On equivalent methods for functional determinants

TL;DR

This paper demonstrates the equivalence of the Gel'fand-Yaglom theorem and Green's function methods for computing ratios of functional determinants of one-dimensional differential operators, offering insights into handling eigenvalues.

Contribution

It unifies two prevalent methods for calculating functional determinants via a contour integral approach and discusses their treatment of special eigenvalues.

Findings

Both methods are shown to be equivalent for one-dimensional operators.

The Green's function approach naturally handles vanishing and negative eigenvalues.

The contour integral argument provides a unified framework for these methods.

Abstract

Computing functional determinants of differential operators is central to any field-theoretical calculation relying on a saddle-point expansion. A variety of approaches is available for the computation that avoid having to know the eigenspectrum of the operator, and in particular the Gel'fand-Yaglom theorem and the Green's function method. In this note, we show how both approaches can be constructed using a contour integral argument and conclude that these are completely equivalent for computing ratios of determinants of one-dimensional operators. Furthermore, we comment on the presence of vanishing as well as negative eigenvalues and show how the Green's function method provides a natural prescription for handling them.