48 Nuclear Physics B (Proc. Suppl.) 18B (1990) 48-52

North-Holland

LIE BALLS

AND RELATIVISTIC QUANTUM FIELDS

R. Coquereaux

Centre de Physique Theorique. Section 2. Case 907, Luminy. 13288, Marseille. France

Abstract

We investigate some aspects of the complex domain S0(4, 2)/(S0(4) SO(2)) in relation with relativistic quantum mechanics and conformal invariance.

1. Geometrical aspects of Lie Balls.

In the present paper, we are mainly interested in the four dimensional (complex) Lie ball that we shall denote by D. This smooth manifold can be written as SO0(4,2)/SO(4) x S0(2) or as SU(2,2)/S(U(2) x U(2)). Because of the local isomorphism between G = SU(2, 2) and S0(4, 2), D is a bounded non compact symmetric domain of type I and IV. D = G/H is a Kahler manifold for its G-invariant metric (which coincides with its Bergman metric) and is also Einstein. D is in particular a complex manifold with (integrable) complex structure j0. It is also a non compact Hermitian homogeneous manifold for the action of the conformal group S0(4, 2) of Space-Time and is of rank two as a homogeneous space. Moreover it is a symmetric quaternion-Kahler manifold (hence quaternion hermitian and quaternionic) but not hyperkahlerian. As such it admits a twistor space which is also a complex manifold fibrated as a bundle above D with CP1 fibers. As a topological space, D is homeomorphic with R8 and is therefore a manifold without boundary. However, from its realisation as a bounded domain of C4 or from its realisation as a subset of its compact dual, the Grassmanian S0(6)/(S0(4) x SO(2)), via the Harish-Chandra embedding, it admits a weak boundary which is stratified under the action of the stabiliser S0(4) x S0(2). One of the strata is actually a singular four dimensional orbit and is of special interest for us since it is diffeomorphic with S3 X|Z2 S1 i.e. with compactified Minkowski Space-Time. This particular orbit is both a quotient of S(U(2) x U(2)) by a diagonal SU(2) and a quotient of the conformal group itself by the semi-direct product of a Poincare subgroup times the subgroup of dilations. The metric of D is euclidean and blows up near the boundary ( as in the usual geometry of Lobatchevski) but, what is of particular interest here is that it induces a conformal Lorentz structure on the boundary. The domain D is a Lie ball in the

sense that it is a ball for the Lie distance defined as follows. Let ...

(snip)

... The study of analyticity properties of n-points functions is a rather traditional field of research in Relativistic Quantum Mechanics and in Quantum Field Theory. Many results are known thanks to the work of a generation of theoretical physicists (and in particular thanks to the efforts of Raymond Stora [9-12] who has always been a master in this area (and in others..!)). What we suggest here is a kind of different game: analytic continuation from the real line to the complex plane, with its flat euchdean geometry, is not the same as going from the real line (or from the circle) to the Poincare upper half-plane (or to the disk), with its curved Lobatchevskian geometry. This one-dimensional (complex) comparison (and contrast) sits at the roots of the message carried by the authors of [3]. Our belief is that Physics is "simple" (and euclidean) in the domain D and that many of the difficulties of classical or quantum physics arise because we try to go to the "boundary" and to formulate the laws of Physics there. Many mathematical properties of the space D (and of other classical domains) are already known but the use of those properties in Physics is a new and open subject. In particular, new physical intuitions have to be developed (for instance, everybody knows what the Fourier transformation is, and understands its physical interpretation, but here, what we have is rather a (Radon)-Gelfand-Graev transformation -i.e. integration over horocycles- and its physical interpretation is quite different and not necessarily familiar...). Much remains to be done. Here, we have only sketched a few properties of the appropriate mathematical structures (more can be found in [3]). The present article is also a kind of invitation to further study.

 

 

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References

[1] V. Avanissian. "Cellule d'harmonicite et prolongement analytique complexe". Coll. Travaux en Cours. Hermann, Paris 1985.

[2] F.A. Berezin, "Quantization", Math. USSR Izvestja Vol. 8 (1974), No. 5. "Quantization in complex symmetric spaces", Math. USSR Izvestja Vol. 9 (1975), No. 2.

[3] R. Coquereaux, A. Jadczyk, "Conformal Theories, Curved Phase Spaces, Relativistic Wavelets and the Geometry of Complex Domains", CPT-89/P.2302, to appear in Reviews in Math. Phys.

[4] A.Grossmann, J.Morlet, T.Paul, "Transforms associated to square integrable representations'' I. J. Math. Phys. 27, 1985, pp 2473-79

[5] A.Grossmann, J.Morlet, T.Paul, "Transforms associated to square integrable representations II. Ann. Inst Henri Poincar~ 45,1986,pp 293-309.

[6] L.K.Hua. "Harmonic analysis of functions of several variables in the classical domains", Translation of mathematical monographs 6, Amer. Math. Soc., Providence, R.I., 1963.

[7] A. Odzijewicz. "A model of Conformal Kinematics." Inter. Jour. of Theor. Phys. Vol 15, No 8 (1976) pp. 575-593

[8] Unterberger. "Analyse harmonique et analyse pseudo-differentielle du cone de lumiere". Asterisque 156. 1987.

[9] H. Epstein, V. Glaser, R. Stora. "General properties of the N Point functions in local quantum field theory". In Les Houches 1975, Proceedings.

[10] H. Epstein, V. Glaser, R. Stora. "Geometry of the N Point space functions of quantum field theory". In Nice 1973, Lecture Notes in Mathematics, Vol. 449, Berlin 1975.

[11] R. Stora. "Some applications of the Jost-Lehmann-Dyson theorem to the study of global analytic structure of the N-point function of quantum field theory". In Nice 1973, Proceedings, Lecture Notes in Mathematics, Vol 449, Berlin 1975.

[12] J. Bros, H. Epstein, V. Glaser, R. Stora. "Some global aspects of the edge of the wedge problems".In Nice 1973, Proceedings, Lecture Notes in Mathematics, Vol 449, Berlin 1975.

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