Hations Educational Scientific and Cultural Organization INTERNATIONAL CEHTRE FOR THEORETICAL PHYSICS

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1 r ^ IC/7V6T / y REFERENCE (Limited INTERNAL distribution) REPORT t-~ : i t,.ul i-j' "! -,., \ct- \,- International Atomic Energy Agency and Hations Educational Scientific and Cultural Organization INTERNATIONAL CEHTRE FOR THEORETICAL PHYSICS CONFORMAL EXPANSION FOR EUCLIDEAN GREEN FUNCTIONS * I.T. Todorov ** International Centre for Theoretical Physics, Trieste, Italy. CONTENTS INTRODUCTION AND SUMMARY 1. A FAMILY OF ELEMENTARY REPRESENTATIONS OF 0 T (2h + 1,1) 2. INVARIANT BILINEAR FORMS AND INTERTWINING OPERATORS 3. THE CLEBSCH-GORDAN'EXPANSION 1*. CONFORMAL PARTIAL WAVE EXPANSION IN A SCALAR FIELD THEORY MODEL REFERENCES MIRAMARE - TRIESTE July * * Talk presented at the Third International Colloquium on Group Theoretical Methods in Physics, Marseille, June 197^; to appear in the Proceedings. ** On leave from the Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria.

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3 INTRODUCTION AMD SUMMARY The equations of motion, of a renormalized quantum field theory can "be written in a form which distinguishes dynamical equations and boundary (or initial) conditions. The dynamical equations are an infinite set of coupled integro-differential equations for the connected Green functions; they involve no parameters (like masses or coupling constants). The "initial" conditions fix the values of the inverse propagator and its derivative as well as that of the 3-point vertex function at some (finite) points. It was observed recently (see Ref.3) that the dynamical equations are invariant under the action of the conformal group of space time, while the boundary conditions are only Poincare invariant. The conformal invariant solution of the dynamical equations (provided that it exists) should be relevant for the small distance behaviour of a realistic quantum field theory. Such a solution is being constructed by using conformal partial wave expansion of Euclidean Green functions (it does not, however, fully account for the permutation symmetry of the problem). The simplest model of a cubic interaction in a 2h-dimensional space-time is considered in Ref.3. Mathematically, the problem is that of constructing explicitly the tensor product decomposition of two irreducible representations of the supplementary series of 0(2h + l,l). (It has been solved for the ordinary Lorentz group (i.e. for h - l) by Naimark.) The solution for the general case, reviewed in this lecture, is obtained in Ref.U, which also contains a bibliography on related work. 1. A FAMILY OF ELEMENTARY REPRESENTATIONS OF 0*(2h + 1,1) The Harish-Chandra construction of induced representations of real semi-simple Lie groups (see e.g. Ref.5) reduces to the following prescription in the case of G = 0 T (2h + 1,1) (the arrow indicates that we exclude reflections of the 2h + 2 axis). Consider the Iwasawa decomposition G * KAN, where K =.0(2h + l) is the maximal compact subgroup of G, A is the subgroup of pseudorotations in the plane (2h + l,2h+2), which play the role of dilatations in the 2h-dimensional space, N is the 2h-parameter abelian nilpotent subgroup of special conformal transformations. Let M = 0(2h) be the centralizer of A in K. Every elementary representation of G is induced by a finite-dimensional irreducible representation of the subgroup MAS. It is labelled by h integers i^,...x (which specify the representation of M 0(2h)) and by a complex number c (fixing the 1-dimensional -2-

4 representation of A); the subgroup N of the inducing group is represented trivially. The corresponding induced representation acts on tensor-valued functions on the homogeneous space G/MAN homeomorphic to the sphere S 1 2h We shall use Euclidean co-ordinates x = (x,...,x ) related to the points of S via stereographic projection. (MAN will then "be the little group of the point x = 0.} We shall "be interested here only in a 2-parameter subset (of the {h + l)-parameter family) of elementary representations X- [t t c] {%= 0,1,...,c e(c) for which the functions f(x) = f (x) are symmetric traceless tensors. Such tensors are in one-to-one correspondence with homogeneous polynomial functions on the complex light cone: The representation "X s L»c! of 0{2h + 1,1) is defined by the following (local) substitution rules for f(x,z) under generic conformal transformations : a) Euclidean ("Poincare") transformations t>) dilatations c) conformal inversion where tlg3b)

