Resummed heat-kernel for surface contributions: Dirichlet semitransparent boundary conditions

TL;DR

This paper derives resummed heat-kernel expressions for Laplace operators with potential and semitransparent Dirichlet boundary conditions, applying them to quantum field theory and exploring boundary condition relations.

Contribution

It provides new resummed formulas for heat-kernels with semitransparent boundary conditions and applies these to scalar quantum fields with Yukawa interactions.

Findings

Resummed heat-kernel expressions for Dirichlet semitransparent boundaries.

Application to bulk and surface form factors in quantum field theory.

Connection established between different boundary condition heat-kernels.

Abstract

In this article we consider resummed expressions for the heat-kernel's trace of a Laplace operator, the latter including a potential and imposing Dirichlet semitransparent boundary conditions on a surface of codimension one in flat space. We obtain resummed expressions that correspond to the first and second order expansion of the heat-kernel in powers of the potential. We show how to apply these results to obtain the bulk and surface form factors of a scalar quantum field theory in $d=4$ with a Yukawa coupling to a background. A characterization of the form factors in terms of pseudo-differential operators is given. Additionally, we discuss a connection between heat-kernels for Dirichlet semitransparent, Dirichlet and Robin boundary conditions.