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On equivalent methods for functional determinants

Matthias Carosi

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.

On equivalent methods for functional determinants

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.
Paper Structure (10 sections, 66 equations, 2 figures)

This paper contains 10 sections, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The $\zeta$-regularisation
  3. A contour integral derivation
  4. The Gel'fand-Yaglom theorem
  5. The Green's function method
  6. The Green's function method revisited
  7. Zero modes
  8. About negative modes
  9. About the heat kernel method
  10. Conclusions

Figures (2)

  • Figure 1: The contour $C_+$ on the complex $\lambda$-plane. The black dots represent the poles of the integrand, while the red wavy line is the branch cut coming from $\lambda^{-t}$, located at an angle $\theta$ with respect to the positive real axis. Adapted from Ref. Kirsten:2004qv.
  • Figure 2: The contour $C_-$ obtained by deforming $C_+$ as shown in Figure \ref{['fig:C+_contour']}. The new contour wraps around the branch cut coming from $\lambda^{-t}$.