Casimir Energies and Pressures for $\delta$-function Potentials

Abstract

The Casimir energies and pressures for a massless scalar field associated with $\delta$-function potentials in 1+1 and 3+1 dimensions are calculated. For parallel plane surfaces, the results are finite, coincide with the pressures associated with Dirichlet planes in the limit of strong coupling, and for weak coupling do not possess a power-series expansion in 1+1 dimension. The relation between Casimir energies and Casimir pressures is clarified,and the former are shown to involve surface terms. The Casimir energy for a $\delta$-function spherical shell in 3+1 dimensions has an expression that reduces to the familiar result for a Dirichlet shell in the strong-coupling limit. However, the Casimir energy for finite coupling possesses a logarithmic divergence first appearing in third order in the weak-coupling expansion, which seems unremovable. The corresponding energies and pressures for a derivative of a $\delta$-function potential for the same spherical geometry generalizes the TM contributions of electrodynamics. Cancellation of divergences can occur between the TE ($\delta$-function) and TM (derivative of $\delta$-function) Casimir energies. These results clarify recent discussions in the literature.

16 pages, 1 eps figure, uses REVTeX4

References in corpus (18)

• New Developments in the Casimir Effect
• Calculating Vacuum Energies in Renormalizable Quantum Field Theories: A New Approach to the Casimir Problem
• Casimir effect for scalar fields under Robin boundary conditions on plates
• Casimir effect for a $D$-dimensional sphere
• The Dirichlet Casimir Problem
• On the ground state energy for a penetrable sphere and for a dielectric ball
• Casimir Energies in Light of Quantum Field Theory
• Identity of the van der Waals Force and the Casimir Effect and the Irrelevance of these Phenomena to Sonoluminescence
• Systematics of the Relationship between Vacuum Energy Calculations and Heat Kernel Coefficients
• Observability of the Bulk Casimir Effect: Can the Dynamical Casimir Effect be Relevant to Sonoluminescence?
• Casimir Energy for a Spherical Cavity in a Dielectric: Applications to Sonoluminescence
• Casimir Effects in Renormalizable Quantum Field Theories
• Casimir's energy of a conducting sphere and of a dilute dielectric ball
• Vacuum densities and zero-point energy for fields obeying Robin conditions on cylindrical surfaces
• Vector Casimir effect for a D-dimensional sphere
• Calculating Casimir Energies in Renormalizable Quantum Field Theory
• The ground state energy of a massive scalar field in the background of a semi-transparent spherical shell
• Unnatural Acts: Unphysical Consequences of Imposing Boundary Conditions on Quantum Fields

Cited by in corpus (57)

• Multiple Scattering Methods in Casimir Calculations
• Casimir effect for thin films in QED
• How does Casimir energy fall? II. Gravitational acceleration of quantum vacuum energy
• Casimir effect in a wormhole spacetime
• Local and Global Casimir Energies for a Semitransparent Cylindrical Shell
• Non-contact gears: I. Next-to-leading order contribution to lateral Casimir force between corrugated parallel plates
• How does Casimir energy fall? IV. Gravitational interaction of regularized quantum vacuum energy
• Vacuum Stress and Closed Paths in Rectangles, Pistons, and Pistols
• Double-delta potentials: one dimensional scattering. The Casimir effect and kink fluctuations
• Vacuum energies due to delta-like currents: simulating classical objects along branes with arbitrary codimensions
• Interactions between delta-like sources and potentials
• Field theoretic description of partially reflective surfaces
• Electromagnetic $\delta$-function sphere
• Electromagnetic field fluctuations near a dielectric-vacuum boundary and surface divergences in the ideal conductor limit
• Local Casimir Energies for a Thin Spherical Shell
• Derivative expansion of the electromagnetic Casimir energy for two thin mirrors
• Vacuum Casimir energy densities and field divergences at boundaries
• Field theoretic description of electromagnetic boundaries: The Casimir effect between dissimilar mirrors from external potentials
• A New Look at the Quantum Mechanics of the Harmonic Oscillator
• Field of homogeneous Plane in Quantum Electrodynamics
• Local and Global Casimir Energies: Divergences, Renormalization, and the Coupling to Gravity
• Casimir effect with dynamical matter on thin mirrors
• Neumann Casimir effect: a singular boundary-interaction approach
• Local Casimir Effect for a Scalar Field in Presence of a Point Impurity
• Magnetoelectric boundary simulated by a Chern-Simons-like model
• Surface Divergences and Boundary Energies in the Casimir Effect
• Zeta-function approach to Casimir energy with singular potentials
• Weyl problem and Casimir effects in spherical shell geometry
• Derivative expansion for the boundary interaction terms in the Casimir effect: generalized $\delta$-potentials
• Vacuum energy for generalised Dirac combs at $T=0$
• Energy conditions outside a dielectric ball
• Lifshitz formula for the Casimir force and the Gelfand-Yaglom theorem
• Casimir effect due to a single boundary as a manifestation of the Weyl problem
• Parity violating cylindrical shell in the framework of QED
• Relative-Zeta and Casimir energy for a semitransparent hyperplane selecting transverse modes
• Dirichlet boundary condition for the Lee-Wick-like scalar model
• Material boundaries in Carroll-Field-Jackiw Lorentz-violating electrodynamics
• First-order correction to the Casimir force within an inhomogeneous medium
• Casimir energy due to inhomogeneous thin plates
• Semi-transparent boundaries in CPT-even Lorentz violating electrodynamics
• Casimir self-energy of a δ-δ' sphere
• Scarf for Lifshitz
• The role of scalar current coupling along surfaces
• Inertial forces and dissipation on accelerated boundaries
• Vacuum polarization with zero-range potentials on a hyperplane
• Comment on "Casimir energies with finite-width mirrors"
• Recent Developments in Quantum Vacuum Energy for Confined Fields
• Casimir type effects for scalar fields interacting with material slabs
• A Two-Gauge Field Model for Magnetoelectric Boundaries
• Quantum propagator and characteristic equation in the presence of a chain of $\delta$-potentials
• Casimir force on a light front
• Casimir traversable wormholes in Gauss-Bonnet gravity
• Casimir effect in conformally flat spacetimes
• The Casimir effect in the presence of infrared transparency
• Casimir energy through transfer operators for sine-Gordon backgrounds
• Vacuum energy of scalar fields on spherical shells with general matching conditions
• Generalized Neumann boundary condition for the scalar field