The Dirichlet Casimir Problem

Abstract

Casimir forces are conventionally computed by analyzing the effects of boundary conditions on a fluctuating quantum field. Although this analysis provides a clean and calculationally tractable idealization, it does not always accurately capture the characteristics of real materials, which cannot constrain the modes of the fluctuating field at all energies. We study the vacuum polarization energy of renormalizable, continuum quantum field theory in the presence of a background field, designed to impose a Dirichlet boundary condition in a particular limit. We show that in two and three space dimensions, as a background field becomes concentrated on the surface on which the Dirichlet boundary condition would eventually hold, the Casimir energy diverges. This result implies that the energy depends in detail on the properties of the material, which are not captured by the idealized boundary conditions. This divergence does not affect the force between rigid bodies, but it does invalidate calculations of Casimir stresses based on idealized boundary conditions.

29 pages, 4 figures

References in corpus (16)

• New Developments in the Casimir Effect
• Precision Measurement of the Casimir Force from 0.1 to 0.9 microns
• Measurement of the Casimir force between parallel metallic surfaces
• The Casimir Effect: Physical Manifestations of Zero Point Energy
• Calculating Vacuum Energies in Renormalizable Quantum Field Theories: A New Approach to the Casimir Problem
• Casimir Energies in Light of Quantum Field Theory
• Casimir Effects in Renormalizable Quantum Field Theories
• Finite Quantum Fluctuations About Static Field Configurations
• Quantum Energies of Interfaces
• Calculating Casimir Energies in Renormalizable Quantum Field Theory
• Heavy Fermion Stabilization of Solitons in 1+1 Dimensions
• Searching for Quantum Solitons in a 3+1 Dimensional Chiral Yukawa Model
• Derivative Expansion and Soliton Masses
• Finite Energy Sum Rules in Potential Scattering
• Unnatural Acts: Unphysical Consequences of Imposing Boundary Conditions on Quantum Fields
• A Heavy Fermion Can Create a Soliton: A 1+1 Dimensional Example

Cited by in corpus (115)

