The Inverse Laplacian: Traces in Infinite Dimensions

TL;DR

This paper develops the theory of trace class operators on infinite-dimensional Hilbert spaces and computes traces of the inverse Laplacian on flat tori, extending previous work in topological quantum field theory.

Contribution

It provides a concise development of trace class operators theory and original trace computations for the inverse Laplacian on manifolds, linking functional analysis with quantum field theory.

Findings

Trace computations for inverse Laplacian on flat tori

Extension of trace class operator theory to infinite dimensions

Generalization of previous quantum field theory results

Abstract

In this paper, we present a concise development of the well-studied theory of trace class operators on infinite dimensional (separable) Hilbert spaces suitable for an advanced undergraduate, as well as a construction of the inverse Laplacian on closed manifolds. With these developments acting as prerequisite, we present original trace computations involving the inverse Laplacian on the (flat) torus, generalizing computations done by R. Grady and O. Gwilliam within the context of topological quantum field theory.