5 (1.3c) We shall assume that the f's belong to the space CL^ of infinitely differentiable {in x) functions which admit for x-+t» an asymptotic expansion of the form oo where H.«is a homogeneous polynomial of degree k in the first argument and of degree X in the second,. It is easy to verify that the space C^. > so defined is invariant under the Euclidean conforms! transformations (1.2). The representations "X are irreducible,except for the following four types of integer points: yj 9 a.?) The space C _ contains a finite-dimensional invariant subspace E^ of polynomials P(x,z)eC satisfying the equation The O 1 '(2h + 1,1) representation acting in the factor space C /E o is X An n equivalent to the one acting in the (infinite-dimensional) invariant subspace F^n of Cy,. of (tensor-valued) funct ions f (x), such that all _ ^ " -.7) Similarly, the space C,_ has an invariant subspace D7 of functions of Xp_ *n the form

6 The factor space C (_/DT is isomorphic to the invariant subspace D- C C, n + X ln * of functions f(x) C (+ such that X X in The representations corresponding to the integer points (1.5) will "be called exceptional. 2. INVARIANT BILINEAR FORMS AND INTERTWINING OPERATORS Two non-exceptional (irreducible) representations X ar *& X are equivalent iff they differ only in the sign of c : (2.1) The corresponding intertwining operators are defined by the invariant kernels: c-t) J W 2. (2.2) d 2h where x._ 3 x. - x o, (dp) = ^r, the normalization constant n(x) is given ^ X 71 C%J= > Us and II (p) is the projection operator on the subspace of "spin" s, satisfying

7 s*< An explicit expression for Hipjz,z ) in terms of Gegenbauer polynomial is given in Ref.U. The operators G : C~ - C and G«- : C -* C«with X A X X X X kernels given "by (.2.2), have the following properties: G T~ - T G, A A A A G~ T T~ G«, G G««G«G = 1, for non-exceptional x's. The situation is more complex at exceptional points. In that case G + vanishes on E- and maps isomorphically G _ ^o n onto the invariant subspace F. of C +. The inverse mapping in ^ *t* has a kernel with a logarithmic behaviour t <2.T> The operator G (+ (with normalization (2.3)) can only be defined in the X Jtn invariant subspace DT of C,_ and maps it isomorphically on C /+/DT. * D X An X to in Similarly, G, maps DT onto G /-/Dp. The inverse mappings can x n * n X Kn in also be constructed, as veil as a pair of covariant mappings establishing the isomorphism (2.8)

8 (see Ref.6). There are also intertwining operators from xz- to Xn* an<3 - from Xo* to Xt t cf ' d'^) and - (l»9)]. For instance,the sequence ^ ( 2. 9 ) % itis exact. Studying the positivity properties of "bilinear forms one can show that there exist the following sets of unitary representations of O^n + l,l) among the 2-parameter family under consideration: i) principal series: X s [X»i<*l, C ii) supplementary series: a) X = [0»c], -h < c < h (c + 0) ; b) X= [i,c], i= l,2,...,-(h - 1) ^ c < h - 1 (c + 0, h > 1) ; iii) two discrete series of representations acting in the invariant subspaces F. C C + and D* c c. 3. THE CLEBSCH-GOKDAN EXPANSION Consider the tensor product space C C (X ft - [0,c a ], a = 1,2) X 01 X 02 Oa of infinitely smooth functions f(x,x ), satisfying asymptotic conditions of the type (l.u) in each variable. If the c & are pure imaginary or real and satisfying the inequality Kl (3.1) then the tensor product representation x m Xno can be decomposed into irreducible components of x = [-»i<?3 o^ the principal series. This means, in particular, that the functions f(x,xg) of the corresponding Hirbert space can be ex P ande<1 in "^be form T- j - -,. - A/ ' (3.2) where 'h.>- & i, *>.>-*%> x >^)r Cx 1>^i). (3.3) -7-. i#t, MiMl ill jt ^ «^ -.,. i 4

