z) = “ J	( ]rnz=y>0) .	(B. 32)
ZW 1 x' - z
It can also be related
to the Green function
ul 2^-1 (Z1’ Z2> = Z7 (k2=i
1
1	11	1’u i'Z2’Z1i
ku	12
(B.33)
- z
(Z1-£1> Cz2-z2} ।	i 2
lZl-*2l
4yiyz
(x1 -*2)2+(^2)2
for the eigenvalue equation of the Casimir operator
Гу12д1
2if у V - i. (t
1 *1
-1)] $ + (z1, z2) = y^o (xx -x2) 5
(yl-y2)
(B.34)
To this end we write down the analogue of (7. 48), (7. 49) and take into
account Eq. (B. 20) .
164
_ _	2/-2	2	f	7	1
(Ф, > Ф ,) =	(z) f,(z) у d z = Jdx dx d z —-------------------—-------
1 Z •' 1 Z	J 1 Z (x_ -z)(x_ -z
1 r -	_	1 2	'1	4
= Jdxi W x12) l°g --------------------------“г f2(x2>
(X12+1O)
In deriving (B. 35) we used analytic regularization of the integral in
The scalar product (B. 26) in is related to the invariant +	i
hermitian form on D as follows: Let ф. (z) = у f. (z), i = 1, 2 ;
then inserting (B. 32) into (B. 26), we obtain: 2£ -2 У
(В. 35)
у and
the orthogonality of f. to polynomials of degree smaller than 2Z -1 (see (B. 20) ) .
165
APPENDIX С.
Positivity of the invariant scalar product in the subspace D of C1^ C.l The problem. Asymptotic expansion of f(p, for p — 0
The positivity of the expression (6. 13) for <P) *s n°t sufficient to conclude that the scalar product (f,	is positive
semidefinite in c'~ , since the Fourier transforms f (p) of elements of are in general singular for p -> 0 and the precise definition of the scalar product may require a regularization, which would destroy its positivity. The objective of this Appendix is to prove that this does not happen. The analysis is based on a study of the asymptotic behavior of f(p) for p -» 0. Because of the isomorphism between the subspace D of C*. and the factor space C /F1. (see (6.19" and the equivalence tv 1 v	tv tv
of the corresponding representations of G, this will be sufficient for establishing the unitarity of the representation in D^
The functions fe are not in general integrable for Re c < 0 (X = [*, c]) and their Fourier transform does not exist in the sense. It can always be defined, however, away from the origin (p = 0) in momentum space and is a fast decreasing function of p for p-> co. (The last statement follows from the infinite differentiability of the functions f(x) e C . ) On
X the other hand, the asymptotic behavior of f(p) for p-> 0 is determined from the asymptotic form (2. 29) of f (x) for x -»oo. If c < 0 , then О Q f(p,^ ) «	(p )	(C.l)
166
where P^ (p,	) is a polynomial of p and , homogeneous of degree t
in . That follows from a general Paley-Wiener type theorem. In order to illustrate this property we shall evaluate the Fourier transform
of functions f t C of the form X
f(x, ) = (2тг)
2	Я1+с+/'
~~2	21
a +(x-bl/
(C.2)
, x } ),
where h^ is a homogeneous polynomial of degree t of its (h+ll(2h+D
2
arguments { 'J- , x} = { 7, , x r(x)^ ; xp xaj. } ,	€ K^(cf. (6. 2b)),
a > 0, and t' > t. We notice that the linear span	of functions of
_	.	X
t
the form (C.2) is О (2h + 1,1) - invariant and dense in . The Fourier transform of f can be defined through analytic continuation in c from the right half plane (Re c > 0). Using Eqs. (3. 915. 5) and (6. 565. 4) of ref. [ G?] , we obtain
,	/	?	\h+c+f'
f(p,Z )=hz({^,iV })(2тт)'Й f —---------A. e'lpX dx
'	P	\a +(x-b) /
,h+c+f , , , .„mi (1-h -ipb “ rh	I, ,(Jp|r)dr
= 2 hpf^.rV }),|p| e Г 777^h^7	h‘1
r	0 (a +r )
= 2h/({ ,iVp})e-ipb пыТ+Г)	Кс+Г(а,Р,)-	{C’3)
The right-hand side of (C. 3) decreases exponentially for |p[
oo as
167
anticipated. For non-integer c the small p asymptotic behavior of f(p, J, )
is given by
~	h/({^’iVp})
P’^ p~0 Г(Ь+с+Г)
( 2xc+/ ' Г(с+Г ) + O(p) + r(-c-?)fej (HO(p
(C. 4)
(Otp) denotes as usual a quantity of the order of magnitude of p ).
For negative (non-integer )c the second term is dominant. If c+f'is a non-negative integer, then we have
/ i \/	r .	/ 2	4
(C. 5al
i
Finally, if с + I is a negative integer, we find
h ({^V })	/n2\C+r
fTh+c+T) (|c+r|-D' (4- )	(1+O(p)).	(C. 5b)
p->0	X '
C.2 Existence of a non trivial positive semi-definite hermitian form
(f., g’+ f ) on C'"
1 l v 2 ------- f v
Let now у - xy - [f + v, 1 - i - h] . Consider the quadratic form
on C.
1 v
(f, G’^f)= lim f <f(p), G'* (p) f(p)>	.	(C.6)
El° p2>e	(2я)2Ь
168
If we allow a priori the value + co, then the limit in the right-hand side always exists, since the integrand is non-negative. We remark that the set of f 1 s for which this limit is finite forms a linear manifold. This follows from Schwartz's inequality for the integrand, which implies that
<'l*'r G'A »l*y> ^2(<fro'L <1> +	f2>) 
The form (C. 6) vanishes for f e F^ (Q C*^ ), since according to the results of Sec.	the integrand in (C. 6) vanishes for such f's.
Proposition C, 1. The limit (C. 6) exists for every ft c'^, . It gives rise to a positive definite scalar product on the factor space	•
Proof. Since (f, G* f) = 0 for f e F' , it would be sufficient ------	Lv	Lv
to prove the existence of a finite limit (C. 6) for a suitable representative in each coset in c'” /F'. . To each such coset we shall assign a
L v L v
representative f satisfying the p-space equation
(pD	= 0	(C. 7)
(where D is given by (A.47) (A.49). It is easily seen, that a representative with this property does exist and is determined uniquely by (C. 7). Indeed the Fourier transform of very element f e C* can be
L v
written in the form f = f^ + f^, where
169
/ t + i/s	Ш t + v s
fl(p) = s?0 П (P> f(P’- f2(p’ =s£+l П (p)f(p).	(C. 8)
It follows from (5. 21) (5.14) and (6. 6) that f e F^ and f2 satisfies (C. 7).
On the other hand (5. 21) also implies that if fe Fj satisfies (C. 7) then f = 0.
Our next objective will be to replace the finite dimensional scalar product < f(p), G’ (p) f(p)> by a form suitable for exploiting (C. 7'.
Lemma C.2. Define the M(=SO(2h)) average of f(p) x f(p) by
F(p;|4 ) =
f f(mp, m ) f (mp, m j.) d m,
(C. 9)
where dm is the normalized Haar measure on then the angular
integration in the right hand side of (C. 6) can be expressed in the form
JdlL <f(p), G1^ (p)f(p)> = <г2М tr[G^ (p)F(p)]	(C.lOa)
where
(Г
v
is the surface of the unit sphere, given by (A. 21), and
tr[G^(p) F(p) ] =[(f+H!]'2 G^ (p;9	9^) F(p;^.,J )	(C.lOb)
, 2 is a function of p only.
The lemma is a straightforward consequence of the rotation
invariance of the integral over the unit sphere in momentum space and of
the covariance property G' (mp; mt, , m£,_) = G' (p;£. , £. ) of the * P	1	i V X
170
intertwining kernel.
Eq. (C.l) implies that the M-invariant function F can be written
in the form
P(P;3J) = 4(?*n -33 .	?1ГТ = ?г35~ (РЗХРЗ) (с.И)
where ф is a polynomial in all three arguments, homogeneous of degree with respect to the first two of them. Furthermore according to
(C.7) , it satisfies the equation
(Уо/Ф (фЧТ ,J3 ; рг) = О
(С. 12)
where('pD)1’ is given by (A. 55) , (A. 60) .
We shall show that any polynomial solution of (С.1.?) is bounded by C(p ) for p -> 0. Indeed every polynomial ф , satisfying, the above homogeneity conditions, can be written in the form
Ф(рг53-(РЗ)(РЗ> >33 > Рг) =<33 >4 Ог/| • 35 ; рг) +
+ Ф/р'пфзз^р1) >
(С. 13)
where
171
I (?ln ’ 55 ; P*)| * c Cp2/+1
(C. 14)
and Ф^ is a homogeneous polynomial degree f of the first two arguments and a polynomial p of overall degree not exceeding 2£ .
Since Eq. (C.12) is satisfied identically with respect to p, it has to
be satisfied by each of the two terms in the right hand side of (C.13).
Lemma C.3. The only solution of the equation
(P<D)V{ (jj )v<fy	p2)} - о
(C.15)
satisfying the above conditions is
Ф{, = 0.
If Lemma C.3 is proven,
then the validity of Proposition С.I
will follow from the behavior of
G
for small p and
from (C.IO), (C.ll), (C.13), and (C.14).
Proof of Lemma C.3.
Assume that there is a non-zero polynomial 4^
(of degree I ) satisfying (C.15). Let s be the maximal non-negative
integer for which the coefficient (p1) in the expansion
(зз )у<^(р1п>зз >рг) = 2 %(p2> (Vn) (33 k
does not vanish. Here a^(p2) is a polynomial of degree 1~\г. and
6 < /	. Inserting (C.16) in (C.12) and taking into account the equations:
Р^З^РзЗ'З ) = 0 . (Р3з>1(зз)3= 7~, (P3)L (33^	= 0
(P-^)J - H)* Нтчт (P1)'^)* (Pln^21
J	Q -11) 
• □ (c-17) э/зз )J = -у (?1ГГ/ 1 (35 ? (рз )г , Я
л"(/П);(35Л-Л1’”')к к-(”’’
172
we obtain (pl) )V (jj )v cj> (p1 П t 3 j ; p1 ) =
= £0 £’k(p1) k ' ^'V~K'> ' б?з)и£ Cf-V (?liT)k 7зз/ к+' •
*7v НГ^у!	___________________i
m nj (w-m-n)’n.! (i-n-j ) ! (m-L+ n +j ) ! (к-m-t+M+-j)1 (^-к + i-j )!
. »s(T*> („-(35/-‘Ite/c'ggW bM>W53>‘--X'r‘>’.--о
For the sum in the first term we have (see Appendix E below, Eq. (E.10)
22 .£21.W.dj.•	„ (/, + rs-i),	(C.19)
m (y~m).(s-Tn)'rn!
Thus the coefficient of ds(pl)(f>2n' )s is not zero, which implies as (-p2- ) = о , contrary to our assumption.
173
PART TWO CONFORMAL PARTIAL WAVE ANALYSIS
SYNOPSIS
We study the tensor product decomposition of two unitary irreducible representations of the Euclidean conformal group, O+1, 1 )	. Conditions are found under which only principal
series representations contribute to the decomposition. Clebsch Gordan kernels are evaluated which satisfy a completeness and orthogonality relation. Convenient normalization conventions are discussed for Clebsch Gordan kernels which effect the de
composition of the tensor product of two class I ("scalar") representations, and identities are found which are satisfied by their analytical continuation at partially equivalent integer points.
The results are applied to write down conformal partial wave expansions for 1-particle irreducible n-point Green functions
in conformal invariant quantum field theory.
Special cases of these expansions are used to derive an expansion of the type
(0)
Here	are labels for infinite dimensional symmetric
tensor representations of the Euclidean conformal group, X4=[-C,-c4]	, the constants C ( jjq ) are real, and	and
X
have the properties of vacuum expectation values of field
products. The starting point is an infinite set of coupled
non-linear integral equations for Euclidean Green functions
in 2h space-time dimensions of the type written some 15 years ago by Fradkin and Symanzik. The Green functions of the corresponding Gell-Mann - Low limit theory are expanded in conformal partial waves. The dynamical equations imply the exis
174
tence of poles and factorization of residues in the partial waves as functions of the representation parameters. In proving the validity of (0) we use some differential relations between partially equivalent exeptional representations of 0+(2h+l,l), established in Part One.
175
IV. CLEBSCH-GORDAN EXPANSION OF THE TENSOR PRODUCT OF TWO UNITARY PRINCIPAL OR SUPPLEMENTARY SERIES REPRESENTATIONS
9. The Kronecker product of two elementary representations as an induced representation on G/MA *)
The Kronecker product 'X®’XZ °? two elementary representations
X, = fA , c, J and	® acts in the space of infinitely differentiable functions	°n G*G with values in the vector space ®цА
Considered as functions of the individual variables they share the covariance properties of functions in the elementary representation spaces resp. , cp. Sec. 2.B. In particular,
for p, in MAN.	(9.1)
The action of G on such functions is given by
= (9,2a)
Because of covariance property (9.1), functions f are uniquely determined by their restriction f (x, хг) “	' (X,+ A)
Transformation law (9.2a) reads then
(T(g)f Ж	^Э’Чз'Ч) 	(9.2b)
The cocycles -p(x,cj)e MAN are defined by (1.2/a), viz. g"’nx = n^.,x p(x,j)"'1 • If )(, > belong to the unitary principal series then also their Kronecker product is unitary by virtue of the inherited G-invariant scalar product
О,) =	< f4 (х,хг), ^(x.xp >	(9.3)
<,> is the M-invariant scalar product on	. All this
parallels the discussion of elementary representations in the "noncompact picture" in Sec. 2.D.
We will now demonstrate that the Kronecker product X, ® Xz may also be regarded as an induced representation on G/MA.
We define a map Q of C%®	into a space of infinitely
differentiable functions on G with values in 1НЧ®1/А and covariance property
F(jma)= L(rna),fr(j) wM L(ma') = |a|1	1A (rn) ] .	(9.4)
> In Secs. 9 and 10 we use a different parametrization of the subgroup N. What was formerly called nx is now nex , в = reflection of the 2h-th axis. With the new notation we have «x = wB^w (while with the notation of part one we had •яп^к. = n-x , vS^r = nex).
176
of | ; viz. j (
for та e MA . The map Q is defined by
(o<) > = 4	(?. 5)
w is the Weyl-inversion vVx = 0x/X*; it belongs to the identity component of G as we know. We recall that wmW =0m0€M for tnc M , and v/aK'H= a1 for a. in A. Condition (q. Ц.) is then an immediate consequence of the definition (9• 5) and^covariance property (9.1 )
ma
- [Л™)”®(g,3w ) . x
Since jD(ma)=|af СЭ(|и)Ьу definition (2.28)
we obtain indeed (9. 4).
The space ®$>y) of functions with covariance property (<?. 4) carries a representation of G ,
(Г(д)Н(д'> =	.	(9.6)
It is immediate from the definition (9. 5) of Q.	that
• be. Q. has the intertwining property.
We will show that functions -f in &®£у are uniquely determined by	2
Lemma 9.1: The group G acts transitively on noncoinciding pairs (x^.x^) j X, + Xj.	of points in Euclidean space, with
subgroup of stability isomorphic to MA.
To prove this crucial lemma, consider first the special pair (x = О , X2= *» )	’ • Its subgroup of stability in G is MA.
On the other hand
=(X,,X1) for Х = Х,,у«Х^Х,.
(?. 7a)
Indeed
w<j (O.oe) ” hx wnw3
4 ) = nx (О, у ) « (,
nx Wnw3 («о-0) (X,,xt) .
So every point (x, ,xt) with x,^Xj may be reached from (0,i») by applying a suitable conformal transformation.
This proves transitivity . We remark that also nwy' "x'	> * (X4.X1>	y'^ *'=
Ml
The connection is given by the following corollary of identities (1.27a) and (1.27f):
n.x	= ky, A’’= ^xi^xe , anol y = y+x , x = w(wx-wy')
Explicitly h2 = mzax with [az| - z1, mz = - '"j'zlS , f(z)'Mj>-	+2х^х„ /г1 
Now we want to reconstruct from Q-f . It suffices to determine	(nx , nx ) for x, 4. xx . Pairs
which cannot be written in the form (ixx p, , n^p*1) with p,, pz in MAN and x, =Axx form a lower dimensional submanifold of <SxG ; and	was assumed infinitely differentiable. Consider then
(Qf	, их nM;( w)
One has nWjW. un^ ,	» n^p'’	Pwy e P and given
explicitly by (1.27e). Covariance property (9.1) gives then
^QP^xV - [^®£*4Pwy)]f(W	\=x , x^ ун-х .
J	J	к 7
The matrix in the brackets has an inverse £ ® J)X1(pwy)”'	, and so f-
is uniquely determined by Qf .In other words, we can define an inverse Q* to Q. by
ffoxj = ('Q*f)(Kx,’"x2? = Г1® (pWy)”]r ("x"^)
If one writes	then ^(-p^)) by definition.
There is an equivalent formula which is based on (9.7b). It is derived from (9.9a) by exploiting (9.8):
(Qf	- L(h)(Qf)(Xxn^
by covariance property (9Л). Since L (k) = Dx<(к)® 3>*1 (kw) one has ь(1,)[1.®^(рИу)]-Эх<Ск)®2/г(^Ц)	• By (9.8), /Л-	, s.
1>X1 (AwhW3 ) • 2)	. Altogether
(Qf ) (пиу'%<) - [®X,(k)®3»X4Auy)]f (x,xx)
with у '= X;l , x'= i^(n/x1-^xl) and h - h^,	= kx, hux> ; x »	, у ~ xx- x( .
We consider now the special case that L(ma) is unitary. This is
178
true if^-c, is pure imaginary. This includes the case when %* and belong to the unitary principal series, i.e. c, and сгаге pure imaginary. A G-invariant scalar product may then be defined by
(f^r)»	< p, ("»)Л
Ji	(Я .10)
N К
We show that this agrees with the scalar product (9. J) for the Kronecker product X ® Xx an case that x, and XL belong to the unitary principal series, viz.
