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

North-Holland

**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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