9 Here the V's are the invariant below j Clebscbr-Gordan kernels to te -written down 1=0 X where p. (cr) is the Plancherel measure on the principal series (X> =[*,* C3.5) (n(x) "being the normalization constant (2.3) of the intertwining operator (2.2)). The kernel V is fixed up to a factor by Euclidean conforms! invariance: k> X t,c t i X 3 %. z)=r J+C.-I where x ik x i - x^, i,k = 1,2,3, (3.6) (3.T) The normalization constant N is determined from comparison of (3.2) and (3.3) and from the symmetry properties -8-

10 and Its explicit expression (as a square root of a ratio of T-functions) is derived in Ref.^t. k. CONFORMAL PARTIAL WAVE EXPANSION IN A SCALAR FIELD THEORY MODEL In order to give an idea of the physical applications of the above results,we consider a simple example of (Euclidean) Green functions'" partial wave expansion in a theory of scalar fields <p.(x) of dimensions h + c. with a trilinear interaction g : <p. cp^tp'. (For a systematic treatment of such applications we refer. the reader to Ref.3.) For -j/v < c : < 0 t 1= 1,2,3 (*.i> the 1-particle irreducible Green functions G. and the Bethe-Salpeter kernel %(x x ; x_xj,) appear to satisfy the integrability condition necessary for the application of an- expansion formula of the type (3.2). The result for the 4-point function is: and a similar formula for the Bethe-Salpeter kernel B(x, c, x^, c^ ; x_, c, x^, c ) with g(x) replaced by b(%). The conformal partial g(x.) is expressed conversely in terms of G. : <*.3> where CT 1 " (t) is a Gegenbauer polynomial and the constant A can be expressed as a ratio of T-functions. -9- Jit tip.!

11 vaves; The Bethe-Salpeter equation is diagonalized for confonaal partial (.%) = U%) + Ux)}<x) I i(x)= iffl/. ik. Its solution Ut) has a pole for b(%) = 1. On the other hand, the bootstrap equation for the (invariant) 3-point function, gives Thus the dynamics provides information about the singularity structure of the conformal partial waves. Assuming that they are meromorphic functions of Xa ~ [4»25-h] in the right (complex) half plane 25-h, we can derive by shifting the integration path in (4.2) (and in analogous equationsfor higher point Green functions) a discrete operator product expansion formula 3) The relations among the values of g(^) at exceptional points are relevant for such a derivation. They follow from the existence of (singular) intertwining operators xj n * X^ * xj n a nd from identities such as for the Clebsch-Gordan kernels(see Ref.6). -10-

12 ACKNOWLEDGMENTS I -would like to thank Professors H. Bacry and A. Grossmann for their hospitality at CNRS in Marseille,where this talk was presented. I also wish to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics in Trieste, where the present note was written. REFERENCES 1) K. Symanzik, in Lectures in High Energy Physics, Ed. B. Jaksic", (Zagreb 196l) pp.u85-51t- 2) G. Mack and K. Symanzik, Commun. Math. Phys. 2J_, 2U7 (1972). 3) G. Mack, J. Phys. (Paris) 3k_, Cl, Suppl. No.10, 99 (1973). k) V. Dobrev, G. Mack, V. Petkova, S. Petrova and'i. Todorov, "On Clebsch-Gordan expansions for 0(2h + 1,1), JINR, Dubna,preprint (197 1 *). 5) G. Warner, Harmonic Analysis on Semi-Simple Lie groups, Vol.1 (Springer-Verlag, 1972), (see, in particular, Sec.5.5). 6) V. Dobrev, V. Petkova, S. Petrova and I. Todorov, "On exceptional integer points in the representation space of the pseudo-orthogonal group 0*(2h + 1,1)" (in preparation); see also I.T. Todorov, "Conformal expansion for Euclidean Green functions ", preprint SNS 13/7^, Scuola Normale Superiore, Pisa (1974). 7) T. Hirai, Proc. Japan Acad. 1+2, 323 (1966). 8) G. Mack and I.T. Todorov, Phys. Rev. D8_, 176U (1973). -U-