https://doi.org/10.1140/epjp/s13360-026-07346-9
Regular Article
Two-point functions and the vacuum densities in the Casimir effect for the Proca field
Institute of Physics, Yerevan State University, 1 Alex Manoogian Street, 0025, Yerevan, Armenia
a
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Received:
20
July
2025
Accepted:
20
January
2026
Published online:
3
February
2026
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. The Casimir–Polder forces acting on a polarizable particle are repulsive with respect to the nearest plate for PMC conditions. For PEC conditions, those forces are attractive for 
and vanish for 
. For 
, the vacuum energy density is positive for the PMC conditions and negative for the PEC conditions. For 
and for PMC conditions, the vacuum energy density is negative in the region between the plates and vanishes in the remaining regions. For and PMC conditions, the energy density is negative everywhere. In the case of PEC conditions, the vacuum energy density is positive for and vanishes for .
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
