Casimir effect for scalar current densities in topologically nontrivial spaces

Abstract

We evaluate the Hadamard function and the vacuum expectation value (VEV) of the current density for a charged scalar field, induced by flat boundaries in spacetimes with an arbitrary number of toroidally compactified spatial dimensions. The field operator obeys the Robin conditions on the boundaries and quasiperiodicity conditions with general phases along compact dimensions. In addition, the presence of a constant gauge field is assumed. The latter induces Aharonov-Bohm-type effect on the VEVs. There is a region in the space of the parameters in Robin boundary conditions where the vacuum state becomes unstable. The stability condition depends on the lengths of compact dimensions and is less restrictive than that for background with trivial topology. The vacuum current density is a periodic function of the magnetic flux, enclosed by compact dimensions, with the period equal to the flux quantum. It is explicitly decomposed into the boundary-free and boundary-induced contributions. In sharp contrast to the VEVs of the field squared and the energy-momentum tensor, the current density does not contain surface divergences. Moreover, for Dirichlet condition it vanishes on the boundaries. The normal derivative of the current density on the boundaries vanish for both Dirichlet and Neumann conditions and is nonzero for general Robin conditions. When the separation between the plates is smaller than other length scales, the behavior of the current density is essentially different for non-Neumann and Neumann boundary conditions. In the former case, the total current density in the region between the plates tends to zero. For Neumann boundary condition on both plates, the current density is dominated by the interference part and is inversely proportional to the separation. a

Hadamard function and the vacuum currents in braneworlds with compact dimensions: Two-brane geometry

“…As it has been shown in [16] for the Minkowski bulk and in [3] for the geometry of a single brane on AdS bulk, unlike the VEVs of the field squared and of the energy-momentum tensor, the current density is finite on the branes. For the geometry under consideration, the VEV of the current density on the brane is obtained from (4.3) with z = z j .…”

“…In the limit of the large curvature radius, a ≫ y j , m −1 , one has z ≈ a + y, z j ≈ a + y j , and both the order and arguments of the modified Bessel functions in the integrand of (4.3) are large. By using the corresponding uniform asymptotic expansions (given, for example, in [36]), to the leading order, the result for the geometry of two parallel Robin plates on the Minkowski bulk with the topology R p+1 × T q (see [16]) is obtained:…”

Vacuum currents in braneworlds on AdS bulk with compact dimensions

“…The expression in the right-hand side coincides with the boundary-induced part of the current density for the geometry of a single Robin plate in (D + 1)-dimensional Minkowski spacetime with spatial topology R p+1 × T q (see [46]). …”

“…Here, the first term in the figure braces and the part with the first term in the square brackets come from j l 0 . The expression on the right of (3.34), divided by the conformal factor (z/a) D+1 , coincides with the current density in the region between two plates on Minkowski bulk with Dirichlet boundary condition on the left plate and Robin condition (2.8), with β → β − M , on the right one (see [46] for the problem with Robin boundary conditions on both plates). The fact that the problem with a single brane in AdS bulk in the L-region is conformaly related to the problem with two plates in Minkowski bulk is a consequence of the boundary condition we have imposed on the AdS boundary.…”

“…The vacuum current densities for charged scalar and Dirac spinor fields in de Sitter spacetime with toroidally compact spatial dimensions are considered in [44]. The influence of boundaries on the vacuum currents in topologically nontrivial spaces are studied in [45,46] for scalar and fermionic fields. The effects of nontrivial topology induced by the compactification of a cosmic string along its axis have been discussed in [47][48][49].…”