Local Neumann semitransparent layers: Resummation, pair production, and duality

We consider local semitransparent Neumann boundary conditions for a quantum scalar field as imposed by a quadratic coupling to a source localized on a flat codimension-one surface. Upon a proper regularization to give meaning to the interaction, we interpret the effective action as a theory in a first-quantized phase space. We compute the relevant heat-kernel to all order in a homogeneous background and quadratic order in perturbations, giving a closed expression for the corresponding effective action in $D=4$. In the dynamical case, we analyze the pair production caused by a harmonic perturbation and a Sauter pulse. Notably, we prove the existence of a strong/weak duality that links this Neumann field theory to the analogue Dirichlet one.