Scalar Field with Robin Boundary Conditions in the Worldline Formalism

TL;DR

This paper extends the worldline formalism to compute the heat kernel expansion for scalar fields with Robin boundary conditions, providing explicit calculations of boundary contributions to key coefficients.

Contribution

It introduces a method to incorporate Robin boundary conditions into the worldline formalism and computes specific heat kernel coefficients for scalar fields.

Findings

Derived boundary contributions to heat kernel coefficients A_1, A_{3/2}, and A_2

Demonstrated the applicability of the worldline approach to boundary problems

Provided explicit formulas for Robin boundary conditions in the heat kernel expansion

Abstract

The worldline formalism has been widely used to compute physical quantities in quantum field theory. However, applications of this formalism to quantum fields in the presence of boundaries have been studied only recently. In this article we show how to compute in the worldline approach the heat kernel expansion for a scalar field with boundary conditions of Robin type. In order to describe how this mechanism works, we compute the contributions due to the boundary conditions to the coefficients A_1, A_{3/2} and A_2 of the heat kernel expansion of a scalar field on the positive real line.