• The Casimir effect: Recent controversies and progress
• The Casimir Effect and the Quantum Vacuum
• Casimir forces between arbitrary compact objects
• Scalar Casimir effect between Dirichlet spheres or a plate and a sphere
• Worldline algorithms for Casimir configurations
• Casimir Forces between Compact Objects: I. The Scalar Case
• Attractive Casimir Forces in a Closed Geometry
• Casimir force on a piston
• How Does Casimir Energy Fall?
• Casimir Effects: An Optical Approach I. Foundations and Examples
• Casimir piston for massless scalar fields in three dimensions
• Casimir Forces in a Piston Geometry at Zero and Finite Temperatures
• Attractive and Repulsive Casimir Vacuum Energy with General Boundary Conditions
• Comment on the sign of the Casimir force
• Casimir Energies and Pressures for $\delta$-function Potentials
• Casimir pistons with hybrid boundary conditions
• Quantum vacuum energies and Casimir forces between partially transparent $\delta$-function plates
• Nonsmooth backgrounds in quantum field theory
• Finite temperature Casimir energy in closed rectangular cavities: a rigorous derivation based on zeta function technique
• Casimir Energy for a Purely Dielectric Cylinder by the Mode Summation Method
• The perfect magnetic conductor (PMC) Casimir piston in d+1 dimensions
• Casimir Energy for a Dielectric Cylinder
• The Casimir force on a piston in the spacetime with extra compactified dimensions
• Recent Developments in the Casimir Effect
• On the vacuum energy of a spherical plasma shell
• The Casimir Energy for a Hyperboloid Facing a Plate in the Optical Approximation
• Casimir forces between arbitrary compact objects: Scalar and electromagnetic field
• Effective Actions, Boundaries and Precision Calculations of Casimir Energies
• How does Casimir energy fall? II. Gravitational acceleration of quantum vacuum energy
• Green's Dyadic Approach of the Self-Stress on a Dielectric-Diamagnetic Cylinder with Non-Uniform Speed of Light
• Casimir effect in a wormhole spacetime
• Casimir stress in an inhomogeneous medium
• Local and Global Casimir Energies for a Semitransparent Cylindrical Shell
• The Dirichlet Casimir effect for ${\varphi }^4$ theory in (3+1) dimensions: A new renormalization approach
• A new renormalization approach to the Dirichlet Casimir effect for ϕ^4 theory in (1+1) dimensions
• How does Casimir energy fall? IV. Gravitational interaction of regularized quantum vacuum energy
• Casimir densities for a spherical boundary in de Sitter spacetime
• Cancellation of nonrenormalizable hypersurface divergences and the d-dimensional Casimir piston
• Effective description of brane terms in extra dimensions
• Finite temperature Casimir effect in piston geometry and its classical limit
• An exact solution for the Casimir force in a spherically symmetric medium
• Gravitational and Inertial Mass of Casimir Energy
• Vacuum Energy and Renormalization on the Edge
• Local Casimir Energies for a Thin Spherical Shell
• Boundary divergences in vacuum self-energies and quantum field theory in curved spacetime
• Casimir Piston of Real Materials and its Application to Multi-Layer Models
• Radiative Correction to the Dirichlet Casimir Energy for $\lambda {\varphi }^4$ Theory in Two Spatial Dimensions
• Electromagnetic Casimir energy with extra dimensions
• Geometrically confined thermal field theory: Finite size corrections and phase transitions
• Electromagnetic Casimir densities for a wedge with a coaxial cylindrical shell
• Casimir effect for scalar current densities in topologically nontrivial spaces
• Casimir Self-Entropy of a Spherical Electromagnetic $\delta$-Function Shell
• Slab Bag Fermionic Casimir effect, Chiral Boundaries and Vector Boson - Majorana Fermion Pistons
• Casimir Pistons for Massive Scalar Fields
• Geothermal Casimir Phenomena
• Field of homogeneous Plane in Quantum Electrodynamics
• Local and Global Casimir Energies: Divergences, Renormalization, and the Coupling to Gravity
• Surface Divergences and Boundary Energies in the Casimir Effect
• Casimir energies with finite-width mirrors
• Casimir energies in spherically symmetric background potentials revisited
• An approximation to the Woods-Saxon potential based on a contact interaction
• Unparticle Casimir effect
• Neumann Casimir effect: a singular boundary-interaction approach
• Zeta-function approach to Casimir energy with singular potentials
• Weyl problem and Casimir effects in spherical shell geometry
• Casimir effect in 2+1 dimensional noncommutative theories
• Three dimensional Casimir piston for massive scalar fields
• Resolution of an apparent inconsistency in the electromagnetic Casimir effect
• Scalar Casimir densities for cylindrically symmetric Robin boundaries
• Local Neumann semitransparent layers: resummation, pair production and duality
• Lifshitz formula for the Casimir force and the Gelfand-Yaglom theorem
• Scalar-Mediated Quantum Forces Between Macroscopic Bodies and Interferometry
• Something Can Come of Nothing: Surface Approaches to Quantum Fluctuations and the Casimir Force
• On Casimir Pistons
• Parity violating cylindrical shell in the framework of QED
• The Dirichlet Casimir Energy for ${\varphi }^4$ Theory in a Rectangle
• Casimir effect due to a single boundary as a manifestation of the Weyl problem
• Numerical Analysis of some Generalized Casimir Pistons
• Casimir buoyancy
• Wightman function and scalar Casimir densities for a wedge with two cylindrical boundaries
• Unruh-DeWitt detectors as mirrors: Dynamical reflectivity and Casimir effect
• The Casimir effect for fields with arbitrary spin
• Evaluation of the Casimir Force for a Dielectric-diamagnetic Cylinder with Light Velocity Conservation Condition and the Analogue of Sellmeir's Dispersion Law
• Attractive Electromagnetic Casimir Stress on a Spherical Dielectric Shell
• Vacuum local and global electromagnetic self-energies for a point-like and an extended field source
• Inhomogeneity-related cutoff dependence of the Casimir energy and stress
• Regularization versus renormalization: Why are Casimir energy differences so often finite?
• Perturbative Roughness Corrections to Electromagnetic Casimir Energies
• Casimir effect with machine learning
• Gedanken experiments with Casimir forces, vacuum energy, and gravity
• Casimir Forces in CFT with Defects and Boundaries
• Worldline formalism for a confined scalar field
• Isospin Invariance and the Vacuum Polarization Energy of Cosmic Strings
• Scarf for Lifshitz
• Casimir effect at finite temperature in a real scalar field theory
• Local thermal behaviour of a massive scalar field near a reflecting wall
• Negativity of the Casimir self-entropy in spherical geometries
• Casimir effect with nonlocal boundary interactions
• Vacuum fluctuations in the presence of nonlinear boundary conditions
• Weighing the Vacuum Energy
• Casimir type effects for scalar fields interacting with material slabs
• One-Loop Correction to The Casimir Energy in Lifshitz-Like Theory
• Recent Developments in Quantum Vacuum Energy for Confined Fields
• Searching for Novel Objects in the Electroweak Theory
• Thermodynamics of the scalar radiation in the presence of a reflecting plane wall
• On the Problem of Vacuum Energy in FLRW Universes and Dark Energy
• Local and Global Casimir Energies in a Green's Function Approach
• Dirichlet Spheres in Continuum Quantum Field Theory
• Domain wall interactions due to vacuum Dirac field fluctuations in 2+1 dimensions
• The covariant electromagnetic Casimir effect for real conducting spherical shells
• Vacuum Polarization and Casimir Energy of a Dirac Field Induced by a Scalar Potential in One Spatial Dimension
• Æther Coupling Effects on Casimir Energy for Self-Interacting Scalar Field within Extra Dimension
• The Casimir effect in the presence of infrared transparency
• On the Casimir energy for scalar fields with bulk inhomogeneities
• Path-integral approach to Casimir effect with infinitely thin plates