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

Abstract

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.

arXiv comment25 pages, 4 figures. Added a discussion on the renormalization process. Added some missing factors in formulas (72) and (73), App. A

References in corpus (35)

• Heat kernel expansion: user's manual
• Perturbative Quantum Field Theory in the String-Inspired Formalism
• Stimulating uncertainty: Amplifying the quantum vacuum with superconducting circuits
• The Casimir effect: Recent controversies and progress
• The Casimir Effect: Physical Manifestations of Zero Point Energy
• Numerical methods for computing Casimir interactions
• Current status of the Dynamical Casimir Effect
• Calculating Vacuum Energies in Renormalizable Quantum Field Theories: A New Approach to the Casimir Problem
• On the ground state energy for a penetrable sphere and for a dielectric ball
• Roughness correction to the Casimir force : Beyond the Proximity Force Approximation
• Heat Kernel Expansion for Semitransparent Boundaries
• Fluctuations, dissipation and the dynamical Casimir effect
• Quantum vacuum energies and Casimir forces between partially transparent $\delta$-function plates
• Model for resonant photon creation in a cavity with time dependent conductivity
• Nonsmooth backgrounds in quantum field theory
• Quantum dissipative effects in moving mirrors: a functional approach
• Exact Casimir Interaction Between Semitransparent Spheres and Cylinders
• Quantum mechanical path integrals in the first quantised approach to quantum field theory
• A simple model for the dynamical Casimir effect for a static mirror with time-dependent properties
• Hard and soft walls
• Worldline Formalism in Snyder Spaces
• Electromagnetic $\delta$-function sphere
• Stress tensor for a scalar field in a spatially varying background potential: Divergences, "renormalization," anomalies, and Casimir forces
• Connection between $\zeta$ and cutoff regularizations of Casimir energies
• Boundary divergences in vacuum self-energies and quantum field theory in curved spacetime
• Derivative expansion for the Casimir effect at zero and finite temperature in $d+1$ dimensions
• Quantum Effects in the Presence of Expanding Semi-Transparent Spherical Mirrors
• Black hole collapse simulated by vacuum fluctuations with a moving semi-transparent mirror
• Quantum dissipative effects in moving imperfect mirrors: sidewise and normal motions
• Propagator from Nonperturbative Worldline Dynamics
• Dynamical Casimir effect for semitransparent mirrors
• Dependence of the vacuum energy on spherically symmetric background fields
• Semi-transparent Boundary Conditions in the Worldline Formalism
• Casimir energy due to inhomogeneous thin plates
• Non-perturbative gauge transformations of arbitrary fermion correlation functions in quantum electrodynamics

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