AND RELATIVISTIC QUANTUM FIELDS
Centre de Physique Theorique. Section 2. Case 907, Luminy. 13288, Marseille. France
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.
sense that it is a ball for the Lie distance defined as follows. Let ...
... 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 . 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 ). The present article is also a kind of invitation to further study.
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