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

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.

Figures (7)

• Figure 1: Vacuum energy density (in units $1/a^{D+1}$) as a function of $z/a$ for the Proca field in the limit $m\rightarrow 0$. The graphs are plotted for $D=3,4,5,6$. The product $a^{D+1}\left\langle T_{0}^{0}\right\rangle$ increases with increasing $D$.
• Figure 2: The VEVs of the electric ($U=E$, full curves) and magnetic ($U=B$, dashed curves) field squares (measured in units of $m^{D+1}$) as functions of $z/a$ for the $D=3$ Proca field. The graphs are plotted for $ma=0.75,1,1.25,1.5$. The ratio $|\left\langle U^{2}\right\rangle |/m^{D+1}$ is a decreasing function of $ma$.
• Figure 3: Vacuum energy density (in units of $m^{D+1}$) versus $z/a$ for a massive vector field in $D=3$ spatial dimensions. The graphs are plotted for $ma=0.75,1,1.25,1.5$. The ratio $\left\langle T_{0}^{0}\right\rangle /m^{D+1}$ decreases with increasing $ma$.
• Figure 4: The Casimir pressure for PMC conditions as a function of the distance between the plates for $D=3,4,5,6$.
• Figure 5: VEVs of the electric and magnetic field squares (in units $m^{D+1}$) for the Proca field with PEC boundary conditions in 3-dimensional space as functions of $z/a$. For the parameter $ma$ the values $ma=0.75,1,1.25,1.5$ are taken (the ratio $|\left\langle U^{2}\right\rangle |/m^{D+1}$ is a decreasing function of $ma$).