CONFORMAL
THEORIES, CURVED PHASE SPACES,

RELATIVISTIC WAVELETS AND THE GEOMETRY

OF COMPLEX DOMAINS

R.
COQUEREAUX and A. JADCZYK

*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

Summary:

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 manifolds2.2. Boundaries of Cartan domains

2.2.1. Classical domains as differentiable manifolds

2.2.2 Boundaries of Cartan classical domains as subspaces ofC^{n}

2.2.3. Boundaries via the Borel-Harish-Chandra embedding

2.2.4. Geometry of the Shilov boundary of Cartan domains2.3. Global and local charts on Cartan domains

2.4. Cayley transformation for A and D series2.4.1. On A

_{1}= D_{1}

2.4.2. On D_{n}

2.4.3. On A_{n}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 H^{2}_{1}(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 D_{1}and D_{4}3.5.1. Bergman kernels

3.5.2. Szego kernels3.6. 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 D_{n}4.5.1. General remarks

4.5.2. The Lobatchevski metric on D_{1}, the product metric on D_{2}=D_{1}xD_{1}and the Bergman metric on D_{n}

4.5.3. The connection coefficients

4.5.4. The Riemann tensor on D_{n}

4.5.5. Geodesics4.6. Conformal transformations on the Lie Ball D

_{4}4.6.1. Action on D

_{4}(direct calculations)

4.6.2. Killing vector fields

4.6.3. Conserved quantities4.7. Cones, spheres and hyperboloids

4.7.1. Spaces of two-spheres in S

^{3}and in B^{4}

4.7.2. Spaces of light-cones

4.7.3. Spaces of hyperboloids

5. Cartan
Domains as Spaces of Symmetries, D_{4} 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. D_{4}is the harmonicity cell of the Euclidean ball B^{4}

7.5. The Shilov boundary of the Lie ball D_{n}(Lie spheres)

7.6. Noninvariance of construction, physical meaning of the ball B^{4}

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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