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Functional Determinants in Quantum Field Theory

Gerald V. Dunne

TL;DR

This work advances the computation of functional determinants in quantum field theory by extending the Gel'fand–Yaglom approach from 1D to higher-dimensional, radially separable operators. It provides explicit, renormalized determinant formulas valid in two, three, and four dimensions, expressed through boundary data of radial solutions and Jost-function techniques, thereby avoiding explicit spectral computations. Two key applications—false vacuum decay and massive quarks in instanton backgrounds—demonstrate how these determinants yield physically meaningful fluctuation factors and are compatible with traditional Feynman-diagram approaches after renormalization. The results enable direct, numerically efficient treatment of quantum fluctuations around nontrivial backgrounds and point toward generalizations to non-separable cases and Dirac/finite-temperature settings. Altogether, the work broadens the class of solvable fluctuation-determinant problems in QFT and provides practical tools for semiclassical analyses.

Abstract

Functional determinants of differential operators play a prominent role in theoretical and mathematical physics, and in particular in quantum field theory. They are, however, difficult to compute in non-trivial cases. For one dimensional problems, a classical result of Gel'fand and Yaglom dramatically simplifies the problem so that the functional determinant can be computed without computing the spectrum of eigenvalues. Here I report recent progress in extending this approach to higher dimensions (i.e., functional determinants of partial differential operators), with applications in quantum field theory.

Functional Determinants in Quantum Field Theory

TL;DR

This work advances the computation of functional determinants in quantum field theory by extending the Gel'fand–Yaglom approach from 1D to higher-dimensional, radially separable operators. It provides explicit, renormalized determinant formulas valid in two, three, and four dimensions, expressed through boundary data of radial solutions and Jost-function techniques, thereby avoiding explicit spectral computations. Two key applications—false vacuum decay and massive quarks in instanton backgrounds—demonstrate how these determinants yield physically meaningful fluctuation factors and are compatible with traditional Feynman-diagram approaches after renormalization. The results enable direct, numerically efficient treatment of quantum fluctuations around nontrivial backgrounds and point toward generalizations to non-separable cases and Dirac/finite-temperature settings. Altogether, the work broadens the class of solvable fluctuation-determinant problems in QFT and provides practical tools for semiclassical analyses.

Abstract

Functional determinants of differential operators play a prominent role in theoretical and mathematical physics, and in particular in quantum field theory. They are, however, difficult to compute in non-trivial cases. For one dimensional problems, a classical result of Gel'fand and Yaglom dramatically simplifies the problem so that the functional determinant can be computed without computing the spectrum of eigenvalues. Here I report recent progress in extending this approach to higher dimensions (i.e., functional determinants of partial differential operators), with applications in quantum field theory.

Paper Structure

This paper contains 9 sections, 46 equations, 2 figures.

Table of Contents

  1. Introduction : why are determinants important in QFT?
  2. What we know
  3. Gel'fand-Yaglom theorem: ordinary differential operators
  4. Gel'fand-Yaglom theorem: partial differential operators?
  5. Comparison With Feynman Diagram Approach
  6. Removing Zero Modes
  7. Application 1: fluctuation determinant for false vacuum decay
  8. Application 2: determinant for massive quarks in instanton background
  9. Concluding remarks

Figures (2)

  • Figure 1: Comparison as $\alpha\to 1$ of the exact numerical result [dots and solid blue line] from (\ref{['dm-answer2']}) with the analytic thin-wall limit result $\frac{9}{32}\, \left[1-\frac{2\pi}{9\sqrt{3}}\right]\approx 0.16788..$ of kr.
  • Figure 2: Plot of our numerical results for $\tilde{\Gamma}^{S}_{\rm ren}(m)$ from (\ref{['answer2']}), compared with the analytic extreme small and large mass limits [dashed curves] from (\ref{['masslimit']}). The dots denote numerical data points from (\ref{['answer2']}), and the solid line is a fit through these points.