(9.Ц)
where the l.h.s. is defined by (9.10) and the r.h.s. by (9. $). One speaks of a "unitary induced representation on G/MA" if the inducing representation L(ma) is unitary and the scalar product is defined by (9.10). [ Note that dndn is an invariant measure on G/MA, and functions F are specified by llwir values on t/N = G/MA.J
Theorem 9.2. The Kronecker product of two representations
X< and °t the unitary principal series is unitarily equivalent to a unitarily induced representation on G/MA with inducing representation L of MA given by (9. 4).
Proof: The equivalence is effected by the map Q • It only remains to prove (9-il).
&уг'х'Пу”,У> Ufy.'YS»  We write
We have, using (Mj), h и w = и w и = и и n a I.r(x) j	7	"	у r'f" A »v A
Also olx = (x*)2**dx «idKl^dz j where px=mxaxn	as usual.
If all this is inserted and use is made of the covariance condition (9«l) equality (9.11) is readily established.
179
We note also that the inducing representation L(ma) of MA
is not irreducible unless О or * О the trivial representation of M. As a first step towards the decomposition of the induced representation one can decompose L into irreducibles.
7*	1
We write U1» in place of V 1 if we want to consider this vector space as carrier of the representation (•’') = » (дтв )	of M. Let us decompose
= 21	, sum over j€M such that	•
4
Define Clebsch Gordan maps	and-their adjoints, the
M-invariant imbeddings C (4, А.» |
= V"**®	; C(/,Z1;p* =
Then F.	= id; and
d
j-	- ^4<m) C(U^,i ).	(9.12a)
We may then decompose
; rJ(p=
j	(9.12b)
and covariance condition ($. 4) becomes, for mo. € MA
F"j (gwa) = L? (ma ) ’	(3 )
with	J (*«a) = |al C1'C< DJ (m),	(9./3)
It will be convenient to introduce the map
Q. = c«4;j )»Q
J	(9.14)
and Q*Q*o	j )*	, cp. C?. 9b). If
FJ = Qj f	then j = Z. Gj	(9.15)
since Q* is the inverse of Q and the CG-map	is unitary.
180
The sum runs over UIR's j of M such that j c 7,®
If %, and the adjoint of	are in the unitary principal series then	is the operator (2j .
181
10. Construction of the Clebsch Gordan expansion
IQA Clebsch Gordan kernels
We seek G-invariant maps
V ' Х.ЙХ'г1—* X	C°’ 1>
from the Kronecker product of two elementary representations to another elementary representation X • Since all concerned are function spaces, the maps will be effected by integral kernels. If	> Xx=I4.>cil and
X*	then XfgJXx. is made up of functions with
values in V 1®V 1 while % consists of functions with values in the finite dimensional representation space of M. In physical applications one works in the "noncompact picture", i.e. with functions on x-space. The kernels are then maps
V(x, X, «хХг. • xx > '	V<	(Ю. г )
with covariance property
(p3> V (x/x,<Xti xs' X ) I®*’ (p,) ® 1>Xx <PP I = У (XX xiXt i хзЮ for x/.	» Pi = p	, g in G.
As is usual in representation theory, we start by defining the map V on infinitely differentiable functions. We are looking for a complete set V Is of maps (IO. 1 ). The meaning of the
labels j,s will emerge later.
n л
We make use of the result of the
. , « o*
(e the induced representation
preceding subsection. ZL+ cr=c4-cl on G/MA which is induced
by
the representation L-* of MA and consists of functions
on G with covariance property(9.
3).
Let
(io. 3 )
a complete set
(labelled by s) of G-invariant maps
„ "J from
° J
( i 1,1
tis = ^-X
to the elementary representation X .
182
Then a complete set of maps (40- 1) is given by
Vjs = t‘S »Qj
Qj r-.jr t^s X,®*,. >-* г? —» X
(40. У)
since ¥ ® ¥ = £ Q ZJ<r by (.10.1).
1	1 j J	-s
So we have to find the maps -fc' first. They are inter
twining (i.e. G-invariant) maps between two induced representations. As such they can be found by a standard procedure due to Bruhat.
L |0*
Functions in й admit an integral representation
° $ olmola. L'(ma) MA
This makes covariance property (9.2J) manifest for arbitrary F .
Of course F is not uniquely determined by F .
The map t»* will be implemented by an integral kernel
> 3% e 6 >viz*
F <зг>•	(w,6)
G
It will be required that this be independent of the choice of F provided it satisfies t/OJS)- In particular let K. in MA. Then F and f'(j)= L(A)F(j£) determine the same F • Therefore, we must have	) F (jx) «
s	L(<) F (j') .
Since F is arbitrary this requires
> ЗаА *	)=	for 4. in MA,
(10 7^
Conversely, covariance property (40.,..4a) makes it possible to rewrite the r.h.s. of (10. 6 ) in terms of F itself. This will be exploited later on, cp. Eq. (40.29) below. For now we continue with the determination of kernels (J	• We
have to meet two more requirements beyond (40.7a).
i) The map must be G-invariant.
183
ii) P ,F must belong to for F in E > so it must
have covariance property (£Рг)(^р)«1)\р)*,(4*1р) (g) for p in MAN, or км'л^(1М)
Since F is arbitrary we deduce the covariance requirement
(p)	for pc MAN.	(.YQ.Zb)
Condition i), viz. G-invariance of the map t, means that
=	) F (g, )	The l.h.s. is
(+Pr)(g’g1) by transformation law (2.15) for elementary representations. Concerning- the r.h.s. we note that
“	41) “ L? (ma)F (j дгтл). Equality of both sides reads then
(3 3- •d»>	’ Jda»^ 1*<ма)	э***)
The last equality is based on G-invariance of the Haar measure
ot^i, • Since F is arbitrary
t >3<3x)’(3-',3x) for	in g . (yo.zc)
Our problem is to find the most general solution of Eqs. fJOJa,b,c).
The most general solution of (.70,7.0 ) is
js	JS
* Qh-3x) =	>	WO. 8)
where V are matrix-valued generalized functions of one dr
variable g in G. Covariance conditions C70,7;b,c) become in terms of -t*
for c H = MA J pe p = MAN (ЧО.Э) dt о
Here and in the following we abbreviate the groups MA = H, MAN = P. We have to find the most general solution of covariance condition (70.3).
Let us define an action of the group HxP on the
184
manifold G by
(Л,-р)3 =r	{1010)
This satisfies the group's composition law (k.,,
We may therefore decompose G into orbits on which H*P acts transitively.
We show that there are three orbits P, wP , and the remainder G' of G. G' is the only open orbit, the others have lower dimension.
It is clear that P andwP are orbits; for they are invariant (e.g. UwPp"’i w k^P - w P since kw - 'e Нс P ) and right multiplication with P alone acts transitively already.
r
It remains to be shown that hxp acts transitively on G • G' is union of cosets nx P with X t O, oa	. Right multi-
plication with 7s acts transitively on individual cosets. Left multiplication with H gives к P = P . H = MA acts transitively on the pointed x-space IR.2*'4 (oj. Therefore kx is arbitrary nonzero for any given X + 0	. This proves transi-
tivity.
Corresponding with the three orbits there may be three classes of invariant kernels with support concentrated on P , w P and the closure G' = G of G' respectively. We shall want to extend the map V to measurable functions later on (when we deal with Kronecker product of principal series representations). Kernels concentrated on P or wP cannot be used then and it suffices to know the kernel	for in G'.
*They can however be found by the Bruhat method also, cp.£Wil.Alternatively one can use a physicists method which works in practice and which will be implicit in our later arguments. In the square integrable case (principal series) the singular functions on G' specify invariant distributions defined by an ordinary integral. These distributions are mesomorphic in the continuous parameters с, Сц c^, and the invariant kernels with support on
"P or wP appear as residues at the poles in these parameters (if the appropriate normalization is chosen).
185
We determine the little group of G' in Hxp . We choose a standard point n . , x - (0,1)	. Let the "rotation group"
UcM consist of u in M such that их - x .	(10.11)	
Certainly кп£-р*л=п_*	is true for (к, p ) * (u,u) ; ueU Conversely hn_£p'*P » - kx	is equal to ri_*'P « -x only if UU , and u ri -	* ri •	only if -j/- e Therefore	« pu » tx.	if k = u • In conclusion, the little group of n G G'	consists of		pairs
(u , u ) « И x "P	-u-i M	u e U Because of transitivity of HxP on G', every g in G'	(and	(10.12) there-
fore almost every in G) may be written in the form £ =	with k in H , p in "P . Covariance condition (10.9) says that fAn_.p'’) = 3) (p)^ LJ(k) ' with P -	- ) For consistency,	must be U- -invariant		do.1?) (10.14a)
P =3>*(и)Р LJ (и)"’ в	DJ(U)''	for u in U .	(10.14b)
Conversely, given iJS which satisfies (10.14b), a kernel -t* (g ) is defined by (10.14a) which in turn gives rise to an intertwining map	in manner explained above.
We classify matrices P ; they are U -invariant maps tP-»1P . Let us decompose UIR’s j and I of M into irreducible representations s e. U of U ,
« Z IXs	£. Vt‘
sc Cl ’	s«u	(10.15)
jcz	scj
Consider the projection operators ъ (l,s} and their adjoints, viz. imbeddings T*((s) ,
TT(/s) •.	,• тг (Is}*,	.	(10.16a)
186
Every U. -invariant map from to is a linear combination of matrices
* i s	л
t - TC (<S ) TC ijs ) j S€ U . set , Л j ,	CfO.fgb)
This solves the classification problem since the Clebsch-Gordan maps	are determined by 1	. The labels j, s run through
j e M , S e 1Л and such that c Z,®	, s c j and -see .
The symbol C is to be read as "is contained in...".
It remains to derive explicit formulae. We apply t</. j S	* 15
(TOJ'/a) to express the kernels -fc' in terms of f1 . * #
Given (> ' we have to find h in H and p in P such that p'’	. Every g in G1 таУ be wi'itten in the
form q = пито.	with n e W etc. The factor trio. is readily
°	-1	-1.
absorbed into p .It suffices therefore to consider g « nx d
We seek и in H, p in P such that
"x’’"wj .	SJo.iT^)
The l.h.s. is n|4 kp'( . By Uniqueness of the Bruhat nx •
decomposition we must have hx = x and kp ’ = nw^ . The solution of QO.ITa.) is thus
(Ihp)’ (H»),nw h(s)) with hUkH such that "A(x)x=x .	(^ОЛ/Ъ)
With that (h'’hv,1 ) =	( A n • ’ -p'1) -	* l-4h) whence
s -lJS(x)	{10.18)
independent of . By definition (10.18), /«	U r-tr 6	A IS f .1	5 Is A ?
P O^nx) = Ul	DJ(m) ;r(x)sV C1O\I2’)
is
Let us now go back to Eq. (ЛО.ба) for t .We insert
(ЗО.&) and make a change in the variable of integration.
187
This gives
Corresponding to the Bruhat decomposition пи »na. the Haar measure factorizes as we know, ctjx “ dmUnctn dn . So (|lsp) (3, ) =
cinofn olmota. (a'm n'* n
by covariance (70, 10). The integration over MA can now be performed with (JO, 5), whence
(*’’И(зз •	5°*"
We write и = и wu. ( n = и , oin = (иЪ o(u toln . c(i< and insert
expression (70.15) for to obtain
(t'Sr)(nWJnz) = ^(vT24duo(x V (x) F	),
We simplify the argument of F by using identity (9. 8) to reexpress nznWir . This gives n лг.ик»-мх -• nWj Mw(z+U) h"x = Hwy + wtz+o) "z' ЧЛ with z\(v as indicated below. H -covariance (S«13) of F gives then finally
(t'5 f) ( hw пг ) » j (и1) olu dx (*> L F (hwj + v(z+u) Иг'+Ах ) «J
with z w (wz - w(z+u) ) ,	= bo+i f>Wir .	(,10.20a)
This formula will be saved for later use. In the special case
Wsj • z - О	we obtain instead
(Р*р)(е) « (u-r^dvc/x V’(x) Р(п^ипх ) r
(70.20b)
Eq. 00.20b) supplies us also with a formula for V •**	.
Let F • Qj I for in	Using the explicit
formula ^9. 9c) for Qj	we obtain
for Vis|, tiSQj^
(V‘V Xе’ =	)®Ли>-)]Р(х,хг) (10 21)
188
with 'Af'H x,, xx functions of x^y' as given in (9. 9c).
We reinsert the definition {10.18) of (x) and use the
M-invariance (3‘12a) of the Clebsch-Sordan map	) to write
P(x')C«Aij )[®x’ (1>)®эХг(кн>1) bx>x(Mx')) Vc(<£ ^)[з>х,(Ь1х’)’^)®1>Хг(^*')'Чиу)]
with -f, (x')s wb(x') W ’ ,kas before.	(J0.2.1&)
We split & (x’> - ft (x’> m(x' ) with o.(x ) £ A and m(x)e*4 .Then
|o.(x ) | « | x’t ftnci n»(x>x=x/|xl for our standard X « ( 0 i1 ) .
Similarly h * aWy> ) h = am . By the definition (5.19c) of iv-v	,	-
Q =	= a*. aWXi •_ m > mx, m . By definition,® (мл) = |«l cDc(m) etc.
We insert this in (70,1/e). We have I« (x‘>l - t x'l - |	- wxjГ = ix,iixtUx1-xiul'1
and |a|s	a |xxt* J*,-**!'*'	, and I - I xxГ1 . So
r.h.s. of (102.1a) = 1x^4 Ci * IxJ 11 C lx,-Xxl 1 * C •	07 0.21b)
• з/(т(х')) PSC«Kij (xVV)®l/‘(wm(x'r,W'"WXi)]
with x»w(wx,-Wxx) as before, and m» mx, mwx .
We abbreviate
•t j a t C i j ) • * C^4 )*(j*) C (itliij \	(.10.Z1 c)
Matrices t are U. -invariant maps V^*®V * •—*	. This
follows from the U-invariance of Г and (9.22a) upon noting that Э * (ч) si)’ (0“ 0 ) » 1> (u)	since elements
of IA ("rotations”) commute with the "time-reflection" в
We introduce an M-covariant version (*) by
fj (mx) « l/(m) -t [D^1 (m)'1 ®	(m)*4 ]	(10.21ti)
189
" Js
and the additional stipulation that T	is to be independent
of the length of x, i.e. depends only on x/ixl . -tJS (x) is invariant under the little group Ux of x in M in the sense that
Let us abbreviate Wm (x’w~4 = m (xf)
. We note the identity
hi (x'j 4m » m (x ) 4 mfor m = mx,	,
Indeed, for m in M and x arbitrary, hi m = m	with
'	X mx
£ = wmw’1 by ( IO-27*).
So m(x') " mx, m wx »	hiwx ini (x ) *%x, since hi(x')x>x' and M j»l .
We insert this identity and (fO-2ld) into (MZib) to obtain
-k*c-c,-c ,	+ C, - C .	3b+c, *-C, + c
r.h.s. of (-{02icO » lx,I	)хг1	|x,-x.J
iJS(w(wx,- vxt>) [l/4 (mwxJ® j/1 («vxj]
This can be inserted into (tO.^) to give the final result. We define the C<5 -kernel 0?4>?£) by
) ("„ ) =	(х^х^г» XX ) /(x-xj.	(W.2.^)
The Clebsch Gordan kernel for x = XA= [L-iJ,X =	then given by
V*S(X.X. X1X11 °x) = !х,Г*’+Ч'Ч-С 1ххГ!'"с,4С1-'с lx,-xzrt'+c’+c’-4t
with vnWK =-0r(x) € tA , ”(x)/*v =-5Av + ix/*xv/lxl’- .
Y js
We have made use of the fact that x (x) is independent of the
190
length of its argument by definition. The kernel (10.2 ) is translation invariant, so it suffices to know it for x- О	.
Summing up we have
Theorem io.I , Let "U ~ Spin (2^-1) c ft the centralizer in M ® Spin (2h) of reflections в . Denote the set of their игв'з by U , M as usual.
The Clebsch Gordan kernels V^s for the Kronecker product of two elementary representations	[2,c, ] ®	•“* L^c I
of (5 v Spin (2h + 1,1) are labelled by j,s ,
j i M , s € u and such that j e	, sc j anol sc t}
with the mirror image offe’M.The kernels are given explicitly by Eq. (70,23) above. The M-covariant projection-imbedding-operator p (x) is independent of the length of its argument by definition and is given in terms of Clebsch Gordan coefficient (9 «72 a) for M and projection operators on UlRs of IL (ср. (10.1fc)) by Eqs. (£0.27c,d), with (m ) s "1/ ( в* в)	, Ц)1 the repre-
sentation matrix for UIR I of M acting in vector space V%
As an example, consider the special case 2. - О (trivial representation). Then	« о , so j «• о and s«o , and t
must be a completely symmetric tensor representation in order that set . Vector spaces V1' and are 1-dimensional.“ Linear maps V ‘ ।—» V	may thus be identified with
vectors in V*" , so CG -kernels V ( ... ) take values in , and t°° is a normalized U -invariant vector t° »uoelT^ .
Completely symmetric tensor representations 2 of M have a unique (normalized) U -invariant vector г>0 . The components of Ig (mx)«5)‘(,")tf0 are called (zonal) spherical functions.
Corrollary 10.z. In the special case 2, = /х« 0 (trivial representation) a nonvanishing CG -kernel exists only for where S. is a completely symmetric tensor representation of Mo
191
It is unique up to normalization (having j»s»o), viz.
v (x. x хх Хл j 0 x ) = Iх,1 t + t’ Cl lxJ ’ 1 <x,-x4l	x
-for х^1°сД . wiK >>=	(7V7O.
For completely symmetric tensor representations £ , vectors in may be considered as homogeneous polynomials in a complex isotropic (5% О )	X&. -vector j , cp. Sec. and Appendix A.
The spherical functions X>'e
So the C6 -kernel of Corollary IO.Z becomes in this language
v kx, xAx > °X )" ci 'M " C*C* 1	‘
*1 лг
do. 2 ‘r)
192
•OB- Application of the Plancherel theorem to the Kronecker product of two principal series representations.
In this section we consider the tensor product of two principal series representations of G. The expansion formula for this case will be deduced from the Plancherel theorem for the (left) regular representation, cp. Sec. 8.
We will restrict our attention to even number of spaceftime dimensions 2h for now.
