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 analog Dirichlet one.

Figure 1

Pair-production probability/rate per unit area in the massive case. The left panel compares the harmonic and Sauter cases (denoted as $H$ and $S$ respectively) for different values of masses and as a function of the corresponding frequency. Notice that in the harmonic case, the pair-production rate should be understood. The panel on the right is a density plot of the pair-production probability per unit area for the Sauter pulse as a function of the frequency ${\omega }_S$ and the mass $m$. In all the cases, the amplitudes ${\eta }_0$ are set to unity.

Figure 2

Probability of assisted particle production for a Sauter pulse per unit area, ${\zeta }^2{P}_S^{\zeta }/A$. The left panel shows its density plot as a function of the frequency ${\omega }_S$ and the background $\zeta$. The right panel displays its behavior as the frequency varies, considering different backgrounds. We have set the amplitude of the pulse (${\eta }_0$) to unity.