Scalar heat kernel with boundary in the worldline formalism

TL;DR

This paper advances the worldline formalism by developing non-recursive master formulas for scalar heat kernels with boundary conditions, enabling efficient computation of heat-kernel coefficients on the half-space.

Contribution

It introduces an improved worldline formalism with master formulas for boundary conditions, facilitating automated calculations of heat-kernel contributions.

Findings

Derived non-recursive master formulas for heat kernel trace with boundary conditions.

Calculated new heat-kernel coefficients a_4 and a_{9/2} for the half-space.

Demonstrated the formalism's efficiency through computational examples.

Abstract

The worldline formalism has in recent years emerged as a powerful tool for the computation of effective actions and heat kernels. However, implementing nontrivial boundary conditions in this formalism has turned out to be a difficult problem. Recently, such a generalization was developed for the case of a scalar field on the half-space R_+ x R^{D-1}, based on an extension of the associated worldline path integral to the full R^D using image charges. We present here an improved version of this formalism which allows us to write down non-recursive master formulas for the n-point contribution to the heat kernel trace of a scalar field on the half-space with Dirichlet or Neumann boundary conditions. These master formulas are suitable to computerization. We demonstrate the efficiency of the formalism by a calculation of two new heat-kernel coefficients for the half-space, a_4 and a_{9/2}.