In Sec. 9 we showed that a Kronecker product of two principal series representations Х,®Хг may be considered as a unitarily induced representation on G/MA. Because the inducing representation may be decomposed into irreducibles, it suffices to consider representations S4*” which are induced by unitary irreducible representations L-*(>na) = of MA with ®"= с,-сг .
We show that every such representation is contained in the regular representation of G in the sense that the regular representation may be decomposed in a direct integral of such induced representations, and the integral runs over all of them, with multiplicity	0
The regular representation consists of complex valued square integrable functions -f on G. Let je M Ыг J UIR of M in 1)1 and ir^’an orthonormal basis in 1)1 . .We consider the Fourier-Mellin transform of -f for imaginary o"
|at"r ^0")<
mA	uuZS)
Functions	take values in 1)1 and have covariance property
oc
^<x	~ ,e 'DJ(m > j* (j ) ”	f* ($K
(1Ш)
This shows that they transform according to induced representation on G/MA- Mellin-inversion formula and Peter-Weyl theorem (i.e. orthogonality and completeness of representation functions
193
for MAJ provide the completeness relation
100
J£M -io* 6/MA
(10.27)
2 , > is the scalar product on V J integrand on the r.h.s. depends on <|
j = 3MA • If
factors q - n и m
c<<| - olnoln oimctcu theft - olndiL
and olj - сЪл* V J , Tfte only through its coset in terms of the Bruhat
We may conclude that almost every unitarily induced representation on G/MA decomposes into irreducibles which appear in the decomposition of the regular representation. Гог even dimension 2h this means that only principal series represent-
ations appear in the decomposition. Moreover we can apply the expansion formula for square integrable functions f (j) on G which effects their decomposition into functions which transform irreducibly. Applying to it the integral operator E^°" of (10.2$) we will obtain an expansion formula for functions
j0" in	which is certainly true in a distribution
theoretic sense, i.e. (roughly) after smearing over <r with a test function ij(o-) such that 5c,<r	f ^*3' >
are square integrable. We shall take it for granted that it holds also pointwise in <r whenever it makes sense. (A rigorous argument is known for the case of complex semisimple groups ws J.)
Define
in
:5_CiWlet	hx.
ЬУ "/j (j) »	(3,5)
<10-28)
Let then F-l|>, ;h
F j
and { J
are
functions on G with covariance property
LJ (mo.)’’ pl (3) « |ftlql)i(m'’)F-l(3> ? same for f. . fl0.29)
194
The expansion formula alluded to above reads according to Sec. 8A
•HpO
Witt V[<c] ,pXS£^ [?(l,c)dc
(10.30a)
where p is the Plancherel weight given by Theorem 8.1.
0^ is the character of the principal series representation % ; it is given by (3«18) and (3.9).
(f) = F* (M) with к
•tt F^X (k4 ,t?1) = vkr(h)r(2h) 1 famdadn |d| ^(m) f(/et nma kJ) (10.30b) where ij^(m) = 4rD^(m)
if Haar measure is normalized as i/g = dndn dm da. in terms of the Bruhat factors. К is the maximal compact subgroup of G .
It is convenient to translate formula (10.30b) to the noncompact picture (cp. Sec. 2D).
We write te= kzrn with m e /А	and	a(z) * nz
|a(z)| = (l+z1)'1 • This is the Bruhat decomposition of keK as we know, cp. (1.31) and (1.30c). Normalized Haar measure on К becomes dk = т"1 Г (zk ) Г (A ) 1 | a (z ) | dz dm Inserting this in (10.30b) gives
(f) = J la(z)l2l< dzdmdadn [a lh'c^(m) f (nza(z)'1nj n man^^nj ) .
We make a change of variables in the n. -integration. Write first n'= n.~'n n- , with m = wmW* then dn =dn' . Put now * ma z
n"= a(z)'1 n'a(z) , then dn" = / a (z ) I 2k dn' . Since man» »	.< ma	this gives
z m a z
&%(/)= f dz J dmdadn [a | C ^(m) f (nzn man'1) ,	(10.31)
JK2J' MAN’
where (m) = ttlpQm)	the character of -(e M as before.
This is the desired formula for the principal series character
in the noncompact picture.
We will apply this to the function
defined by (10.28).
J
195
This is a vector valued function, with values in . The above formulae apply to its individual components in an arbitrary basis. We shall use vector notation, 0^) is then also a vector in VJ . Writing n» n.wu. , dn = (у1)'2*1 o/v , etc. we get
2hdvdi ^dmda	\tznwirlnci 	(Ю.32)
We pose ourselves the problem of expressing this in terms of the Clebsch-Gordan kernels which were determined in the * )
previous section . This will be done by using covariance condition (10.29) to carry out the integration over a subgroup Uz of M.
Let UZCM be the little group of z , it consists of n in M such that uz = z , hence un"* = n.j1u . If we write m = m, u. then by (10.29)
”J(rlwy\tzr‘wlzma\1) = DJ(u)
(10.33a)
for m - m,u , ue Uz .
The coset space M/uz ~ S2^’’ is the unit sphere in 2h dimensions, with normalized measure dm. An integral of the form
= J du <f(mu) Uz
jc/m Cp (m) « dt
1Л	M/u2
and (р(т') depends
We shall use
Lemma 10.3«	Let
map from 11 to defined by (10.16a,b) and (10.18’). The adjoint map (xhermitean conjugate matrix) from to is denoted by t (z )	. With this notation
n tp(m') defines
(ю.ззь)
on m only through the coset rn = m Uz , the following identity
z = z/jzl and i^S(z) be the Uz-invariant
*)
It is implicit in the following considerations that the Clebsch Gordan kernels could have been deduced from (10.32) saving part of the labor of Sec. 2C.
196
Uz	scj,<
Summation is over UIR’s s of tained both in the UIR’s j and representation s .
U2“Spin (2h-l) which are con-l of M; d(s) is the dimension of
The proof of this lemme is relegated to Appendix D , it is a straightforward application of standard orthogonality relations of representation functions of compact Lie groups.
We make use of (10.33b) to split the m. - integration in (10.32). We insert (10.33a) and carry out the integration over Uz with the help of lemma 10.3. This gives
0X J(v2)'2k^c/z	'
( 5 -summation as in Lemma 10.3)>	(Ю.З^а)
The argument of the integral depends on itl only through the coset m. = m Ux .
From here on until Eq. (10.36) there is some amount of straightforward computation to be done. We will determine the Bruhat-decomposition of the argument j of , use the covariance (10.29) of F^ and make some changes of variables in the integrations. In this we shall use the fact that MA acts transitively on the pointed Euclidean space K2’*'4 {0}	. Finally, by comparison
with Eq. (10.20a) of Sec. 10A we will be able to express the result in terms of Г.
Let us got to work then. We want to find the Bruhat factors of
J = nwy rlwu-mQ	in the order nnma ; n e , etc.
First we consider nx+zn-wir • Identity (9.8) gives
PL И = n ,	. n. A	wi-t/i wz'= tv(x+y)-iv(x+z*u-) .
X+Z WIT tW(xi-Z+ir) "x' vx+z + u- Wir	’
We insert this in g and switch the H -factors through n'1	.
This gives
q = n. ,	. ,, PL , ,	1 -fl fi,,	, z’ as above, x’=maz.
3	lw(x+z + v) + wy z - nx+z+u.hwu.x x+z+u w«-	’	’
We insert this in (Ю.З^а) and make use of covariance (10.29) of
197
= Ip1) pm (da |a|k‘c£, ol(s)''l^(z)*^(m)l}S(z)
J JJ	J J	SCj,f
lai J)3(m) L (hx+x+l, hW(,) F1 (nw(x+z+iz)4.w^ hx+2+^wvx )
with wz's w(x+z) - iv(x + z+v) , x'= maz , z = z/lzl .	(lO.jAb)
Next we use homogeneity and M-covariance (10.18') of i3S(z) to scale its argument and pull O3(m)"' through it:
iJS(z/lzl)*I>f(m) I3 (z/lzi)D3(m) '= |х|Л|а|Ь+<: ’’P (z)* i3* (maz) .	(10.35)
We also change variables of integration: The argument depends on m and a. only through maz= x' , apart from a remaining explicit factor I a | .MA acts transitively on pointed x-space RZl' ' {0}	, so integration over M/U2 and A may be
replaced by integration over x'. M / Uz » S231"1	, the sphere,
so dm = dClx, is the angular integration (normalized to 1). Also da = lar1dlal= |xT1c(|x'|	, whence dmda = lx'l'<df)x,dfx'l = |х'Г231 dx'
and integration over x' runs over all of R2,1 (the missing point 0 has measure 0).
Since / а I2*1 |z |231 = Imazl231 = |x'|2tl we get
0X Ю ^(^d.dzdx'E d(s)<	L3(hXz.J 1
, F ^w(x+z + v) + wy n*'-Rx+2+l>wu.x') • with wz = ,v(x+z)-w/(x+z+u)
By comparison with (10.20a) we see that 0 (f.) may be expressed is	* J
in terms of id F^ as
0* Ф	^S(z)*(iIS3rJ)Cnwy«x+z)	(10.36)
We now go through some of the steps of Sec. 10Д in backward direction, expressing	through the kernel	>§z)
by (10.18), (10.8) viz. .	-s
iJS(z)= (n;’)=	, nwynz+x) .
Inserting the hermitean conjugate of this in (10.36) and using a new variable of integration zz= z+x we get
198

Finally we use conformal covariance (of	, .) and P
which transform contragredient to each other) to write
J $<-<, J
(10.37)
Checking this amounts to a direct verification of the conformal invariance of the scalar product for a unitary principal series representation in the noncompact picture, cp. Sec. 2.D. The result (10.37) combined with (10.30a) (and 10.28) is the sought for expansion formula for (functions in the representation space of)  j®' the unitarily induced representation	on G/MA.
FJ(g) = Z2 cftsfjdz P (g, nz) ) (\) 	(10.38a)
sc	1R2k
We have derived this for g of the form extends to general g by writing down of g and using МА-covariance of F^
3 =	6	• It:
-J	wy *
the Bruhat-decOmposition and P (. , . ).
1 Is — Is
The map t  с, X is an
isometric map between unitary
representation spaces, so it has
an adjoint
Ps*
which effects the G-invariant imbedding; its integral kernel is just (•>•)*	. Expansion formula (10.38a) may then be written
for short
Г ’ = f	,d(s) 1 P F1 .	( я = [(, c ] )	(10.38b)
In our last step we use the connection, established in Sec. 9, i jv
of the unitarily induced representations c. and the Kronecker product %,® %x of two unitary principal series representations.
The connection is effected by operators Qj , cp. end of Sec. 9 : Let / in X, ® and r = Q f	• Then / = Zj Qj FJ
Furthermore the CG-map V^s = tJS о Qj , and Q* is the adjoint of Qj . Expansion formula (10.39b) gives then
199
{ = Z 0*FJ • i fa d(s)-4Q* P5* PSQ^
That is
f 4pX?^ ^C3’" V^* viS f	(Ю.39)
where the sun is over UIR's j of M and s of U such that jC /, ®	and sc^scj . Writing this out we have our final
result
Theorem 10.4. Consider functions -f(x,Xt) of two Euclidean variables, contained in the Kronecker product	^w0
unitary principal series representations of G - Spin (2h+l,l), h integer. [That is, = [-C, , c, 1 , X2 = !A > ci 1 with c1,c2 pure imaginary; | takes values in	the vector space in
which acts UIR ( of Ms Spin (2h); -f is square integrable in the sense of (9.3) and transforms as in (9.2b)] . Then •[ admits a conformal partial wave expansion in terms of principal series representations %= [f,c] as follows. Let У^(") be the CG-kernels of Theorem 10.1. Then
Q’f(x) = f ,, takes values in , and -f admits expansion
Summation is over UIR's j of M which are contained in the Kronecker product 0 C2 , the mirror image of A,e M , and over UIR's s of U« Spin (2h-l) which are contained both in UIR's j and { of M; ol(s') is the dimension of the UIR. s .
£dX = L. (ЫУ' f
-tw with p the Plancherel weight of Theorem 8-1 the star * denotes hermitean conjugation of a matrix (= adjoint map
200
10.C Odd space time dimension 2h
In our discussion in Sec. 10.В we assumed that the number of space time dimensions 2h was even, viz. h = 1, 2, ... This implied, by the result of Sec. 8.A	that only the principal
series of unitary representations entered into the decomposition of the regular representation of G * Spin (2h, 1) or SO (2h, 1). The same holds then for the Kronecker product of two unitary principal series representations.
For odd number of space time dimensions, the situation is different in general. According to the results reviewed in Sec. 8.В , the regular representation, and therefore the Kronecker products of principal series representations also, decompose in general into principal series and discrete series contributions. The contribution of the principal series is of the same form as before, only the explicit form of the Plancherel weight p(f,c) changes (it has poles now). So it remains to discuss the possible contributions form the discrete series.
We consider the special case that x • ГЛ >c, 1 and
Г^2'С11 are class I representations, viz.	- 0	the
trivial representation of M. They are of course representations of SO(2h+l,l). We show that in this case, discrete series representations do not appear.
The proof makes use of the following known fact reviewed in Sec. 7.
1)	All elementary representations of G are simply reducible with respect to the maximal compact subgroup К = SO(2h+l). (In other words, each irreducible representation of К enters at most once in a given elementary representation
y of G - cf. Corollary 2.1).
2)	Any discrete series representation U of G is inequivalent to its mirror image (i.e. to the representation obtained by space reflection) (Theorem 7«3)«
201
3)	It follows from -1) and 2) that each irreducible representation of К which belongs to a discrete series representation U of G is inequivalent to its mirror image.
The Gel'fand-Zeitlin patterns for S0(n)(which provide a simple rule for the M = SO(2h) content of К = SO(2h+l)) tell us that if an irreducible representation of К contains the trivial representation of M, then it is equivalent to its mirror image. Together with 3) this implies that a discrete series representation U of G never contains the trivial representation of M.
finally we shall make use of the following
Reciprocity Theorem (see 'Warner {V/31 vol. 1, Theorem 5. 3. 3.1).
Let G be a Lie group and H be a closed subgroup of G. Let L and U be differentiable representations of H and G, respectively. Let Tu be the representation of G(differentiably) induced by L. Define the intertwining number	of two representations
T^ and T^ of G, acting in the Fr&chet spaces and S? , as the number of linearly independent invariant bilinear forms B(f^, f^) on S^. Then the intertwining number i-C’L.U) is equal to i. (L, U|H) ( 1* I и	being the restriction of the
representation U to the subgroup H of G).
Assume now that the induced representation <e on 6/мд contains a discrete series representation U of G. Then there is a (projection) map of Hilbert spaces, ТГ =	, which
commutes with the action of the group; 7TTuCj) = UfglTT .
Using the scalar product on , we can define an invariant bilinear form 3 (f .<? > an	where W. is the complex
conjugate representation of U. Restricting the representation and U to corresponding (dense) Frechet subspaces and Su of X* and (in order to comply with
202
the differentiability assumption of the Reciprocity Theorem) we end up with a (non-trivial) bilinear form В on ® S-
We now apply the Reciprocity Theorem, choosing for H the inducing subgroup MA of G. We conclude that the number of nontrivial (linearly independent) invariant bilinear forms on
i (l. u(MA)
is equal to the intertwining number
, where L is the one-dimensional representation
(9.1A-) of MA and U1МД is the restriction to MA of the discrete series representation U of G. According to 4) above, this number is zero, since L is trivial on M. Thus, no invariant bilinear formJB(A9>) exists, contrary to our assumption. This proves that 2,^°" contains no discrete series representation.
The results of this section are summed up in the following
Theorem /0,5. The tensor product of two class I (zero spin) principal series representations of the (generalized) Lorentz group G is decomposable into principal series (unitary) representations of G only.
203
1O.D. Analytic continuation in and c^ .
In our applications to conformal partial wave expansions in QFT we will need the decomposition of the Kronecker-product two elementary representations for two special cases
1) % = [o,c, 1 , [о,сх] and in the complementary series ( real)
2) % = [o,ct] and in complementary series, in the unitary principal series.
We shall now derive expansion formulae for these cases from the result of Sec. 10.В by analytic continuation in and c2« The procedure will be somewhat heuristic.
Consider Schwartz test functions of compact support xx) with values in V * <g) 1)	. They may be considered as elements of
the representation space for the Kronecker product %, ® Xz~ ГА'c.l® for arbitrary c^, c2> They are distinguished by the fact that their asymptotic expansions as x, -» «> and/or xz M vanish identically. Nevertheless they form a dense subset of the Hilbert space which carries the unitary representation in cases 1), 2) above. We consider the expansion of such functions first, and discuss extension to the whole Hilbert-space later on.
We shall use the notation
- [-g ,-c ]	* - R, * ]
(10.40)
c is the complex conjugate of c . We note that % = x* if (and only if) x belongs to the unitary principal series. Using this we can rewrite expansion formula Theorem 10.4 as follows too
f (*<xz) = f 2d J р(Ас)<^£1 d(s)"'Jdx vj	**")* 
^dx'olxi V^(X'X, xixz ! f >
If F( c ) is a matrix valued analytic function of c then so is	.
204
We understand the integral jc/x/c/x/ (—) in the distribution theoretic sense (to start with, it is defined by an ordinary convergent integral). The kernels V^s(...) are distributions meromorphic in the complex parameters c, clt c^, with poles as specified below. The integrand of the c-integral in (10.41) is therefore a meromorphic function of c, clt and C£.
[One should discuss convergence properties of the integrals, but we will content ourselves with these remarks: The x-integration is well convergent at large x because it is conformal invariant and can be rewritten as a compact integral in the "compact picture" of Sec. 2C. The c-integration is also well convergent at large c because the partial waves of smooth functions fall off in lc| quickly] .
We can then analytically continue equation (10.41) in c^ and c^. It will continue to hold as it stands as long as no poles of the integrand cross the path of the c-integration. If such a crossing occurs, however, this will produce an extra contribution according to the residue theorem. (We are not interested in formulae where the path of the c-integration is deformed).
Lemma 10.6. For % - [-£,,	the kernel V x*xx > x% )
is a distribution meromorphic in c, c^, c^ and without poles in the half plane T?e ( h + c( +	) > о	(l0.42)
We omit the proof of this lemma; it follows the lines of ref. [M7[. Let us explore its consequences. It implies that no pole of the integrand in (10.41) crosses the path of the c-integration so long as we stay with c^ and Cj in the halfplane (10.4-2), and also with ~C]_ and -C2 •
Proposition 10.7. The Kronecker product	°f ^w0 unibary
elementary representations = [Д, с, ] ,	, cx] in the
supplementary or principal series decomposes into principal series representations only for even number 2h of space time dimensions provided / 7?e c, | + /Rec* | < к .
