Functional determinants in the presence of zero modes

TL;DR

This paper introduces a straightforward contour integration approach to compute functional determinants, exemplified with the Laplacian under Dirichlet conditions, and extends to more general operators and boundary conditions.

Contribution

It provides a simple, accessible method for deriving functional determinants that can handle zero modes and various boundary conditions.

Findings

Derived explicit formulas for Laplacian determinants with Dirichlet boundary conditions.

Extended the method to more general operators and boundary conditions.

Clarified the use of contour integration in functional determinant calculations.

Abstract

We present a simple and accessible method which uses contour integration methods to derive formulae for functional determinants. To make the presentation as clear as possible we illustrate the general ideas using the Laplacian with Dirichlet boundary conditions on the interval. Afterwards, we indicate how more general operators as well as general boundary conditions can be covered.