On Virtual Representations of Symmetric Spaces and Their Analytic Continuation

Abstract

In this paper we study a problem in group representation theory motivated by relativistic quantum field theory. One of the most powerful approaches to constructing models of relativistic quantum fields relies on Euclidean field theory, a description of quantum field theory in which time is purely imaginary. The basic objects of Euclidean field theory (the Euclidean Green's- or Schwinger functions) can often be expressed in terms of functional integrals. The construction of quantum field models is thereby reduced to quadratures. A well-known, simple example of this method is the reformulation of quantum mechanisms by means of Wiener integrals. (The fundamental solution of the Schrodinger equation, analytically continued to imaginary time, is given by the well-known Feynman-Kac formula; see e.g. [15], [19].) In this approach one faces the problem of analytic continuation back to real time. While this problem is elementary in non-relativistic quantum mechanics, it is rather intricate in relativistic quantum field theory. For some class of relativistic quantum field theories, a fairly general solution to this problem was given in papers by Osterwalder and