The same is still true for odd dimension 2h if = C* = 0 .
205
Our considerations so far show that this proposition is correct for the subspace of functions -f(x,xx) as specified above. We extend the result first to all functions which belong to the representation space Cv resp. of Sec. 2.D when considered Л1	Aj.
as functions of the individual variables x^ resp. x^.
This means that we give up the requirement that their asymptotic expansion at large x vanishes. In this way we obtain a space which is invariant under the action of G (i.e. a true representation space). To make the extension, one needs to observe that the xl', x2' -integrations will still converge at large x^’, x^' because they are conformal invariant and can be rewritten as compact integrations in the compact picture of Sec. 2.C. The point at °° is no special point then any more.
Further extension to the whole Hilbert space is made by completion in the norm given by the appropriate scalar product. For the Kronecker product of two class I (scalar) complementary series representations it is given by
Cf, - ^dx.dx'dx^	)	(io,4ja)
and for the Kronecker product of a class I complementary series Xo with a principal series one has
(Ых) = jdx,dx^dX1 <f<(x1xi) ,	(10.43b)
p 6	6
where < , > is the scalar product in V * ® V z « V 1 here» The intertwining kernel G ~ - G ~ is given in Sec. 4.B , only the scalar case is needed here.
Conclusion 10.8. The expansion formula (10.41) holds true for functions with finite norm (f,f) given by (10.43a or b) if the hypothesis of proposition 10.7 is satisfied. Convergence is strong convergence in Hilbert space.
206
11. Special cases and further properties of the expansion formula
11.A The Clebsch-Gordan kernel for two class I representations .
Symmetry and normalization.
In this section we shall start from special cases of the results of Sec. 10 and consider the Clebsch-Gordan decomposition of two class I ("scalar”) representations
X( = [°, c, 1 .	= [°« ci 1	(11.0)
of G ( с1 and c2 satisfying c < A 1 , i, z	, in other words each
cfe is either pure imaginary or belongs to the interval -
Secs. 3.D., 5«C). For ease of future reference we shall first restate the results of Sec. 10. We shall however use different normalization conventions for the Clebsch Gordan kernels in order to exhibit symmetry properties of these kernels in the most convenient form.
Let 1 (x ,x2)c $ (the space of infinitely smooth functions which vanish at coinciding arguments and are obtained by restriction to NxM of C°°-function / on G*G having covariance property (9.1)). The Clebsch Gordan kernels V (x, X, x2 ; x allow to define vector valued functions
(x) - z [[ V	i	J dx,dx2
(11.1) such that
[Tx(g)Fx ] (x) =•£• ffy	> xx)[T(g)f ] (X',xz-)dx,o/xz (ц.г)
where T = T * . The intertwining property (11.2) implies that V has to satisfy the covariance condition
[	(g) V ] (x,% хг%г ixx) »
(11.3)
Such invariant 3-point functions have been constructed in the course of the study of conformal quantum field theory (see, e.g.[Mio,ri|]). In particular, it was shown ] , that if y, and y2 are the zero spin representations (11.0), then a nonvanishing conformal invariant J-point function only exists if у is a type I (symmetric tensor) representation.
207
According to Corollary 10.2 and Eq. (10.2A-) the general conformal invariant 3-point function for the above class of representations can be written in the form
(11 Ла)
Л/г(с^.С.,с)
|- Шс-frc , i(bc-O-e-1
(!)	fcj
(ИЛЬ)
. rr where V. =
<- Эх.
Л* , С+ = |(с+сг) , and is the polynomial
D{ («,X; b//t) = Д	H V'*
(a-*,b-0 / ,u->
P( 4+X
(11.5)
[.We have used again the notation (5.11); is
the Jacobi polynomial.]
Here x, and are the class I representations (11.0), j belongs to
the complex light cone	(see (2.11)),
X -	= 2	- ^2-')
*U = *i. *
(П.6)
The harmonic extension of
0
(A3)
is given by (A.15) (n=2h)
= t ' (Xyu/

4 (w - (А(4,	, 2-h-£;
<? s	(v .
We observe that Eqs. (11.1) (11Л) define a Cw-function ^(*>3)(cCx) provided that the exponents of x and	in V do not assume
any of the positive (half) integer values
(h+c- t )±c_ =h + k, k = 0, 1, 2,...
(П.7)
208
The kernels V satisfy symmetry conditions which relate their values at equivalent points in the representation space. Indeed, it follows from the covariance properties of the V's and the intertwining operators G^ (4.19) that (for example) the kernel
f (x, - <) И(х1'£ , x3Xx •
should be proportional to V(x,%, ,хг%1; x3% ) etc. We shall choose the normalization constant in (11.4) in such a way that the proportionality constant is equal to one. In other words we require
/б- (x,-x;) V'(xl'%l,x1x1 .x3x)^; > V(xJ, .x^x .x3%)	о
Xi	(11 • oa
and similar formulas for the arguments 2 and 3.
We shall further demand that h/j is analytic in С, , сг , and reduces to a phase factor for imaginary с, , сг , c , viz. principal series representations. Thus we demand that for imaginary c
(-C+ ,-c. ,c) h/f (c+,c. , c) - 1
(11.8b)
Such phase factors (and their analytic continuation) cancel out in the expansion formulae (Theorem 10.4 and Eq. (10.41)) of Sec. 10.) The constants appearing in (10.24) were omitted in (11.4). They will be absorbed in the Plancherel weight.
The expansion formula (10.41) and reality properties of kernels (11.4) imply then orthogonality relations of the following form
= S (%,%') S(x-x') (3'3'/+ £(%,%')	J, j') J
here
(x.x') = &ee'	 for x( [S( \ (a1'5] .
rf I )
(11.9)
(11.10)
209
and p^(c) is the Plancherel weight:
p,(c) - —С^Л) | £..(£• i+c)i [(h+£-i')1-c1] rQltp. nfaWx)
~e	2 (2ir)hX! 1	Г’(с) I	2(2n)4!	(11.
(for C = Iff ).
The sum of two terms in the r.h.s. of (11.9) arises in the following way. Because of equivalence of principal series representations
X and x t every UIR appears twice in the partial wave expansion (10.kl); because of amputation convention (11.8a) on leg 3, both contributions are equal. This fixes the relative size of the two terms on the r.h.s. of (11.9).
In deriving (11.9) from (10.kl) one also needs the overall constant in the Plancherel weight (8.6). Conflicting answers to this are given in the literature. We have used the constant found by explicit calculation in [D.51 •
Expansion formula (10.kl) implies that the mapping given by (11.1) can be inverted with the result
f	c/% Jc/x , хгх2 j XX )	(x)	(11.12)
where	M L0Q	л 00
px -,L /af.M
J	1= 0 -LK>	X. - v-oo
This is valid provided
I Re c | + I Rec. | < к , 1 (1
It is known for the ordinary Lorentz group that for + сг > к the complementary series representation С = C1 + c- к may contribute to the expansion [N1] . This suggests that at the boundary C1+ci= h the expansion in principal series representations
will still hold
210
In the rest of this section we shall demonstrate that Eqs.( 11.8)and (11.9) are compatible and fix (c^, c , c) up to a sign.
We start with the exploitation of the symmetry property CH.8) .The
calculation is based on the integral formula

61^/2-^363) =
.	Г (6 )	Г (6,)	Г (5,)
1 Г	1_____________4_____________5_______
(2ir)h [Ifx-xp2] 1[|(x-x2)2]	2[| (x-x3)2]	3
00 daj 00 da^ 00 da3
61 62 63 °T a2 a3
0	1 0	2 0	3
{222 ala2X12+ala3X13+a2a3X23
2 (к^а^+к^а^+к^а^)
(k. > 0, Sk_> 0)
rfh-Sp r(h-62)	r(h-63)
a 2 h'6la 2 h'5Za 2 ,h'63
(2 ^З1	(2X13> зХ12
for 6j+ 62+ 63 = 2h (11.14)
(see|D2] [S8] ) and on the identity
____
(|х23)“(|х23)Р
- У I nkJ 1 -
~k=0	’ lk' Г (a+t-к)Г (p+k)
V , .А, Г Г (P) .	'g+J-k,...	_ ,k _z_ P _
~k=0(-1) ( к Г (p+k) ^~2 }	13*	V3* ( 2 } “
X13
X23
(11.15)
X13	X23
— - c ), where ^3(^3) differentiates with
h+c-f
(used for a = —
respect to x3 to the left (right). Using the first equation^! 1.15Jand (11.14)
+c , p =
we find:
211
(р,,/ i Г (c +k-j)r fay--- -C +j) v IO	4 Lj	&	£	*t"
Changing the order of summation and using the sum rule
V ( nk J x/kxEk+k) _ , iJJ x JJp-aH-j) г (or+j) ’ к j Г (p+k)	r(p+f)T(P-a)
(11.16)
(see Appendix E) for a = c^-j, P = —2  " C
we obtain
J dx2 V(X1X1*X2’X2’X3X>) \(X2'X2} .T i	> h+c+f v ^h-c+f
N. (с , с , с) Г -с ) Г f—~- + c ) ____________XT-__________L______+____________L___________-	_________ h-c+f_______________________________________________________________h+c-f	h+c-f
n Л^Л-c+f .	XT,h+c+-£ W1 2 x 2 +C-,1 2 X 2	+C+,1 2 X 2 ’C
(2тт) Г f +CJT f—— -c_)(j x12)	(гх1з)	^X23
N^fc^c .clr^^-с^)Г	•+c )
= T^	\ v<xi	•	^^)
N1(C-’ C+’ С)Г(2 +С+)Г f—2— -c_)
Applying the obvious symmetry property
j, Nx<c+. c • c)
V(V1'X2X2'X3X^ = H) N;7r;r,^) v(x2 x2’xlxl’x3 X>>	(^.18)
212
we can derive from (11.17) another relation of that type, involving inte
gration over the first argument of V.
Combining these two equations and comparing with (11.8) we obtain
Ml (ct,C-,C)	_ Г’(~Т~"	2
W<(-c+,-c_, c) r(Lij±X-C+)r(b±^--c+)
The consequences of the amputation identity (11.8a) on leg 3
can be derived in a similar way. It gives, for imaginary c
Mt(c+,c_,c) = r(bc±l,cJr(A±c±-L + c_)
Ni(c+>c-,~c) Г	- CJ r(A -c	„ c_)
(11.19)
(11.20)
We shall not reproduce the necessary computations here. (An independent derivation is contained in [D5]). In any case, (11.20) and the relative size of the two terms on the r.h.s. of (11.9) are merely a convention.
Eqs. (11.8b), (11.19) and (11.20) for the normalization factors are solved by
z
(11.21)
The sign of the square root will be fixed in the following i and c^ be fixed real numbers such that
|c, I +lctl <
We regard as an analytic function in c with cuts along
vals on the real c-axis for which the expression under the square root sign in (11.21) is non-positive. We demand further that for real c, for which
Let c^
(11.22)
inter-
Iе, I lcxI + lc I <
(11.23)
213
is positive. This rule determines completely in the cut c-plane. We note, however, that in most applications only various products of the N£'s enter, which do not involve square roots.
11.B. Identities for the Clebsch-Gordan kernels at exceptional integer points
In the physical applications which we have in mind (see [М2, d4[ and Chapter V) one needs analytic continuation of V not only in c^ and c^, but also in c to real values. In that case some differential identities between the V's at the exceptional integer points (3.3) play an important role. We shall describe below these identities excluding for the sake of simplicity the case c_= О, V odd.
We shall first establish the relation
J 3	(11.24)
= sign [(1^- + с_)у]^(х,%1,хгХг,х3%')(з)
between the V's at exceptional points with c < 0.
To prove (11.24) we use the representation (11.4b) for V and remark that the operator	is equivalent to - (3 F, + J-) when applied to a trans-
lation invariant function of x^Xj.x (which only depends on the coordinate differences). Then (11.24) is a consequence of the identities:
V?-'*'- -A (t> '), МЧ	
(11.25Ta)
+	I7l -v -\]1	(11.25b)
------------------ = (-' ) t-5—+	’
214
where
V s,„ - ¥ . <	(11-20
for both x^ and (where S% is defined in (11.5)) and we use the relation
(see Eq. 8.962.1 of ref. [G7]). Inserting (11,25) in (11.4b) we obtain
1-(з^^^)Г7(хд,л2%г.х3%^3)=!ф^2.('ЬГ + е.)/(х1Х,,хг%г)хзХ';5)	(11.27)
*1x4)
where we use the short-hand notation N(x ) for the normalization constant (10.21). According to (10.21) that leads to (11.24).
Now we shall consider exceptional points with c > 0 . Tor such points we shall derive the following relation;
In order to prove (11.28) we shall use the x-space relation between
the bilinear forms 3/ and St ((6.27) and (6.24)), which follows
lv	i*
from (6.32)
and the analytic continuation in CpC., and c of (11.8) (with xlx1 replaced by x^ X ) to real c's for which %, and . are complementary series representations and %	>:t 
7/	(*зз' > Ц ИП 'Мр *з
-	ГГ^-с.)г('-^- + с_)и(х,%,,^гх1 ^з'хХ2;)
(77)!	(Xj3' > *5 ’ (ХЛ, - X2 %г > *3 %£к"3 ) ^З
(11.30а)
(11.30b)
215
The sign factors in (11.30) appear in accord with the definition of in the cut c-plane (see Eq. (10.21) and subsequent remark). To establish (11.28) we apply the integral operator with'kernel
(6-v).11


to both sides of (11.2A-) and integrate the left hand side by parts.
Using (11.29) we obtain
(2h +	’хз'Х'1„3^с/хз =	(11.3D
=	^Эз)'/^^’х^’хзугйЗ)Лз
(/+ *)! *
In order to obtain (11.28) it remains to substitute (11.30) in (11.31)
taking into account
We note that although 3^ is only defined as a bilinear form on the invariant subspace	of C^_	, Eq. (11.31) makes sense
as an operator equation on	.
216
We can use (11.2A-) and (11.28) in order to derive similar identities for the "conformal Fourier transform" F^ (x,j), defined by (11.1). For example, applying the differential operator (jj-V)v to both sides of the equation
tv
(which is (11.1) analytically continued in c (for c_^ 0)), and using (11.2A-)(with %, replaced by Xi t * = 1, 2) we obtain
(з-^Л (X,3)=	[(11^ -с.\]к	(x,3) .
/у
Similarly, as a consequence of (11.28), we find the relation
= (11.2
217
11. С. Tensor product representations and Clebsch-Gordan expansions
for distributions
As already noted (see conclusion 10.8) the expansion formulae (10.41), (11.1) and (11.12) are valid for all square integrable functions. That includes the 1-particle irreducible Green functions to be considered in Chapter V below. We shall, however, also need to apply the intertwining differential operators of Secs. 6 and 11.8 to such functions. That is indeed possible if we regard them as distributions and use the notion of derivative of a distribution. The above expansion formulae (and related equations) can be extended from the space of test functions to the dual space of distributions in the line sketched below.
Consider the space S' of continuous linear functionals on the test function space £>K° , defined in Sec. 11A. We shall use the distribution theoretic notation for the functionals $ of SK :
(Ф, f) = J/ф (x,,x2)f (х,.х2)с/х,с/х2 (feS^) .	(11.32)
We define the dual T'
to the representation
(acting in
) by
(r'(3)£	) = (£,T(g)f )
(11.33)
(the
denoted by Т'(3)ф
right-hand side is
a continuous linear functional on £K (sf) which is The representation
regarded as an extension of the representation T«= T~ ®T~	. Indeed, it is
easily verified that Ss C SL and that the action of T' defined ко Ko
by (11.33) t coincides with f on S>~ (where is SK° with K,-» %, , X2—►%
- cf. Proposition 3.6.
Consider the topological vector space Cg of (test) functions
F,(x) which can be presented in the form (11.1) with f (x, , x, ) e A	1	’	*O
We shall write
(11.34)
indicating that F^,(x)cCx (for fixed % ). Regarded as a function of
X =	the right hand side of (11.1) allows an analytic continuation in
CZ ° C*

of <?(ex) can be
)
218
c (off the imaginary axis) because of the vanishing of	and its
derivatives for coinciding arguments. It goes fast to zero for J 3m c |oo because of the infinite differentiability of the test functions f . A precise intrinsic charactic of the space Cj- is desirable (in terms of the properties of ^(x) regarded as functions of two (sets of) variables x and у ), but we are not prepared to give such a characteristic here.
Let Cj- be the dual space of (continuous) linear functionals on C<r t which will be written symbolically as
$(F)-^Xp*^(*)Fx(x) (x>e Cz >	€ CZ )	(11.35)
Inserting for ^(х??хг)1п (11.32) its partial wave expansion (10.12) we can define a mapping from g' into ; formally
(x) =	ф (х,,хг)	> ’ЧХг >	£ CS) (11.36)
The corresponding formula (11.1) for the test functions ensures that the above mapping is actually onto and has an inverse which can be written symbolically in the form
£ Сх.,хх) = Ja/x $-(*')
x x)
(11.37)
Thus one indeed has a distribution theoretic counterpart of the tensor product expansion formulas (11.1) and (11.12).
219
V. DYNAMICAL DERIVATION OF VACUUM OPERATOR PRODUCT EXPANSION IN EUCLIDEAN CONFORMAL QUANTUM FIELD THEORY
12. Renormalizable models of self-interacting scalar fields. Dynamical equations for Green functions
12.A. A 6-dimensional model. Euclidean Green functions. Generating functionals.
The simplest model of a renormalizable self-interacting field (p(x) is given by the interaction Lagrangian L^(x) = - Э/з! : <jp3(x): in six space time dimensions. Although this model is manifestly sick (since the corresponding classical Hamiltonian is not positive definite) it can (and does) serve as a testing ground for various quantum field theoretic techniques (apart from its role in the work [s6,M2] which we are going to review, it presents the simplest example of a theory with asymptotic freedom — see, e.g. [Ml]). We shall indicate at the end of this section how one might modify the model, in order to cure it from its obvious disease.
