Two-point functions and the vacuum densities in the Casimir effect for the Proca field

TL;DR

This paper analyzes the vacuum state properties of the Proca field between parallel plates in higher-dimensional Minkowski space, deriving explicit two-point functions and vacuum expectation values under different boundary conditions.

Contribution

It provides explicit expressions for vacuum expectation values and explores the effects of boundary conditions on the Proca field in higher dimensions.

Findings

Vacuum expectation values are explicitly calculated for the Proca field.

Casimir forces are attractive under both PMC and PEC boundary conditions.

Zero-mass limits of VEVs differ between boundary conditions due to polarization mode influences.

Abstract

We investigate the properties of the vacuum state for the Proca field in the geometry of two parallel plates on background of (D+1)-dimensional Minkowski spacetime. The two-point functions for the vector potential and the field tensor are evaluated for higher-dimensional generalizations of the perfect magnetic conductor (PMC) and perfect electric conductor (PEC) boundary conditions. Explicit expressions are provided for the vacuum expectation values (VEVs) of the electric and magnetic field squares, field condensate, and for the VEV of the energy-momentum tensor. In the zero-mass limit the VEVs of the electric and magnetic field squares and the condensate reduce to the corresponding expressions for a massless vector field. The same is the case for the VEV of the energy-momentum tensor in the problem with PEC conditions. However, for PMC conditions the zero-mass limit for the vacuum energy-momentum tensor differs from the corresponding VEV for a massless field. This difference in the zero-mass limits is related to the different influences of the boundary conditions on the longitudinal polarization mode of a massive vector field. The PMC conditions constrain all the polarization modes including the longitudinal mode, whereas PEC conditions do not influence the longitudinal mode. The vacuum energy-momentum tensor is diagonal. The normal stress is uniformly distributed in the region between the plates and vanishes in the remaining regions. The corresponding Casimir forces are attractive for both boundary conditions.