Worldline approach to quantum field theories on flat manifolds with boundaries

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 + × 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 D using an image charge. The main technical difficulty lies in the fact that a smooth potential on + × D−1 extends to a potential which generically fails to be smooth on 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, A3 and A7/2. The calculation becomes sensibly harder as the perturbative order increases, and we are able to identify the complete A7/2 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.