Having in mind models in different number of dimensions, we shall work in a general framework of 2h-dimensional space time (2h=2, J, 4, ...).
In what follows we shall mostly deal with Euclidean Green functions also called Schwinger functions (cf. [S7. 0j>]):
S ( x o' , x „or ) = r ( i-o; x, ,, io- x )
1	"	(12.1)
where T'(x<,...,Xn) = <'”r O’)•••<? Cxn) >o	( у is the interacting
Heisenberg field).
One can define connected, one-particle irreducible (IPI) etc. Green functions without recourse to perturbation theory. The most compact way to do that is in terms of generating functionals (see e.g. [S6, C2]).
Let j(x) be a scalar external source and let /(j) be the generating functional for the s-functions
/(3) = l + £ j-fdx, ...dxn s(x,...Xa) 3(x,)... J(x„)
= < exp Ju(*> ф(х)с!х >o	j dx = dZ^x	(12.2
220
( ф(х) is by definition the Euclidean field). The generating functional CJ (j) of the connected (Euclidean) Green functions G (x-^ ....xn) is defined by
/(3) -	,	s(x)-G(x)=o	(12.3)
The source j(x) is associated with a classical (Euclidean) field фс(х) by
C JJ(X) J ( J Sl(*)
(12.4)
The generating functional for the 1PI Green functions (or proper vertex functions) Г (x^....xn) is given by the Legendre transformation [C2 ]
г(Фс') = $(1)- fol
- £	г’Сх1-Хп)Фс(х,)-^^п) 	(12.5)
To obtain the right-hand side of (12.5) we express J(x) in terms of Фс(х) from (12.4).
12.B.	Graphical notation, li - and 2i - kernels
In order to write down the (renormalized) equations for the model under consideration we shall need some additional auxiliary notions (cf. [F7, S6, М2, F8]).
We introduce the amputated (connected) Green functions
А(хг-хл) - f-.Гс/у G	ff’'(xn-.y„’) 6(y,...yn) ,
J J	(12.6)
where
<S ’* <3 (x^-XjJ = ffl/x G \х4-х)б (х-хг) = ^(x^x^)
(12.7) (We use alternately the notation G(x-^, x^) and 3(x^ - x^) for the 2-point function; that is legitimate, because of translation invariance). We have
= - Г’(х,,’<2) - G'7x,-x2) ? A(x,x1x3) ’ rfx.x.x^ ) .	(12.8)
We shall use the graphical notation
(12.9a)
(12.9b)
221
The 1PI amplitude for the channel 12 -» 3>«>n (n?-A-) is defined by
A1t. (*<>45 Xj-Xn) - A(x1...xrl)-pyJd'yiA(x4xxy<)G(y<.yl)A(ylX3...xn).(i2>i0a)
or graphically
(12.10b)
We define the Bethe-Salpeter (BS) kernel
В (x± x2
as the solution of
the (integral) BS equation
(12.11)
(The factor 1/2 in the right-hand side is necessary because of the symmetry of the theory of a single neutral scalar field). Finally we introduce the "2-particle irreducible" kernel for the
channel 12-*j>--«n
(n>5) by induction in n:
(12.12a)
(12.12b)
The first sum in the right hand side of Eq. (12.12b) (and in (12.12a)) is spread over all 2П~^ -1 partitions of the set of external lines
З...П into two поя-empty subsets. The second sum involves all splittings of these lines into к non-empty subsets (k =	n-j>).
12.C. Dynamical equations
Stress energy tensor. Ward identities
The dynamical equations can be written either in terms of the connected Green functions G(cf. [S6, М2J) or in terms of the proper vertex functions (cf. [F7, F8]) the two forms being
222
equivalent. We shall adopt here the latter form, writing however, the
equation for the 2-point function in a way suggested in [Мб, М2].
In the Gell-Mann-Low limit (in which the renormalization constant = 0	) the dynamical equations for the vertex functions have
the form
(12.13)
(12.14)
(12.15a)
The last equation can be written
equivalently in the form

(12.15b)

and external lines have been attached to the BS kernel in the right hand side (which amounts to the inverse operation of (12.6)).
The bootstrap form of Eqs. (12.14) (12.15) is peculiar to the
Gell-Mann-Low limit theory. In general (away from that limit), there is
an inhomogeneous term in the right-hand side of (12.14) and both
equations require subtractions in momentum space or
( X1~X2\ in such an
in coordinate space. Masses and coupling approach as initial conditions (cf. ref.
multiplication by
constants appear [М2]).
It is convenient to use an alternative form of Eq. (12.15) involving the stress energy tensor f^y(*) (see ref. [Мб]). Let G/Uy(x,x1) x3) be the Euclidean region continuation of <T Cp(xt')<f>(xz') 0цу(х3> >0 and let
% («.*1 J *з ) = f^y,	) G''(*2-5z) (у,уг j x3)
we are using the convention of ref. [S6] , according to which the renormalized field operator <^(x) and coupling constant g are related to the corresponding unrenormalized quantities (f>u and by
<£>(x) - Z/t-ipJx): j : Q’u(x): =-i^-)<Texp(l^u(x)3(xWx)>;’Texp(iJ^(;x)J(x)^x)]3 It should be noted that Eqs. (12.13) - (12.15) do not contain any parameter (like coupling constant), but just relate Green functions among themselves.
223
be the corresponding vertex function. It is assumed the G and	satisfy the following (equivalent between each other)
Ward-Takahashi identities:
4’4/x-xi’xP " g1[5(x3-xl)V^G(X)-x2)-aGfx1-x2)V^S(x3-x;)]	(12.17a)
(x<x2; x3> ='? [<4X3-Xi)^iu G’X*,-x2) + (|+л)б Yx,-x2)V. S(x3-x;)] (12.17b)
7"*	i»1	i	'
The last
(Schwinger) term can be eliminated by multiplying both
sides of each of the
equations (12.17) by
integrating over
x3
with the result
(хт - X^)
(y*/1 )
and
)(x<x)y = (Х,-Х1\Л 6(*<-х2)
J	OXy*
Jdx V* Г (х,Хг ; X) (х„-х)и	= (Xi-X2)v Э
/*
(12.18a)
(12.18b)
This form of the identity has the advantage
to be independent of
the arbitrary constant a .
The set of equations (12.13) - (12.15) is equivalent to the
set
in which (12.15) is replaced by
and the Ward identity (12.17) or (12.18) is
(12.19a)
(12.19b)
assumed to hold.
As
a consequence we obtain an infinite set of additional integral
equations for the n + 1 point functions
the stress energy tensor:
^llIx1...xn ; x) involving
n. = 3,^t ...
(12.20)
224
They also satisfy Ward identities of the type (12.17). To be consistent with the scale invariance of the Gell-Mann-Low limit theory, we have to require that Q/Jv is traceless, so that
Г (x.  x. i x ) = ° ’ G, (*.  *„ j x ) n - 2,3 ...	,
yuyu i n 1	'	Pp ' 1	'	(12.21)
12.D. A more realistic model
Although the above dynamical equations are derived from a renormalizable Lagrangian in 6 space time dimensions, their final form (12.13) - (12.20) makes sense for an arbitrary number 2h of dimensions.
Here we shall indicate how one can fit the more realistic model of a charged (pseudo) scalar field with a quartic interaction in four space-time dimensions
*(<КЮ.$>*(х)) = :	-4 :	(12.22
into the above framework.
The clue lies in the observation (made, e.g., by Symanzik) that the model given by (12.22) can equivalently be described by the Lagrangian
X (<?(x).<?*(x);S(x)) = ; ^Cp*V^Cp-. +-L :Вг: -6,:
of a system of two fields, (p and В , with a cubic interaction. Indeed, varying	B) with respect to В , we find the algebraic
"equation of motion" ДЗ - Vi •• <£*</>: which reduces (12.23) to (12.22). In a canonical perturbation theory the propagator corresponding to the field В would be a S -function. On the other hand, the topological structure of Feynman diagrams in this model is the same as in a theory with Yukawa coupling of a charged field (p and a neutral scalar field В (which will be represented graphically by a cut line----).
Without going into the details of the Green function formulation of this model, we notice that it will involve (a priori)
225
four types of vertex functions
£	;x3.-cB^
SoHx,,-CBi *X,-C3'. X3,-CB)
0) z
b (x , x , * } f>v ' 1 ' 2 >	3 '
_(X) ,
<>	> x3)
(12.24)
(12.25)
(12.26)
(12.2?)
The bootstrap equation for the charged propagator can be equivalently obtained I Мб I from the corresponding equation for the current-field 3-point function
< r Ш) </<4) j,/*? >o	’
which satisfies the Ward identity
(12.28)
vfG,, (x<xx ; x3 ) = e )[S (x,-x3)-S^-x )J J Л*
(12.29)
(e being the electric charge carried by	).
13. Invariance and invariant solutions of the dynamical equations.
Conformal partial wave expansion for the Euclidean Green functions 13.A. Euclidean conformal invariance of the equations .
As already noted,all equations of Sec. 12 only relate Green functions among themselves. They involve no parameters (in particular, no dimensional parameters) and are manifestly scale invariant. Indeed, if we ascribe to the field (f> in 2h dimensions a scale dimension d ц, = h + с (c real), then the Green functions G and Г would have the following transformation properties under dilation:
226
...xn) p"fh+C) G (px,, pxn)	'
r (x,...*n>—* p"(h‘c) r (px,J...,pxn') t p>o ;
in particular,
G '(x,-xt) = - r (x, ,x2) -> pi(h‘c) G'’ (рх,-рхг)
(13.1a)
(13-lb)
(13.1c)
(the canonical dimension for a spinless field <p is obtained for c = - 1). Eqs. (12.13) - (12.15) are obviously invariant under the substitution (13-1)..The Ward identities (12.1?) and (12.30) imply
that the scale dimension of the stress energy tensor the conserved current ju are
and of
d- = 2k , d:* 2k-L	(13.2)
respectively. The dimension dg of the field В in the model, considered in Sec. 12.D, can be ascribed independently.
It turns out that the dynamical equations are invariant under the Euclidean conformal group 0* (2h+l,l) which,as we know from the Bruhat decomposition (Sec. l.B),can be generated by (Euclidean) PoincarS transformations, dilatations and the conformal inversion (cf. (1.29)):
Ux =	x>... + x2\ )
X1
(13.3)
The transformation law of Green functions under
the inversion (13.3) is
summarized by the following rules for Euclidean fields:
и(я)ф(х)и(тг)'’ = (хг)',”с Ф(кх),	(13Aa)
U(R) J/xjU^)-1 » (x2)’2h+' ^(x)Jp(TJx)	(13.4b)
U	’ (xi)~Zh r^, (x)rp, (*) 9^v< (x) ,	(13.4c)
is given by (2.30b):
where r (x)
227
(х> = xl
2х/<ху	?
X1	/»*
(13.5)
The so-called special conformal transformations are given by a translation, sandwiched between two inversions. The R-invariance of the dynamical equations follows from the covariance law for the volume element
c/Px = (хг)'2к det (r^} dx = (xl)~H dx
(13.6)
lj>.B. Conformal invariant 2- and 3- point functions
We shall study in the rest of the paper conformal invariant solutions of the dynamical equations for the models described in Sec. 12. If the Gell-Mann-Low limit is ultraviolet stable (as usually assumed in this approach) then the conformal invariant solution would provide the small distance behavior of Green functions in a more realistic theory with positive mass particles.
The invariance property of the solution allows one to determine the 2- and 3-point functions up to a constant factor without actually solving the equations. Before writing down the corresponding expressions, we shall make a remark about the freedom in the choice of normalization.
In a canonical field theory the field operators are normalized in such a way that the residue in the pole of the 2-point function is one. In a scale invariant theory with anomalous dimensions the 2-point function has no pole and there is no unique choice of field normalization. A multiplicative change (f> (x) -» к(ф(х) in the field (where k=k(c) is some function of the dimension) leads to the following transformation law for Green functions
G (x, •••	-* G (x, ...xa)
Г (x, ...xa)	r(x,...xn)	(13.7)
;X) -* G^y (x, ...X„  x)
[The normalization of	is fixed by the Ward identity (12.1?) .]
228
Thus we can choose the normalization of the 2-point function of
<f> (and B) according to convenience; only the relative normalization of the 2- and 3-Point function will have a physical significance.
We shall choose the normalized invariant 2-point function for a fundamental scalar field Cp to be (A-.19) with normalization (5.25):
G[O,C]<X) ~	•	(13-8)
With this normalization the Fourier transform of G is (cf.(5.23a)):
G(P) = Je^pxG(x)c/x=	(13.9)
The inverse Green function G^tx-^) is obtained from (13.8) by changing the sign of c. We shall say that the field <p is fundamental, if its dimension parameter c=c^ satisfies the inequalities
'1 * c<f 4 °	(13.10)
For a fundamental field <p the two-point function G corresponds to a positive definite Wightman function (cf. the corresponding formula for the composite fields (5.31))
«(p) = -♦ e(po) [g , -<p0*-°) - g (?
(13.11)
= - 2G+(p) ire )°
where ©+(p) = 9(pe)0(p^) >Pm = ?o"E2 • (The subscript M stands for real Minkowski space vectors and scalar products). If we multiply <p by k(c)= [2сГ(-с)]/г we will obtain an elementary field (cf. Sec. 5»D) with positive definite Wightman function for all c which coincides with the 2-point function of a free О-mass field for c=-l.
We shall also need in what follows the conformal invariant
2-point function of an arbitrary rank -C symmetric traceless tensor
field ^...^(x)
normalized 2-point
of dimension h+c, c=c (-Z) . Again we shall use the function (4.19) with normalization (5.25) ( Х = [Дс]):
у ХЛ /	\ IX
(13.12)
With this normalization the momentum space 2-point function becomes
(see (5.23a)):
ff„(p> = (l?l)c Z<xsC‘) пг5(р) a	s« 0
(13.13)
229
where (cf. Sec. 5.B)
""(p -5- ’>  тУМг (-2 c"l‘ (' -are the 8 0(211-1)^ projection operators,
Of, (c j =
(4-c-Os
(13.14)
We know (see (4.11)) that the inverse 2-point function is given by
S'JfxJ ’ G (x„)	(13.15)
А	Д
In order to be able to handle the most general situation (including the model described in Sec. 12D) we shall write down the 3-point function for three different spinless fields with dimension parameters c^, c^ and c^. They are
£(х.,,ха>хэ)» ^/(x.c, jXaca ; x3c3)	(13.16)
г (x,,XvX3) ’	(x,,-C)iXJ,-C2; X,,-C3) ,
where (see (11.4a)):
V/fx,c<;x1cli x3c3) - У(х,%, s xa-y1; x3%3)	,	• [o,c<] .
The function V can be associated with an "infraparticle" triangular
-d-diagram (see [M9], I₽l] , [M? ] ) with scale invariant propagators (-jx^ ) Lk . The parameters d^ satisfy the conservation of dimension law in each vertex of the diagram:
+ dik = dk = 4+cfc > (i.jk)-'Ptrmu.takon f-123)
J	(13.18)
We shall also need the 3-point functions for two scalar (or pseudoscalar) fields of dimension parameters c^ and c^, and a rank Z tensor field O(x,j) of dimension h+c. They are given by
Q (x, ,xa j x3j) = И(х,с, j xxc3 , x3,x,j ) ,	(13.19)
Г (x, ,Xj ; Xj J )	l/(x,,-C, ; Xa,-Ca ; X3,%, j)-,	(13.20)
where (see (10.4a)):
V(x,c,; xxcaJ x3.%,3)- И(х,Х, >х2Гх >• x3,X.3 ) . ’[o.cj , i =	(13’21)
(We do not consider amputation of the external line associated with the tensor field 0). The normalization factor Vj =	(c+ c_ ; c ) is
given by (11.21).
230
The normalization of the 3-point function of the stress energy tensor is fixed by the Ward identity (12.1?).
We have G (x, , хг j )> О -for c1 4 сг , and
<»-22*>
 hs-. (if'»	(15.22b)
L . _ With this G the coefficient to the Schwinger term in (12.1?) is	•
In verifying (12.1?) we have used the relations
O<)’’(w3)[(3V-j£-
-i(2r)-hn(h-<)	&(*п)
13.C. Skeleton diagram expansion
Having constructed the physical propagator and (3-point) vertex function one can expand the n-point functions Г (x., .. • .x ) (n > 4)
*)	In
in terms of skeleton diagrams . It is important to know for the self consistency of conformal invariance that the skeleton diagrams, as well as the graphs encountered in the bootstrap equations (12.14) (12.19) have no ultraviolet (or momentum independent infrared) divergences. Indeed, such divergences would have destroyed even the scale invariance of the Green functions. It was demonstrated in ref. [ M?J, that for a certain range of the parameters c the boson-fermion Yukawa interaction in 4 dimensions is divergence free (for the Gell-Mann-Low limit theory under consideration). The analysis of [M?] is trivially extended to the 2h dimensional models considered in Sec. 12. For the simplest <₽3 model the most stringent restriction on the scaling parameter C=Cif comes from the requirement that the 3-point functions G (ХцХ^.х^)- and Г(x^jX^.x^) are both given by ordinary convergent integrals in momentum space. That leads to
< c *	(13.23)
The 3-point function G/4V(see (13.22)) can be expressed in terms of a convergent p-space integral if c< 0, i.e. if the field is fundamental (see (13.10)). However, it can be continued analytically in c for с>о
’
A skeleton diagram is a Feynman diagram in which internal lines correspond to dressed propagators and points are associated with physical vertices, and which contain no self-energy insertions or (3-point) vertex function corrections. Cf. [В4].
231
to cover the range (13-23), only the point c=0 being excluded.
Coming to the more realistic model envisaged in Sec. 12.D. , we see that the existence of the 3-Point functions (12.24) and (12.25) (complete and amputated on either leg) as convergent integrals in momentum space gives
2|Cvl+	(13.24)
and
V 3h	(13-25)
respectively. The convergence condition for the skeleton diagram
(13.26)
leads to
- 4 < C<f < T
(13.2?)
Assuming that <p is a fundamental field while В is a composite one (cf. Sec. 5.0) so that C<p < О ,	> О , we end up with the following
complete set of inequalities
, 0<cB-2c(/,<k
(13.28)
which guarantee absence of divergences.
