Group Cohomology and Quantum Fields

Abstract

In this thesis it is shown how the 1-Cohomology of groups can be used to classify certain representations of the Canonical Commutation Relations.

First, the algebra of the Canonical Commutation Relations is described in the framework of C*algebras. The Fock and displaced-Fock representations are defined.

A unitary representation of a connected Lie group is introduced into the complex pre-Hilbert space, over which the C.C.R. algebra is built. This group action induces an automorphism of the C.C.R. algebra, and the automorphism is shown to be unitarily implemented in the Fock representation. The question of unitary implementability in the displaced-Fock representation leads to the study of 1-cohomology of groups.

The cohomology of the Poincare group is studied, for various representations of the Poincare group. Also the cohomology with values in the Hilbert-Schmidt operators of the one-particle space is calculated to be trivial.

The results obtained then determine whether or not there do exist representations of the C.C.R. which are inequivalent to the Fock representation and which have a group automorphism unitarily implemented.