Hochschild-Kohomologien von Observablenalgebren in der Klassischen Feldtheorie

Abstract

This is my diploma thesis in german language. In the context of formal deformation theorie of assoziative observables in classical field theory I consider the symmetric algebra S(V) on an arbitrary-dimensional R- or C-vectorspace V as a prototype of comprehensive observables algebras in quantum field theory. In this framework I calculate the Hochschild cohomologies of S(V) with values in S(V)-S(V)-bimodules M. In the case that V is a locally convex vectorspace I compute the continuous Hochschild cohomologies for the (with help of the pi-tensor product) locally convex topologised symmetric Algebra on V and likewise for the completion Hol(V) of S(V) if V is in addition a Hausdorff space and M is complete. For all this cases and in the situation of symmetric bimodules M I prove generalized Hochschild-Kostant-Rosenberg theorems by use of explicite chain maps. Furthermore I have found useful statements about the differential Hochschild cohomologies in the case that M is a bimodule whose right modul multiplication can be written as a sum of the left modul multiplication and higher differential terms.