The skeleton expansion does not satisfy however the dynamical equations for all values of g and c. It turns out [M7, Мб ] that the entire infinite set of equations (12.13) - (12.19) will be satisfied provided that the two bootstrap equations (12.14) and (12.15) or (12.19), (12.18) are satisfied. Since both sides of (12.14) and of (12.19) are conformal invariant they have to be proportional to the 3-point functions (13.21) with 1=0 and 2, respectively. Thus, these bootstrap equations lead to coordinate independent transcendental equations for the two parameters g and c of the theory. As it could have been predicted these equations turn out
232
to be equivalent to the Gell-Mann-Low equation for the coupling constant. That was verified by the £-expansion method for the Cp3- theory in 6+ £ dimensions in ref. [Ml]. Unfortunately, this new version of the self-consistency equations does not seem any easier to handle. That is one reason why a new approach to the whole problem was attempted in [М2, D4] and is going to be pursued in what follows.
lj.D. Conformal partial wave expansion
Eqs. (12.13) - (12.15) can be regarded as generalized (off-shell) unitarity equations. It is well known that in terms of the ordinary partial waves the (elastic) unitarity condition becomes an algebraic equation. It was demonstrated in ref. [М2] that the conformal extension of the partial wave analysis allows to solve the infinite set of dynamical equations for the <p3- model.
Ordinary partial wave expansion is nothing else but the tensor product expansion of two irreducible (positive-energy) representations of the Poincarfe group. Conformal partial wave -expansion is by definition the tensor product expansion of two irreducible representations of the Euclidean conformal group (see Chapter IV).
The proper vertex function T (x^x,,.. .Xn ) ( n Ъ 4 ) considered as a function of the first two coordinates and integrated over the remaining n-2 coordinates with a "nice” test function f satisfies the square integrability condition
where
X, ’ [°, С, ] , Хг- [о,сг ]
= J’-J’^x3...4/xnrfx,...xn){('x3...xn)	(13.30)
in any order of the skeleton perturbation theory [D2]. (This is, however, not true for the 1-particle reducible diagrams, appearing in the right-hand side of Eq. (12.10b)). The integral in (13.29) is nothing else but the scalar product in the representation space of the tensor
233
product of two irreducible complementary series representations %, and xz (as long as - h < ct < к , i= < г which is certainly true if either
(13.23) or (13-2k)-(13.28) take place). If we assume in addition that
(cf. (11.13))
/с I + I c I < к lT?e c, I+- c I < к
(13.31)
which is always true if the convergence conditions (13.23)-(13-28) are satisfied, then we can use the tensor product expansion formula (11.12) with respect to the first two arguments
Г(х,...х„) = fdx	XX )Г’х('х>х3...хп)	(13-32)
For с^С-Дс =0) and Г symmetric with respect to (х^,х^) the sum in (13.42) is over even values of £ only. The conformal partial wave
is, conversely, expressed in terms of Г(хл...хп) by (cf.(ll.l)):
= 1 [dx,[o/xz /fx,.C, ^ZCZ ; Х%)Г(х,...х„) .	(13.33)
In the special case of the 4-point function it follows from conformal invariance that the partial wave (xj x},X^) is again proportional to V:
rx (x;Xj,xM) = J-(x) У(х3,-с3 хм,-с„ > x^)
The conformal ’’Fourier transform’’ y(^) of the 4-point function r’(x1X1X3XIj) also depends on the dimension parameters c^ of the underlying fields, but not on the x’s. The entire space-time dependence of the integrand in (I3.32) is then given by a standard known function of the x’s (the integral in x of the product of two V’s).
Conversely, using (13-33) we can express the conformal partial wave y(^) in terms of Г .We assume at this point that
C,-ct = c3-c^ = 2c_
(13.35)
X,Z Хз*
According to Appendix F.2 the result is
y(x)= ,^l^'Cn'~C3‘”c) fa^f3* fax, fax
2 (2h-2.fe	/ J J	< Wx,£x3V -------’(13.36)
where cik = ^(с^+ск) and the factor is given by Eq. (F.?) of Appendix F.
A similar formula can be obtained by exchanging the roles of (x^) and (x^,x^) . The two expressions are consistent between each other because of the symmetry of r(xl...xlf) with respect to the substitution (*,«,)X^*)5*(XjCj.x^).
The symmetry property (11.8a) (written for the third argument) of the Clebsch-Gordan kernels implies the following relations between conformal
partial waves
Pjf (Х;ХЭ.. ХП) = fayG~ (x-y>ry(y;X3...X„) . y(x)-
(13.37)
(13.38)
234
13. E. Further expansion
Using the results of Sec. 10 one can continue the expansion process (13.32), expanding next the partial wave (xtx3...xn) considered as a function of x , x3 , and so on. In order not to have too many indices we write temporarily r(xy- ...)	in place of
Г^(х,... )	• In the language of Sec. 10, expansion formula (13«32)
reads then ( %* = % = [o,-c ]	, c real)
+too
Г(х,Хг)х3...х„) = Е^ ]’р(Сс)С/срхУ'(х,ХЛ1ХоЛХ)*Г(хХ )x3„.xn')	(13.39a)
-ioo
The only nonvanishing contributions to the sum come from completely symmetric tensor representations Z of The partial waves
r(xx|x3...x„) = 4. px,c/xiV(xlXo’<iXo<xx)r(x1xi, x3...xn)	(13.39b)
V (x % F'fcxl-ja.re vectors in the	. For such vectors
:	> is the scalar product in .
The formula is valid both for even and odd dimension 2h, provided the field dimension cl>j-к . [ Otherwise there is an extra Supplementary series contribution with X=[o,ico] . It may be Called 1-particle reducible and assumed subtracted in Г ; formula (13.39a) is then generally true ].
Conformal invariance implies that also the partial waves are conformal invariant in the sense that, for ge G
rfx'% lx/...х; ) = Г(хх1х3...х„)	(13Л0)
for x / a > x'=	> Vi	’ P = V 3 ) defined by (1 «27а )
235
The expression in [J
is a scalar factor.
Let
,) =
be any test function and define
f(xxj)	rYx%|x3.-X> We make another regularity
assumption which amounts to an a priori constraint on the strength of singularity of the partial wave at x=x^. We assume that
) <oo	(scalar product (10.43b))	(13.41)
This generalizes hypothesis (lj>.29)>
|(x,x3) transforms according to the Kronecker product of the class I complementary series representation x0 with a principal series representation %” [Лс1 .We can again apply the conclusion and expansion formula (10.41) of Sec. 10, for even 2h. The Clebsch-Gordan kernels V are labelled by j,s. However since = О ,
<3 Z - I , and so j = t .We write VS (*X хзХ0 J x X'	in
place of V (хХ,х3Хо	• So the expansion for f- is,
reexpressed in terms of Г’(х%1--)
Г(ХХ |x3--XJ = $dX'X.	хз%»-1 Х'Х')*ГЧХУ% 1х4-х„) (13д2)
with sum over all se. U contained in -t'e for [f'c']	.
The inversion formula is
rS(x'%’x l^-xn) = рхЛ3/s(xx хзХ<>; x\') r(xX lx3--xn)	(13.43)
Eq. (13.42) may be inserted into expansion (13.39a) for the vertex-function. The procedure may then be repeated, i.e. we expand next X X Iх*  ) considered as a function of x' and x^, and so on. At each step the number of x-space variables in the conformal partial wave (cpw.) is reduced by one.
We can also start expanding at the other end. The complex conjugate Г’(хп--*,) has the same conformal transformation
236
law as Г (x, ...xn) because D^»(man) = I a I ^+c° = fa|”Z^+0' is real. So we may apply expansion formula (13.39) to it. Taking the complex conjugate again we obtain a formula of the form
Г(х,-Х„) - px Г(х,...хп_г|хХ)* ^('хп.,Х*х„хГ;*х')
о and the conformal partial wave in this case is a "row vector" in V and has covariance property
{Da°Cp,)  3)*’(p„.2)} r (*,’ *n'-z 1х'х)*Э%)* = r(*, -xn-jxx)* (13.40’ )
(same notation as in (13.40))
The expansion process may be continued step by step, much as before. The expansion processes starting at each end may be combined.
At each step in the expansion process, the number of x-space variables in the cpw. is reduced by one. After sufficiently many steps it will then depend on only two such variables, say x and x’. It will be conformal invariant in the sense that
Э* (?(x.9 >) r (з”х X Х'з”х') DX(?(*'.3)) “ r’(xX-x'x') • (Dots... indicate dependence on further % -variables). It must then vanish unless x and x' are equivalent, and then it is uniquely determined up to normalization by conformal invariance.
Because integrations	could be taken over equivalence
classes of unitary representations * we may without loss of generality put
r(xx ... y'x') oc £ ^(x,x')S(x-x') with ^-function S(x.%')’0 unless x' and ]£olx S (%,x') = 1
In conclusion we find
* As it stands, every one of them is counted twice because Х-У.' but only the sum of contributions from equivalent representations is relevant.
237
Theorem lj.1. The vertex function admits an expansion of the
following form, valid for all integers m in the range
7	7	s"-<	(13.44a)
Jc/x V(xo...xm , X	V'(Xr,-xmM ;xXn-I,Gi-iSn-z"X"'Sm)
where V(—) resp. '/(') are column vectors resp. row vectors in (for %m = [im , cm j ) given in terms of Clebsch-Gordan kernels (theorem 10.1) by
и(хо...хт7 xX4x2sx...%msm) =^n'y1-c(yn, V т(хтхоут.дт., 3 ymXm) •
s. , ~	~	.	(13.44b)
 V (xuXoy, X, J ylXl)'z(x(,XoX1xoi y,X,) 
Every factor herein is a matrix, except the last which is a vector in V; the matrices are to be multiplied in the indicated order.
* denotes hermiteans conjugation of V . Summation over	ranges
over UIR's of U contained both in and	. I
All the dynamical information is in the complex function of X - Хпч ’ and	• Factors- </|i.).((/ims.)' were
absorbed into it.
By obvious modifications of the procedures which lead to Eqs. (13.39) and (13-44) many other expansion formulae can be obtained. For instance
х_?7Л_	= ^X,	Vfi (x3Xo XxXos УгХг )	(13.45)
x; *i	• \/у(х5хох<хозу3хз)г;^(у1х,у2хгу3%э)с/у/уг^у3
Indices a label a basis in l7^(, etc. The partial waves Г (у,%, '/гХгУ3Х3) herein are conformal invariant 3~P°int functions, and therefore linear combinations (with dynamically determined proportionality factors) of the CG-kernels of theorem 10.1.
In applications it is often convenient to use expansion formulae not for the (totally 1-particle irreducible) vertex functions, but for Green functions. They are obtained by first taking out the 1-particle reducible parts in those channels in which one wants to expand, and then proceding as above. We leave the details to the reader.
238
14. Implications of the dynamical equations. Pole structure of conformal partial waves
14.A. Poles in the conformal partial waves implied by the vertex bootstrap equations
We shall start with a brief review of the solution [М2] of the BS equation for the simple <p^ model, and will then extend the results to the more realistic model of Sec. 12.D.
The 1PI amplitudes AK and the BS kernel В satisfy the same covariance and square integrability conditions (with respect to the arguments	as the proper vertex functions Г . We can there-
fore apply Eqs. (13.32), (13.3^) to these functions:
(«Л > хзх*) =	®(X)	> ХзМ	(14.1)
В (х,хг j Mu) =	& (X~) fy > x3xv)
where
f^(x,xtix3x^) = px	(14.3)
and a(x ), b(% ) and F^ also depend on the dimension parameters
*
c^ of the fields . Using the orthonormalization condition (11.9), we reduce the BS equation for the <p^-model of Sec. 12 to the simple algebraic equation
a (%) » <kx> +	) a^X)
for the conformal partial waves. It implies that the partial wave
amplitude
6 M
(14.5)
has a pole for x = X? for which	1
Using the relation
a.(x') = У (%) +	(x~>
(14.6)
where ad (x) is the partial wave of the sum of 1-particle reducible
diagrams	,	2
/	\ +	4	(14.7)
3х	'44	' з
, The function a (x) is related to the partial waves g(x) = g (’X-.ci'l
(. i= 1,2,3,4) of the unamputated Green function used in ref. [М2] by a (%)	=	A similar relation holds for the corresponding BS kernels.
239
which has no singularities in the (12)-channel, we conclude that x(x> shares the poles of a(x) •
On the other hand, the bootstrap equation (12.14) for the vertex function (lj.l?) and the analytic continuation to real c of the orthonormality relation (11.9), imply
g V (*, ,-c . хг ,-c ; x3,-c) - ^(x„) Vfa )Xt.-c ; *31- c) for c
A similar relation follows from Eq. (12.19). Thus,
^(%o)«< f°r %o=[o. "C1	( c = c9> )
$ (Xa.) ’ 1 i°r	। k ]
so that a(x) and y(x> do have poles for % = Xo and X’Xt •
Now we shall demonstrate that the same mechanism also works in the more complicated model of Sec. 12.D.
Let us consider the set of Green functions A4L with total charge zero in the channel (1 2). We shall use the following shorthand notation for the corresponding partial waves;
[= a ( </></>*-» (pcp*^ ) ]
a<f3 ’ a3(t, [= a (BB -* <№*> К ) 1
I = « (BB -* BB ; X ) 1 and similarly for the BS amplitudes. The BS equations are reduced to a system of algebraic equations whose solution is given by
n ж	.	a -	a _ 1 " ^><tV	.
----1	’	-----Д ’ BB-----------Д-----1
Л = A(x) =	-гД
On the other hand the bootstrap equations for the vertex functions (12.2?) and (12.25) give
5, =	(Ш
8° = ^зв (Xb^o + fXB)
The condition that the system (1A-.1J) has a nontrivial solution with respect to the "coupling constants" gQ and g^ leads to the
(14.8)
(14.9a)
(14.9b)
(14.10a)
(14.10b)
(14.10c)
(14.11)
(14.12)
(14.13)
240
equation
Д (''Y.'l = 0
(14.14)
and thus imply the existence of a pole for %= Xg of the amplitudes (14.11).
Similarly, starting from the equations for the J-point functions (12.26), (12.27), which involve the stress energy tensor, we obtain
Л(%г) = 0	(14.15)
for given by (14.9b). Finally, the bootstrap equation for the current-field J-point function (12.28) and the vanishing of	(x3)^a
give
^^,)=O	,	(%,)= <	<=* Д fx,) “ °	(14.16a)
for
X, « [<>	] 	(14.16b)
14.B. Pole structure of the n-point partial waves. Expression for the residues
In this subsection we shall spell out the implications of the dynamical equations for the simplest (<p^) model only. The extension of the results to the (q>,q>* B) model of Sec. 12.D which uses (14.11-14.16),is quite straightforward.
First of all we shall demonstrate that the poles of yfy) (and a (%)	,
corresponding to the points (14.9), are also poles of the n-point partial waves Гу (x, x3 ... x„) (lJ.JJ) for all n 9 4 • We deduce this statement in two steps. It is true, if we replace Гу by the 1L-partial wave
A* (x5 x3	i	xx) A^(x,Xlix3...xn) (14a7)
Indeed,taking the conformal Fourier transform of Eq. (12.12) we obtain
[1- I (%)] A,. (x;x3...xn) = A* (x; X3-..Xn) + £ (% ) X	, X% )Д	fa- X^
+ L £ Г-ру,-^ ^з;(х;у,...ук)б(у|-у,')-б’(у(е-ук/)А(у;.х3...) A(yk'xlk-) Ц4.Х8)
It follows that for each X = for which
(14.19) (and the right hand side of (14.18) does not vanish) the partial wave A1.
241
must have a pole. This is true, in particular, for	and	,
because of (14.9). It remains to show (as a second step) that the conformal Fourier transforms of the proper vertex functions also have poles in these points. That follows from the observation that the difference between AiL and is given by a convergent skeleton diagram, provided that satisfies (13.23) (see Sec. 13.C and ref. [D2])«
We shall assume at this point that we are dealing with simple poles only, so that
4^1 i.
1 dc	(14.20)
The conformal expansion of the 2i-kernel in the dynamical equations (12.13) leads to the relation
r(x; X3 ...Xn ) - 3A%2° (x; x3 ...X„)	(14.21)
Noting the relation between the amputated Green function A and the proper vertex function Г and combining (14.18), (14.9a), (14.20), (12.12) and (14.21) we can express the residue of A^i (or Г* ) at the pole К = %0 in terms of the amputated Green function A:
-o(> T2es 4* (x: x3...xn) =	' Гх (x,x4,-x„> ’ A(x,x3,...,xn’>	(2.4.22)
2 » x=Xo	°
Similarly, the residue of (or ) at in terms of the amputated (n-l)-point function involves the stress energy tensor
X= can be expressed
Ацу(х;х^.. .xn), which
14.C. Basic conformal covariant tensor fields. Analyticity assumption
The proceeding argument can be generalized as follows.
Let <^(x,j) be a conformal covariant rank -I tensor-field of dimension h+c^ for which the 3-point function
< T(p(x<) </з*(хг) Oj (х,3)>о 4=4- 6jx,.Xi!»})
does not vanish. (In writing O^(x,j) in Minkowski space, we can regard
242
j as a real light vector,—cf. [Тб]). Then the conformal partial waves (x; х^...хд) have a pole for	. The argument
is the same as before.
