Spectral functions in mathematics and physics

TL;DR

This paper explores spectral functions in quantum field theory, focusing on heat-kernels, determinants, and spectral sums, and introduces methods for calculating heat-kernel coefficients and analyzing vacuum energies with spherical symmetry.

Contribution

It presents a new method for calculating heat-kernel coefficients and analyzing vacuum energies in spherically symmetric settings using spectral functions and zeta functions.

Findings

Derived a method for heat-kernel coefficient calculation.

Analyzed vacuum energies with spherical boundaries.

Connected spectral functions with zeta function properties.

Abstract

Spectral functions relevant in the context of quantum field theory under the influence of spherically symmetric external conditions are analysed. Examples comprise heat-kernels, determinants and spectral sums needed for the analysis of Casimir energies. First, we summarize that a convenient way of handling them is to use the associated zeta function. A way to determine all its needed properties is derived. Using the connection with the mentioned spectral functions, we provide: i.) a method for the calculation of heat-kernel coefficients of Laplace-like operators on Riemannian manifolds with smooth boundaries and ii.) an analysis of vacuum energies in the presence of spherically symmetric boundaries and external background potentials.