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.

Effective actions, boundaries, and precision calculations of Casimir energies

“…Effective interactions which de- ‡ For recent calculations discussing loop contributions including boundary effects, see [14] and references therein.…”

Energy-Momentum Tensors With Worldline Numerics

“…There remain, however, divergences on the boundary ∂D due to the Dirichlet limit. These can be interpreted as the infinite amount of energy needed to constrain a scalar quantum field on all momentum scales to satisfy the Dirichlet constraint on the boundary 20 . Interpreting the eigenvalue problem (3) as a ficticious quantum mechanical Schrödinger problem with Hamiltonian H = −∇ 2 + σ(x), the Green's function corresponds to a quantum mechanical propagator Fourier transformed to energy space, which can be written as a Feynman path integral in position space.…”

Effective description of brane terms in extra dimensions

“…One example of this situation is the calculation of zeropoint (Casimir) energies of quantum electromagnetic fields in a conducting cavity or in the presence of conducting plates, and toy models related to this situation 1 . If, instead, ǫ −1 0 Λ 0 then the microscopic structure of the branes 1 See [4] for a discussion of effective theories and matching in the context of the Casimir effect, and [5] for calculations with "fat" branes. Note that the localized energy density lies beyond the reach of the effective field theory, and can only be described at the level of the fundamental theory.…”

“…For instance, in the nonchiral class of deconstructed fermions, if we do not fine-tune the mass and Wilson term near the boundaries (as has been assumed so far), both zero modes combine to form a massive Dirac mode [1]. The full set of KK masses is then given by 4) while the wave functions of the right-handed fermion are 5) to first order. The same form of the masses and wave functions, plus one localized tachyon, is obtained for deconstructed even scalars with the mentioned operators adjusted to zero only near the boundary y = 0.…”