Let q>(x) be a free О-mass field; in other words let <p(x) satisfy the D'Alambert equation
V Cp(x) s	» 0	(14.24)
and the canonical commutation relations. Consider the bi-local operator
V*	<?*(*,> «И*»')	(14.25a)
where (a, a; b, S) is the polynomial (11.5), and is a normalization constant (j7 = J0!7-,} In the simple case at hand (in which a=b=h-l) Dcan be expressed in terms of a Gegenbauer polynomial:
э, ...	(lt.25b)
(see, e.g., [G7] Eqs. 8.932 and 8.962.4; we have again used the shorthand notation (a)^ of (5.H))» We note that for the physically interesting case of 4-dimensional space-time h-1 = 2h-3 = 1 and
coincides with the Legendre polynomial. The relevant property of the polynomials for our present purposes is given by the following differential relations, valid for fT’1 = 0 = 571 which can be assumed because of (14.24) :
(h-i ,317 ;	= фД-я)2^., (*>-’ ,3^ >	) Vz	(14.26)
» - (V£D)I)€ (h-1,3^ > h-1 -3^ }
where D is the interior differentiation (A.47), (A.49) on the light cone. Eq. (14.26) implies that the local operator
0- (x,x ) « : О, (4 X i 3 ) : =	(*i -XX > 3 ) " < (XrXi > 3 I
£ k J	J	<	*	(14.27)
is a conserved tensor-current:
(Э-7) O{ (x,3) - 0
(14.28)
Moreover, it is a (weakly [Нб]) conformal covariant basic tensor (in the
243
sense of ref.[T4]). We recall that for a basic tensor Oj (x ) the infinitesimal generators of special conformal transformation vanish at x=0. This means that the Euclidean counterpart of 0^ transforms under an elementary representation of type JQ of 0*(2Ь+1,1). In fact, for any vector ISfi with a finite energy the function (x, j ) =	1 0( (x,j ) I о >
can be continued analytically into the extended (complex) tube (see[B5J)? its restriction to Euclidean x transforms according to the irreducible part of the "canonical representation"
(It belongs to the space ,	—of. Sec. 6. We caution the reader
that the gradient of a basic tensor is in general not a basic tensor— cf. [TA-]). If <p(x) is a complex field ( (? + (f* ) then the J-point function (14.2J) does not vanish for any of the operators ( 14.27). Choosing the normalization constant = e/h-1, where e is the charge carried by <p* , we can identify 0^ with the electromagnetic current:
0, (*.j) =	i	= ie [<j₽*(x>(^u(f’(x^ ' (Р(х)(^<)>*(х1) 3	(14.JO)
If <p is a neutral field (q> = <p* ), then the fields O^fx.j) with I odd vanish. Setting 2h(2h-l)	= 1	we obtain as a special case the
(traceless) stress-energy tensor for ( = 2:
Ог(х.з) "
Qu/x) =	:	<p(*) :
(14.Jia)

(14.Jib)
We shall take the case of a free field as a guide concerning the set of basic tensor fields coupled to <₽ in general. We shall assume, in particular, in the case of the (neutral) model of Sec. 12, that for each even i. there exists at least one basic "composite" field for which the J-point function (14.2j) with <p does not vanish. There is no reason to believe that for Я >/ 4 the dimension of the field in a non-trivial, interacting theory— is canonical (i.e., that is given by (14.2 9)). However, positivity of the 2-poin-t Wightman function of 0/ implies that if	is the	0* (2h+l,l^representation
label for Of , then

(14.J2)
244
(see [B2, F2 ] and Sec. 5.D).
Thus, the dynamical equations and our assumption about the set of composite basic fields imply the existence of a denumerable infinity of
poles	in the conformal partial waves, satisfying (14.32). It is
natural to conjecture that these are the only singularities of Г* in the right half plane c . More precisely, we shall postulate that у (x)
and
•xn) are meromorphic functions of
plane with simple poles, restricted to the real
c in the right half c axis. We remark that
unlike a similarly sounding ansatz about the singularity structure of
scattering amplitudes in the complex angular momentum plane, this postulate is not in conflict with (off-shell) unitarity, since the
dynamical equations are taken exactly into account.
15. Derivation of an operator product expansion for vacuum expectation values
15.A. Another form of the conformal expansion, involving a Minkowski momentum space integral. The Q-kernels.
In order to exploit the postulate about the meromorphic structure of Г it would be natural to try to close the integration path in the representation (13»32) in the right half plane c and then apply the residue theorem. In doing that, however, one encounters the problem of the asymptotic behavior in c of the integrand. A straightforward way to analyze it consists in performing a partial Fourier transform of the vertex functions. To this end we replace the x-space integration in (13.32) by
r(^...xn) =	(15.1)
where G^Cp) is the 2-point Green function (13.13) and
^±(xtxz i'?) = /У(х<-*с<	> *3 )eLP%3 c/x3
-	; C_ . 3^ )
jdu Ji(u)e Cp(uX|‘tX1)
О
244
(see [B2, F2 ] and Sec. 5.D).
Thus, the dynamical equations and our assumption about the set of
composite basic fields imply the existence of a denumerable infinity of
poles	in the conformal partial waves, satisfying (14.32). It is
natural to conjecture that these are the only singularities of Г* in the right half plane c . More precisely, we shall postulate that у (x)
and f^(xjx^...хц) are meromorphic functions of plane with simple poles, restricted to the real
c in the right half c axis. We remark that
unlike a similarly sounding ansatz about the singularity structure of
scattering amplitudes in the complex angular momentum plane, this postulate is not in conflict with (off-shell) unitarity, since the dynamical equations are taken exactly into account.
15	. Derivation of an operator product expansion for vacuum expectation values
15	.A. Another form of the conformal expansion, involving a Minkowski momentum space integral. The Q-kernels.
In order to exploit the postulate about the meromorphic structure of Г it would be natural to try to close the integration path in the representation (13»32) in the right half plane c and then apply the residue theorem. In doing that, however, one encounters the problem of the asymptotic behavior in c of the integrand. A straightforward way to analyze it consists in performing a partial Fourier transform of the vertex functions. To this end we replace the x-space integration in (13.32) by
r(^...xn) =	(15.1)
where (p> is the 2-point Green function (13.13) and
^±(xtxz i'?) = /У(х<-*с<	> *3 )eLP%3 c/x3
-	; C_ .
jdu Ji(u)e Cp(uX|‘tX1) ^+c(fuO-u)xI7) О
245
where D is the ^-th order differential operator (11.5) and A£" is given by Eq. (p.4; of Appendix F.1 , where the last equation is derived. For large £ it is not obvious that the integral in u in (15.2) makes sense. In order to write (15.1) in a manifestly meaningful form, which will be at the same time convenient for studying the asymptotic properties of conformal partial waves in c, we shall assume that
(T1<0j <Гг<0 ; ^>0,..., (Tn >0	, where Oj -	,	(15.5)
x. £ x for / к ,
and shall shift the integration path in the complex ?2^- plane to a contour C around the negative imaginary semi-axis (see Fig.3). In the domain (15*5) the exponential in the right hand side of (15.2; decreases for p2h~* -i°° » so does also f^-(p ;x^,... .xn) because of the spectaal condition. Hence, the shift of the complex energy integration path indicated above is indeed legitimate, and we are led to evaluate the discontinuity
= oLibC [	rp) (f%(p) l(p', X3,-,Xn)],	(15.4a)
^5cf(p) z 0(pa) Um [^p, -ip04]	(15.4b;
We can directly compute (15-4; in the case n=4 using the explicit expression (15.2; for the vertex function. To handle the general case we shall assume that the partial wave for all n^4 has the form
*>,,:>*) = fpp;<	Cu-s)
* ftp; *>>,*.)),
.	gw,
where the functions 4 (b) and t (p) satisfy
246
disc ^(p) = 0= disc^%(р).
This assumption can be justified in the framework of the skeleton perturbation theory by writing
zv	r f	fi
Г*(р; *»>*») - JpxVx* I/
where S(x7,хй, x^,....xn) is a sum of skeleton diagrams. To this end it suffices to express Kv from the identity
2slnlCvKf(t) -XLI.yW -Iv(z)] ,
(see, e.g., Sec. 8.4 of ref. [G?] ) in the representation (15-2) for that gives
i

£ % -
[ Q.i± OWrp) -
where Q^CG^Qp ) is obtained from (15.2) by replacing 2K^ with
“I(+c ~T-t-c •* *	2	2	2	2	2
For p2h -ipoi 0 we have p2-* -p2 ± iOpQ where p2 =pf -p2
=6Л,?)-То evaluate discG^, we use the relations
n^s.s')	$„^)r	<15-7W
(for p2h-»-ip0)
From (15.15) and (15-7) we obtain
disc	}	(15.8)
247
where
с 1 h
WAp)=0+(p)(lP2)CZ ^(с)П (p) } p~?M> Q(P)~0(f>c)&(P3)	(15.9)
Л	J=a
is the Wightman function, normalized by
Wx(p)^(W = G4(p)1 > \^(p)^fp)^(p^ ^(p)	O5-9)
Now we are ready to evaluate A in (15.4); using (15.5), (15.8) and (4.11), we obtain
A = 2ir [fa, хъ“Д tpj	4- , *J -
"Q<- fa^-РЛ) ^М7(Р,Л;Х},..^ >	О5И0)
t PC ./	л
In order to relate the functions and	, and	and
Qj, between themselves we shall use the symmetry property (15*57). First we notice that, according to (15.5),
^сГ^(Р,Х},-.^) = 2l(f-if.; *b ~,xj _	(15.11a)
Taking into account (15.57) we find on the other hand
Z JC Q (p; Xj,xj =	(f) Г% (f; x3/	X,) =
₽-Л'з<иMe;^(Pt .CPi.Kj	(15.11b)
It follows that	jr
and, in particular,
уХ(^)^еЛ)аЦКК1.) t (15И2Ь)
248
where we shall use the following expression for Q +	:
4	'«'	к Гм z
•/Z wr'7’"'" “J	(',5-',2=)
0
and the scalar product in the exponent is *PM = XP -	£
Eqs. (15.11) and (15-6), (15*12) suggest to define the functions Q* (P i x^,...^) by
~ ^SC	-'*>),	(15.13a)
**	'	^nL
or equivalently (due to (15.9 )	)
0+~ disc Q (f; V, **) •	(15.13b)
Eqs. (15.Ю), (15.12) in the above notation and the symmetry of the range of c-integration in (15.1) allow us to derive the following representation for the proper vertex function
= - f	(P„) Qr(fM>  ,*.).	(15.14)
г<*
In the special case n=4, using (13-34) we can rewrite (15.14) in the form	_
r(Kw.) - -w'f м		(15.W'
J(bF/i$c2
Unlike Eq. (15*2) which, as noted, involves a divergent integral
<in u) for / It/2 (.and any c) the integral in the right hand side
j?
of (15.12), which defines the functions Q + , is absolutely convergent for
t+L + Ree > 2/<_/.	(15.15)
Moreover, the representation (15.14*) allows to exploit the analytic
249
structure of у (%) by deforming the integration path in the right half plane c . That possibility is secured by the following character-istic properties of Q+ .
(i)	Q± (x, хг >-p j ) is an entire analytic funtion of p (provided that the integral in u converges, which is certainly true in the range (15.15)).
(ii)	For time-like vectors p, Q* decreases exponentially for Re c -* m • (That property, which will enable us to close the integration path in c, follows from the known asymptotic behavior of the Bessel function J (x) for »-»«> , - see, e.g. [G7] Eq. 8.452.1).
(iii)	For small x12, Q - 2S S2Ven by
Оме
It is natural to assume that in the general case n ? 4 the kernel Qr , given by (15.13) satisfies the properties (i) and (ii). This assumption can be justified again in the framework of the skeleton perturbation theory.
15.B, The vacuum operator product expansion
Now it is legitimate to close the contour of c integration in (15.14) in the right half plane,assuming the partial waves are sufficiently well behaved at infinity. However, transforming the integral over
Vx into an integral over Qx (which vanishes for Re c -» °° ) we have paid a certain price: the appearance of the factor [sin it (€ + c)]”^ which introduces new "kinematical" poles. The main purpose of this subsection is to demonstrate that these poles are actually cancelled out.
First of all, we note that a finite number of poles coming from the sine factor are cancelled by the zeros of the Plancherel weight (11.11) for
c - 0 ... ,4- 2 , h + f - 1
(15.17)
(At this point we assume, for the sake of simplicity, that h is a positive integer, which includes the cases h=2 and h=3 we are primarily interested in. The argument—and the result—can also be extended to the case
250
of half odd integer h). There remain two (infinite) sequences of "kine-matical" poles to be dealt with; they correspond to the elementary representations [Лс] with labels
C * h.-lt С + v , Z = 0,1, 2,...	, v = 1, г , ...
and
c'= к-1+г	, /'= t+v ({- 0,1,1, .... , v = 1,2 , ...)
which satisfy
c'+t1 = ctl = /1-1 + 2/+У
(15.18a)
(15.18b)
(15.19)
The clue to the cancellation problem lies in the partial equivalence of the representations
= [VI (= [W**] ) ancl	(15,20)
exhibited in Sec. 6.B. It leads, in particular, to the following identities among Q-kernels and conformal partial waves (see Appendix G):
(tp3r	,-Р,3) ~	+ с.\]о>('хЛ.,-р,з),	(15.21)
(P,3ix3,...,xn)	(15.22)
where
(15.20’)
Furthermore, we shall use the relation
f(^oj'7?3	(?;<v (15.23)
which follows from (6.32)(15.9) and the identity
р€+Л-1+<)= ~	(15.24)
which is a direct consequence of the definition (11.11) of the Plancherel weight.
Now we are ready to prove that the sum of the residue s in the kinematical poles X(v and 'x'e* vanishes.
Indeed, due to (15.19), for both these poles
251
тг Т?и [Ми т(4с) ]'1 - т1?м [Амт(Лс')]"1 = (-1	1+1/	(15.25)
and we have (according to (15.21 )-(15.24 ) )
[(£+»)! ]'ip(x^)Q^',(xi'xx> -P ’V (p> S A ) Qr^fp-S .xir 'xnb-
AZ **
=	,^)Qr^4pJix3.-,^) • (3.5,26)
This proves the cancellation of the kinematical poles coming from [sin u (f+c)] 1 .
Let us note that in (15.26), Wv’t-(-p) is proportional to the Min-A
kowski space analogue of the kernel (6.28), which defines the hermitean form (6.26). This function is given by a formula similar to (6.231):
Vx;^p)'(-£W*;+)	(6.23-)
Zv	c
Taking into account that the singularity of the integrand for x= % f У
is in fact a double pole, we are led to use Eq. (6.2311) which yields the left hand side of (15.26).
Thus, closing the contour of integration in (15.14) in the right half-plane c we obtain a representation of the proper vertex function as a sum over dynamical poles and poles coming from the normalization factor of Q_x only:
Г (x,x„) =
, p (0 г X	%	1	(15.27)
= 2tE	. -e — J(rfp) Q- (*,Л2;-р)^(р)<Эг (pi хз - хп))
A similar expansion can be deduced from here for the full Green function (see subsection 15»C below). It can be regarded as the result of inserting a conformal covariant operator product expansion (of the type considered in refs. [Вб, G9, Fl, F3, F4, F5, M5, S3]) in the Wightman functions (which is subsequently continued analytically in the Euclidean region). It is, however, important for our derivation that the operator product q> (x^) <p (x2) (which is effectively decomposed) acts directly on the vacuum. (That was used in exploiting the inequalities (15.3) for the Euclidean time-components). It is indicated in ref. [S4] that the general (global) operator product expansion is more complicated. That is why we adopt the term ^vacuum expansion" (of ref. fs4] ) for the situation envisaged here.
252
Remark. It follows from (15.21), (13.34) and (15.22) (for n = 4) that
(15.28a)
as a consequence, (according to Eq.[G3])
for О О .	(15.28b)
On the other hand, if c_= 0 and <p^ (x) = <p^ (x) = 4> (x), then у () = 0 for odd t , and if у (0	f°r even € , Eq. (15.28b)
cannot hold for odd у .
Similarly, the limit of the Clebsch-Gordan kernel V for c.-» 0 does not commute with the one for % -»	• This complicates the treatment of the points	in which the partial waves Q* (p,j ; x1xu')
also have a pole (for c_= 0). The result can also be extended to that case by an appropriate cancellation of the higher order poles of the
integrand.
253
1$.C. Wightman positivity for the 4-point function
The representation (15.27) is particularly convenient in analyzing the positivity properties of the 4-point Wightman function (cf.[M5, 02D. It is, of course, the full 4-point (Wightman or Schwinger) function that exhibits positivity, and not just the proper vertex function Г .So, our first task will be to write down the counterpart of (15.27) for the Schwinger function
s (x, .x,, x3 ,	(x,) <£г(хг> <£, (4’(4 ) >0
where <£, and </>г are spinless Euclidean fields with dimension parameters c^ and c^ (we can have <£, =	as a special case).
Let us assume that there is a scalar (Euclidean) field (x) with c < 0 such that the 3-point function < (x,)4> (xx) <£ (x.) > does not vanish. (For the <p^ model of Sec. 1 we would have 4>, *	'• 4>3
cl=c2=cj)• Then, the "shadow pole" [151
non-amputated li-function
for c
"C3’
Gli	has a
which is cancelled by a singularity
of the 1-particle reducible Green function [М2] . On the other hand,
according to (11.8), (13.33)t (13*34) the conformal partial wave so(%) of the disconnected Schwinger function
^(х,-хэ15«(х1-х1»’* S1Z (x,-xif)s« (X1'X3)	is 1 • That gives
S(X„X2>X3,X4> = S,2 (X,-XX)S):1 (X3-X4) +

where g(%)is the conformal partial wave of Gli(x1,x2;x^,x^) and the sum is over all poles of A£(c+.c- 5c)['l+3<%)] in the right half plane c except the "shadow singularity" for	We note
that for a
generalized free field (for which g(%)”0) the disconnected Green function s« (\'x3)s2i	> + 5,z (*,'*♦	j is reproduced by the poles of Г (к-%* + c+ )
coming from the normalization factors in the Q's. It turns out that for
interacting fields these poles are cancelled by zeros of
1+9 (ОС: c; ) = —V------------
i.e. by poles of -$(х,c,-)
This is suggested
by the analysis of ultraviolet divergences
in the skeleton expansion
of the right-hand side of the equation:
Л(%;С- ) У(х,с3 ,х^с4 ; XX) - f fo/x.fdx^ V(x,.-c,, хг,-сг

(cf. (13.33) (13.34)).
254
Now we are in a position to analyze the implications of the following (special case of) Osterwalder-Schrader positivity condition for the 4-point function [М3]. Consider the space / ’ Д (’r2'1’*	)
of test functions ffx^jX^) of the Schwartz space У’(’RI,|,) which vanish with all their derivatives unless о; > о . <rz > о () and xt xi • Then, for any	, we have
H0xa ,0x,) (x,...x„)f (x3.x4)>0	[	(x ,-<r) J .	(15.30)
Inserting here for s(x.,..,x4) its expansion (15.29) we see that this positivity condition is satisfied, provided that the inequality (14.32) takes place and
- f* P	q (x ) > 0	fl Б 41
ALU TT((+C( )	° '	45.31)
for all dynamical poles ofcjfo), and finally that [ l+g (%)]Т?м r(h+c+-S ) > 0 at the poles of r(h+c+- 8X).
