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

Received 28 December 1989, Revised 24 April 1990

Reviews in Mathematical Physics, Volume 2, No 1 (1990) 1-44
World Scientific Publishing Company



We investigate some aspects of complex geometry in relation with possible applications to quantization, relativistic phase spaces, conformal field theories, general relativity and the music of two and three-dimensional spheres.

l. Introduction

Complex manifolds and in particular classical domains have been studied for many years by mathematicians and theoretical physicists. The very old division of branches of Mathematics between Algebra, Analysis and Geometry is rather arbitrary since all these aspects are interrelated but it remains that it has some deep psycho- logical influence which explains why it is more a classification of mathematicians than a classification of mathematics. For instance, we cannot say that the study of complex domains (and in particular Cartan classical domains) belongs more to the realm of analysis than to the one of algebra or of geometry but, it is clear that most mathematical articles dealing with the subject fall into one of these three families. Often, articles belonging to a given category do not refer to papers dealing with the same subject but written from a different point of view. The same mathematical objects (Cartan classical domains) have been studied -- often without noticing it explicitly -- by theoretical physicists interested in a variety of different topics: particle physics, quantum field theory, quantum mechanics, statistical mechanics, geometric quantization, accelerated observers, general relativity and even harmony and sound analysis. The present paper is written for those who like cross-disciplinarity both in mathematics and in physics. Most of the results that we will give are already known by some people (sometimes by many) but we hope that the references to be found here will help those who believe that looking at a familiar object from a different point of view can be fruitful. Among those topics that we looked at for some domains and have not found elsewhere, let us mention the following: Poincare-Cartan momentum map and action of the conformal group in the future tube taken as a phase-space (contrasted with the conformal group action on space-time), Riemannian geometry of the future tube for its natural Kähler metric, generalization of wavelet analysis to arbitrary dimensions (relativistic music!) in relation with Bergman kernels, Weyl-Berezin calculus on Cartan domains (coherent states). Adopting a particular style for the present paper was not easy because we do not know, a priori, the motivations of the potential reader; also, the main mathematical object to be discussed here has many definitions; our task is precisely to try to present them. Because of a personal prejudice (and because we want to go from the general to the particular), we have decided to start from the geometry of complex manifolds and then specialize to the case of homogeneous spaces for Lie groups; we get classical domains in this way. Their relation to space-time (the Shilov boundary of such a domain) or to phase-space, is discussed at a later stage.

List of chapters:

2. Cartan Domains

2.1. From complex manifolds to classical domains

2.1.1. Terminology and a few standard theorems on complex manifolds
2.1.2. Some results on homogeneous complex manifolds

2.2. Boundaries of Cartan domains

2.2.1. Classical domains as differentiable manifolds
2.2.2 Boundaries of Cartan classical domains as subspaces of Cn
2.2.3. Boundaries via the Borel-Harish-Chandra embedding
2.2.4. Geometry of the Shilov boundary of Cartan domains

2.3. Global and local charts on Cartan domains
2.4. Cayley transformation for A and D series

2.4.1. On A1 = D1
2.4.2. On Dn
2.4.3. On An

2.5. Matrix realization of Cartan domains

3. Geometrical Aspects of Bergman and Szego Kernels

3.1. General vector bundles above Cartan domains
3.2. The spaces H21(D) and their associated kernels 3.3.
3.3. Coherent states on complex manifolds and the Bergman kernels
3.4. Classical domains and the Szego kernel
3.5. The case of D1 and D4

3.5.1. Bergman kernels
3.5.2. Szego kernels

3.6. Square integrable functions, distributions and hyperfunctions on the Shilov boundary
3.7. One-dimensional wavelets and relativistic wavelets

4. Lie Balls and the Action of the Conformal Group

4.1. The conformal group and its subgroups
4.2. Group theoretical aspects of the unit disk
4.3. The Lie algebra SO(4,2)
4.4. Conformal transformations on space-time
4.5. Riemannian geometry of Dn

4.5.1. General remarks
4.5.2. The Lobatchevski metric on D1, the product metric on D2=D1xD1 and the Bergman metric on Dn
4.5.3. The connection coefficients
4.5.4. The Riemann tensor on Dn
4.5.5. Geodesics

4.6. Conformal transformations on the Lie Ball D4

4.6.1. Action on D4 (direct calculations)
4.6.2. Killing vector fields
4.6.3. Conserved quantities

4.7. Cones, spheres and hyperboloids

4.7.1. Spaces of two-spheres in S3 and in B4
4.7.2. Spaces of light-cones
4.7.3. Spaces of hyperboloids

5. Cartan Domains as Spaces of Symmetries, D4 as a Phase-space and Berezin-Weyl Calculus

5.1. Algebra and geometry of the SU(n,n)/S(U(n)xU(n)) family
5.2. Boundary map and Cayley transform
5.3. The momentum map
5.4. Berezin-Weyl calculus

6. Conformal Invariance in Physics and Mathematics

6.1. The breaking of conformal invariance in physics
6.2. Quantum mechanics in arbitrary frames
6.3. Conformal structures and diffeomorphisms
6.4. The bundle of conformal frames and the compactified tangent bundle
6.5. Conformal invariance and accelerated observers

7. Harmonicity Cells, Lie Spheres and Lie Balls

7.1. Definition of harmonicity cell
7.2. Complex cones and spaces of spheres
7.3. Lie norm, Lie distance and Lie balls
7.4. D4 is the harmonicity cell of the Euclidean ball B4
7.5. The Shilov boundary of the Lie ball Dn (Lie spheres)
7.6. Noninvariance of construction, physical meaning of the ball B4
7.7. The physical meaning of the Lelong map

8. Other Aspects of Classical Domains:

-Field theory of extended objects
-Super-Hamiltonian formalism and Schwinger proper-time formalism
-Field theory at finite temperature
-Jordan pairs

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