Worldline approach to quantum field theories on flat manifolds with boundaries

TL;DR

This paper develops a worldline approach to quantum field theories on flat manifolds with boundaries, addressing technical challenges with non-smooth potentials and successfully computing heat kernel coefficients.

Contribution

It introduces a method to handle non-smooth potentials in the worldline approach, enabling calculation of heat kernel coefficients on manifolds with boundaries.

Findings

Recovered known heat kernel coefficients on manifolds with boundaries

Computed two additional heat kernel coefficients, A_3 and A_{7/2}

Demonstrated the viability of the worldline approach in boundary scenarios

Abstract

We study a worldline approach to quantum field theories on flat manifolds with boundaries. We consider the concrete case of a scalar field propagating on R_+ x R^{D-1} which leads us to study the associated heat kernel through a one dimensional (worldline) path integral. To calculate the latter we map it onto an auxiliary path integral on the full R^D using an image charge. The main technical difficulty lies in the fact that a smooth potential on R_+ x R^{D-1} extends to a potential which generically fails to be smooth on R^D. This implies that standard perturbative methods fail and must be improved. We propose a method to deal with this situation. As a result we recover the known heat kernel coefficients on a flat manifold with geodesic boundary, and compute two additional ones, A_3 and A_{7/2}. The calculation becomes sensibly harder as the perturbative order increases, and we are able to identify the complete A_{7/2} with the help of a suitable toy model. Our findings show that the worldline approach is viable on manifolds with boundaries. Certainly, it would be desirable to improve our method of implementing the worldline approach to further simplify the perturbative calculations that arise in the presence of non-smooth potentials.