16	. The problem of crossing symmetry. Concluding remarks
16	.A. Crossing symmetry and duality
The vacuum expansion (15.27) or (15.29) of the product q>(x^) ч>(х^) which satisfies the dynamical equations (in the (1,2 )-channel) is not
symmetric with respect to a permutation of the arguments x^ and x^
with any of the arguments x
3
,x . We are stuck here with n
of a familiar problem of ordinary partial wave analysis: it
the analogue simplifies
the unitary equations but complicates the crossing symmetry condition. Yet, it should be stressed that the conformal expansion (as pointed out in ref. [М2] ) solves an infinite set of coupled non-linear (integral)
equations, while the problem of crossing symmetry can be reduced to a set of non-coupled (a finite number for each n) linear (integral) equations for the conformal partial waves.
In order to exhibit these symmetry equations we shall introduce
another bit of graphical notation.
We shall represent the Clebsch-Gordan kernel V by
V (x, >	; xx , -ct ; x , у )
(16.1)
* In the original paper of Osterwalder and Schrader condition (15.30) is only assumed to hold for test functions f(x.,Xg) which vanish unless <r, <	. The stronger form of the positivity condition used
here is a consequence of analysis of Glaser and Mack [G6,M3] .
255
and will write Eq. (13.42), (15.1) in the form
Then the crossing symmetry condition for the special case of the 4-point vertex function assumes the form
In the case of the <p <p* -> <p <p* vertex function for the model considered in Sec. 12.D. we would have (%) = Y)[( (x ' ln the case of the tp^-model all	should be the same:
У,2 (%) = У,3 (x> •	(x) = 2Г (%) fa =	(164)
In order to find the crossing symmetry equation for y(%) we expand the kernel F(x^x^jx^x^) (14.3) in conformal partial waves:
\ (*< .*з ;	 S ) = fa' c (,X''X') rx‘ (X<’X* > хз-х* )	(16.5a)
(16.5b)
An explicit expression for the crossing kernel C(%,%') can be obtained
from the following consequence of the orthonormality relation (11.9):
(16.6)
Multiplying both sides of (16.5) by V(x1c1,x2c2;x / " ) and over x^ and x2 we obtain (after replacing %" by %' )
integrating
C (%, %')
(16.7)
256
From the involutive property of the crossing operation we deduce that
if	- Sfx.x')	(16<8:
Inserting (16.5) into (16.5) in the symmetric case (16.4) we obtain the following linear integral equation for y(%):
y(%)»	y(%')	(16.9
We could have alternatively formulated the crossing symmetry condition as a duality property for the discrete vacuum expansion (15.27) (or (15.29)). To do that we first need to continue both sides to Minkowski space arguments with space like separations (since the inequalities (15.3) for different channels contradict each other). That form of crossing symmetry makes obvious its relation to the local commutativity of the underlying fields. For further discussion of this duality property and its extension see [M5'J •
The difficulty in treating the duality relation of type (16.3) comes from the fact that an approximation of Г involving only a finite number of poles in a given channel would not do. The reason is that the poles of the conformal partial wave in the cross channel are reflected in the divergence of the infinite sum over residues in the direct channel.
16.B, A crossing symmetric representation for the 4-point function
If we forget for a moment about the dynamical equations, it is not difficult to write down a crossing symmetric representation for the conformal partial waves. It can be based on the known Mellin-transform representation of conformal invariant Green functions, proposed by Symanzik [S8] (see also [D2]) and Mansouri [M8p. For instance, the general
conformal invariant 4-point function, with dimensions restricted solely by Eq. (13.45)t can be written in the form
257
where с., =	(c.+c, ) as in (13.46). The right-hand side of (16.10) is
1л 2	1 л
independent of the real parameter о provided that K( z , w) is analytic in a strip domain along the real axes which includes the points
z-tT-i'5 , w = r - i4 and that no poles arising from the x-dependent
factors prevent us from shifting	the integration	path. In	the	<p^ model
under consideration we have cife - c? The representation (5.10) can be	made manifestly	crossing	(16.11) symmetric	
by setting 5=ih-tc^,	and к (<r, t ) = f ( с, r , - o’ where	. ,	f , f (<*>/3, у ) - ( (/3 ,<*, у ) Inserting then (16.10) into Eqs.	-r ) ’	У./3 ) • (13.33) (13.34)	defining	y(x)	(16.12a) (16.12b) we ob-
tain a conformal partial wave (depending on an arbitrary symmetric function f(a,B, у )) which satisfies automatically the crossing symmetry equation (16.9)	• The difficulty now is to construct an f consistent
with the pole structure of iffy) implied by the dynamical equations.
This problem is not yet solved.
16.C. Summary and discussion
Our aim in this Chapter has been to construct a conformal invariant quantum field theory satisfying
(a)	the dynamical equations (12.11) - (12.20) (in a given channel),
(b)	Wightman (or Osterwalder-Schrader) positivity, and
(c)	crossing symmetry.
We were able to solve (a) and to incorporate some consequences of (b) by using the vacuum operator product expansion, which can be written in the form
^(Xx>4’<(x,)lo> » S(x,-xpio> +
ZC(%PpxQj(x,.xtix) 0%z(x)lo> ,	(16
where we have set
(for Minkowski space
coordinates with space-like x12)
fl )
' Using the integration formula of ref. fS8] we obtain:
и. (с,г, с'1, с )&г)1Ь T<=/<* T± ‘k'fe у	r (-*-/*^c„)r^^e)
Г (71-	— с,3+Г4ув)г(А-^^+<г+<Х-т)Г^—^-.гп-о<-о-)г(5.-с13-о<-/3-<г-,г)
258
- { r (h’Sx'c+')r'(ln’	r'(h-§x+C-)r(h-S5i-c-)r(h-S-+c_)) Q*(*, ,хг ,х)
-@i) ’	'	(16.14)
0 and
л(х )= JZlfitlSd т?и [r(h-Sx^+)r(h-S^^e+) (i^fy))
W U-пфС') %-%/ k	J	(16.15)
[Лс] , Sx - |	[l.ct1 , ct =X(c,±cJ
are local (hermitian) tensor fields, whose two-point functions are given by the Fourier transform of (15.13):
< Vx)® %'(x’)>o = \(X'X')S^'S^	(16.16)
In the simplest qr"’ model the sum in (16.13) is over even values of t only and the first dynamical pole of g("X ) for I = 2 comes from the stress-energy tensor. The positivity condition implies the inequality (1A-.32) and the reality of the coefficients C(%^) (16.15).
The sum in (16.13) is over all poles of the expression in square brackets in the right-hand side of (16.15) with positive c^ , except for the scalar shadow pole (c_ = -c in the иЛ model) which is omitted. We U	ф
notice that the above construction automatically insures the positivity of the energy spectrum (cf. [L3j).
The crossing symmetry condition implies a set of uncoupled linear integral equations for the conformal partial waves. The simplest of these equations—for the partial wave у (%) of the 1PI A—point function r(x^...x^)— is given by (16.9) (16.7). It is satisfied by the general crossing symmetric conformal invariant A—point function (16.10-16.12). However, the problem of displaying simultaneously the pole structure, implied by the dynamical equations, and the permutation symmetry, reflecting the local commutativity of the underlying fields, is not solved. We conjecture that in carrying out a construction which takes into account all three requirements (a)(b)(c) one should be able to discard the q>^ model as inconsistent. The difficulty of this constructive problem should justify further study of simple soluble models [DI, S4] from the point of view of global operator product expansions presented here.
259
APPENDIX D
Proof of lemma 10«3»
Let us abbreviate
A(m,z) = Jo/u ^Qmu) (u’*)	(d
Uz
Let z = (1000) and m(z) e M such that ziz/lz|=m(z)z .
From definition (D.l) we deduce the covariance property
A(m,z) = DJ(m(z)) A (m(z)"' m mfz) , z ) DJ (m (z))"'
Because of definition (10.18') of Р*(х)	, the r.h.s. of the equation
of lemma 10.3. has the same covariance property. It suffices then to check it for z = z .
Let us introduce a canonical basis I vj I in	etc.
Here and later on, letters a,b,... label UIR's of the stability group U = Spin (2h-l) of z . Matrix-elements
• 111 articular,
(0.2)
are matrix-elements of UIR's of U and independent of j .
In the following we shall omit indices /* >v .
In this basis m (ja )bc = 8ai) Sac for о,Ь,с C j 
Therefore
AC“
I , ® оль for a c / , a c i	« We must then show that
be Qb Qc
 Id“cX
But
-^A.s. = fda	(m) d (u ) с/° (u"’)
’	^cc fa ) =
because of Schur orthogonality relations for representation functions (a, be j).
260
APPENDIX E
A summation formula involving ratios of Г -functions
Eqs. (11.16) and (C.19) can be derived from the following
known formula for the value of the hypergeometric function F = at the point x = 1:
,	“ Г(а+m )Г (b+m) !*(<:)	_ r(c)r(c-a-b)
; c i 1 ) = r(a) г (Ь)Г(с+т) m (	Г (с-a ) Г (с-b)	(E.l)
(see e.g. [G?J Eq. 9.122.1).
In order to reduce Eq. (11.16) to the form (E.l) we set k-j=m, ^“j=n, and continue to non-integer n , writing it in the form
* Г (m-n ) г («+j + m )	r + n't Г (ix+j )
£0 Г(-п) m ! Г^+jtm) "	П(/3+рп) r f/3-o()	(E.2)
Here, we have used the identity
, m Г(n+<)	r(m-n)
(-1)				-	 r (n- m t f )	- r(-n)	<E-3)
Eq. (C.19) is established in a similar way.
There exists also a direct elementary proof of Eqs. (11.16) and
(C.19) which exploits their similarity to the Newton binomial formula.
In the above notation, Eq. (11.16) assumes the form
f-. (ЕЛ) where a = a+j, b = B+j (ns I -j) and
r(x) ° x(x+N  (x+fe-<)	(E.5)
is the finite-difference counter part of the power x^. In order to
prove (E.4), we evaluate the finite difference fn(a,b) - fn(a,b-l) using
261
(x )fc - (х-1)к = [ х+к-1- (х-1) ] (х)к_т = к(х)к_1 .
Thus, we find the recurrence relation
f (a,b) - f (a,b-l) = nf _(a,b) n ’ n ’	n-1	’
with the initial condition
f1(a,b) = b-a .
In order to fix fn(a,b) uniquely we have to evaluate it particular value of b. For b = a we have
f Jo.a) = (Q) £ Н)” (Д) = (а) (<-<)" = О n	m«0	n
It is easily seen that the only polynomial solution of (E.7-9)
by the right-hand side of (E.4).
To reduce (C.19) to (E.A-) we multiply both sides by (-1)
and obtain (using (E.j>)):
£	(2-R- (-v+m)v_m = (2.k.t.y + s)v ,
m® о
(E.6)
(E.7)
(E.8) for a
(E.9) is given
(E.10)
which is (E.4) with a = -s, b = 2-h-l-у
n
v
262
APPENDIX F
Partial Fourier transform of VCx^Xp^^) and related formulas Pel Fourier transform in xz ----------------------------------------------
To evaluate the Fourier transform of V in the third argument, it is convenient to use the representation (11.4b) involving the differential operator Dg— (11.5).
The calculation of the Fourier transform of V is reduced to
the application of the following known relations:
—f / 2. \A [ i / -ip*, .
J lx,J (Ц/ £ dx5 -
= \du г-'/Ъ \cU Sexf>f-l(U2+ ; D
(F.2)
(see [G7j Eq. 3.471.9). The result is
V? (xltxx;	; x2,c2; xJC.j) e =
= S' С,№хк-,}Ъ'Л-С-.^) (fif'd
X ^du	1(
й
where we have used the equation -l-C and have set
/?£ - =

(F.F)
Л* = A£ (ic+ ,±c_,c) =	*<( (tc^.tc-.c)
r(S%+f+c_)r(5x+f_C_)
A^(±c+,±c ,-c)
263
For Хд*о we can evaluate the main term in (F.l) exactly. The
result depends on the sign of Rec-£ . If V corresponds to a physical
3-point function then the Wightman positivity condition (14.32)
implies that for h > 2 , c - f is non-negative. In this case c- 6
2 c<z) ~ r'Cc'^) (t)	for z “* О and the small distance
behavior of
V * is given
P'3)x^oA^
s (А+СИ)
r(c-t)r(h.c*ty t
by h- Sx-
3}	)eLp*:
4K-Sy*Cj. fl
11
(F.5)
F.2, Derivation of Eq. (I3.36) for the conformal partial wave
We shall first derive Eq. (I3.36) for у (я) for
4?ec»«	(F.6)
and then proceed by analytic continuation.
We start with Eq. (13.33) for n = 4, and after insertion of (13.34) integrate both sides with respect to x . The result is given by (F.5) with p = 0 and c+ -» - c3lf : c_-»-c_ on the left hand side. Noting that (because of (13.35))
= [r’(h-Sx->ca)r(h-£;4-ca)r(h-Sx4e34)r(h-Ssvc34?) g , c ; c ) ^(-c3Vc-;c)'[r(h-Sx-clt)r(h-S5-c,1)r^-Sx-c3k)r(l,.J5;-c^)J° 1 n’ 3* ’	(F.7)
[ Cife - "X (CL + CI« 5 ] we obtain
(З^^Х.ХЛХ,)	(y<8)
Finally, we apply to both sides of (F.8) the operator
<	/ хпЭ У
! (h-1 )g ' x3(,
where D is the interior differentiation (A.47) (A.49) on the complex light-cone,and use (A.13) (n = 2h)
(F.9)
Noting the normalization condition
. 2ki><
(F.10)
264
for the Gegenbauer polynomials we end up with Eq. (13.36).
F.3. Splitting of V'X(x1,x2; p) into two Q functions Using the known relation (see, e.g. [G?] 1 Eq- 8.485) 2^Z>’ -!,(*>]
between modified Bessel functions, we obtain a splitting of into two Q-functions, which have the properties (i) - (iii) of Sec. 15.A. In order to prove Eq. (15.3), (15.4), we will establish the following relation
IZ. (п-Ik
xfc.0<	 (F.ll)
ii	-f '	6
where t is related to the integration variable u in (F.3) and (15.4) by t = 2u-l, and
/	+C" fl-i V1”’’0-
= 1 (V)	(~)	,	(F-12)
xt = ~~ x< + ~T’*2. * -z (xfl-xz) ~i'ixiz	(F.13)
If we assume that (F.ll) is true, then multiplying both sides by / i \c+ + 4
~	and using (F.4) we obtain the counter-
part of (15.3) for (xltx2; -p) replaced by V+ (xltx2;p).
The proof of (F.ll) is rather tricky, since the equality does not hold for the (t-) integrand. We shall only verify it for the leading terms in both sides for x. -*• 0 . The validity of (F.ll) for arbitrary 44	%
and Xj would then follow from the covariance of Q+ under the
semigroup S defined in ref. [ L3] (S consists of those transformations of 0 + (2h+l,l) which leave the sign of the Euclidean time component x2h invariant).
To find the small	behavior of each side of (F.ll) we use
the power series expansion
265
Iy(z)
fe!r(v+fe+()
(F.14)
of the Bessel function and the relations

1	(-nf i1 ^ЛУ2РЗР-х,лгрмл-г), pS-xu >
= fh-c-<)£'2- 7 \ p2 / i ' ' p-J p xj
A'c- -3^) [tfA'ipXt] -+
= Z7h+e.ck(ip3/'fc6xll3)fee'L'pX+Vfek’^>(^ ♦	,	(F>.
<X* = Ste+k-1±C_ , fe- 0,-f.t .
Here P^a,B\t) is the Jacobi polynomial; Eq. (F.17) is a consequence of (13.13)	(b)	) ; in deriving (F.18) we used the following relation
between the Jacobi polynomials and the hypergeometric function
= (-t)" ^3-" F(n + tf+/3+i , -n j /3+4; 1^)
(F.19)
(see Eq. 8.962.1 of ref. [G?]). Using further the integration formula (we caution the reader that the corresponding equation 7.391.3 of ref. [G7] contains an error):
1И(тГ(тГ'₽Г/!^)';В^П+,’/3+"И) {or Р = И (F.20a)
a 0 {or v = 0, ... , и - i ,
(F.20b)
we obtain the following small	expression for the right-hand
side of (F.ll)
266
./А \^'С> (£l)C£ (£)	f2M ?Х|Л \ х ,>	(г.21)
W 121 to r(c+fc.f+<) l-V-) (	3 
It follows from (F.15-17) and from the identity
,1	(Cfl,h-2)
< 	(oo ) я (c + 1~ I ) Г (A+C-1 J - t у C + F - I. j —~	)
( w= 1.	)	<r-22)
k	?3 -px.i 1
(see Eq. 8.962.2 of ref. [G?I ) that the left-hand side of (F.ll) is also given by (F.21) in the small	limit. This completes our
proof of the representation (15.3).
267
APPENDIX G
Identities between Q and x functions for partially equivalent representations
Eq. (15.21) is equivalent to (11.24). To see this note that
= ty№)lv (Л	(G.d
because rTj= p.
Now it remains to prove
where S^z. —~t is given in (11.26). But thia is in fact what we have proved in the derivation of (11.24) (with G-»-£*) noting
(G‘5)
Passing in (13.33) to pseudo-eucltdean p —*pM (cf. 15.A) and using the definition (15.13) for the kernels Q*({э;х^,... ,xn) and Eq. (15.9х) we obtain
Qr Ли, лз,"-*«) = JIQ.
This equation together with (15.21) leads to (15.22) (in the same way, as we derived (11.2^) )•
In the special case n*4 using (13.J^) and comparing (.15.21) and (15.22) we find
«-5)
We note that for c_=0 Eq. (G.2) as well as (15.22) require a modification, since
~q for odd V,	(G«6)
and the sign function in the above formulas is not defined (cf. the remark at the end of Sec. 15.B).
268
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FIG. I
Figure 1. Quartet diagram of intertwining operators at exceptional
points.
279
FIG.2
Figure 2. The quartet diagram for the Lorentz group (h = 1) and
1 > 0.
280
FIG.3
Figure 3
Deformation of the integration path in the complex energy-plane (Sec. 15.A). Original paths the real Pg^-axisJ deformed path: the contour C.
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