Effective action for delta potentials: Spacetime-dependent inhomogeneities and Casimir self-energy

We study the vacuum fluctuations of a quantum scalar field in the presence of a thin and inhomogeneous flat mirror, modeled with a delta potential. Using heat-kernel techniques, we evaluate the Euclidean effective action perturbatively in the inhomogeneities (nonperturbatively in the constant background). We show that the divergences can be absorbed into a local counterterm and that the remaining finite part is in general a nonlocal functional of the inhomogeneities, which we compute explicitly for massless fields in $D=4$ dimensions. For time-independent inhomogeneities, the effective action gives the Casimir self-energy for a partially transmitting mirror. For time-dependent inhomogeneities, the Wick-rotated effective action gives the probability of particle creation due to the dynamical Casimir effect.

Figure 1

Contribution ${\mathrm{\Gamma }}_{TI}^{(2)}$ to the effective action per unit time for Gaussian inhomogeneities: the left panel is a density plot as a function of $\zeta$ and $\sigma$, while the right panel corresponds to a plot as a function of $\sigma$ for several values of $\zeta$. All the dimensional quantities are measured in arbitrary units, and we have chosen ${\eta }_0=1$.

Figure 2

Nonlocal contribution to the effective action (in the “null-mass” renormalization) per unit time ($\frac{{\mathrm{\Gamma }}_{m=0}^{(2)}}{T_0}$) for Gaussian inhomogeneities: the left panel is a density plot as a function of $\zeta$ and $\sigma$, while the right panel corresponds to a plot as a function of $\sigma$ for several values of $\zeta$. All the dimensional quantities are measured in arbitrary units, and we have chosen ${\eta }_0=1$.

Figure 3

Imaginary part of ${\mathrm{\Gamma }}_{M,G}^{(2)}$ for Gaussian inhomogeneities. The left panel corresponds to a density plot as a function of $\zeta$ and ${\sigma }_t$ for ${\sigma }_s=0.1$, while the right panel is a double-log plot as a function of ${\sigma }_s$ for fixed $\zeta =1$. and ${\sigma }_t=0.05$. All the dimensional quantities are measured in arbitrary units, and ${\eta }_0=1$ is chosen.

Figure 4

Plot of $\mathrm{Im}({\mathrm{\Gamma }}_{M,H}^{(2)})$ per unit time and area for harmonic inhomogeneities. The left panel corresponds to a density plot as a function of $\zeta$ and ${\omega }_0$, while the right panel is a plot as a function of ${\omega }_0$ for several values of $\zeta$. All the dimensional quantities are measured in arbitrary units and ${\eta }_0=1$ is chosen.