Hochschild-Kohomologien von Observablenalgebren in der Klassischen Feldtheorie

TL;DR

This thesis investigates Hochschild cohomologies of symmetric algebras modeling classical observables in field theory, extending known theorems to locally convex and complete topological settings.

Contribution

It computes Hochschild cohomologies for symmetric algebras in various topological contexts and proves generalized Hochschild-Kostant-Rosenberg theorems using explicit chain maps.

Findings

Computed Hochschild cohomologies for symmetric algebras in locally convex and complete topological spaces.

Extended Hochschild-Kostant-Rosenberg theorems to new topological and bimodule contexts.

Provided insights into differential Hochschild cohomologies with higher differential terms.

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.