Boundaries Immersed in a Scalar Quantum Field

Abstract

We study the interaction between a scalar quantum field \documentclass{article}\pagestyle{empty}\begin{document}$\hat \phi (x)$\end{document}, and many different boundary configurations constructed from (parallel and orthogonal) thin planar surfaces on which \documentclass{article}\pagestyle{empty}\begin{document}$\hat \phi (x)$\end{document} is constrained to vanish, or to satisfy Neumann conditions. For most of these boundaries the Casimir problem has not previously been investigated. We calculate the canonical and improved vacuum stress tensors \documentclass{article}\pagestyle{empty}\begin{document}$ \langle \hat T_{\mu \nu } (x)\rangle\$\end{document} and \documentclass{article}\pagestyle{empty}\begin{document}$ \langle \Theta _{\mu \nu (x)} \rangle\$\end{document} of \documentclass{article}\pagestyle{empty}\begin{document}$\hat \phi (x)$\end{document}; for each example. From these we obtain the local Casimir forces on all boundary planes. For massless fields, both vacuum stress tensors yield identical attractive local Casimir forces in all Dirichlet examples considered. This desirable outcome is not a priori obvious, given the quite different features of \documentclass{article}\pagestyle{empty}\begin{document}$ \langle \hat T_{\mu \nu } (x)\rangle\$\end{document} and \documentclass{article}\pagestyle{empty}\begin{document}$ \langle \Theta _{\mu \nu (x)} \rangle\$\end{document}. For Neumann conditions. \documentclass{article}\pagestyle{empty}\begin{document}$ \langle \hat T_{\mu \nu } (x)\rangle\$\end{document} and \documentclass{article}\pagestyle{empty}\begin{document}$ \langle \Theta _{\mu \nu (x)} \rangle\$\end{document} lead to attractive Casimir stresses which are not always the same. We also consider Dirichlet and Neumann boundaries immersed in a common scalar quantum field, and find that these repel. The extensive catalogue of worked examples presented here belongs to a large class of completely solvable Casimir problems. Casimir forces previously unknown are predicted, among them ones which might be measurable.

Wightman function and scalar Casimir densities for a wedge with two cylindrical boundaries

“…The formulae for the Wightman function and the VEV of the field square in Neumann case are obtained from the corresponding formulae for Dirichlet scalar by the replacements sin(qnφ) → cos(qnφ), I qn (jx) → I ′ qn (jx), K qn (jx) → K ′ qn (jx), j = a, b, and with the term n = 0 included in the summation. In the expressions for the VEVs of the energy-momentum tensor this leads to the change of the sign for the second term in the figure braces on the right of (35) and to the change of the sign for the off-diagonal component (36).…”

“…Having the vacuum energy-momentum tensor we can derive the vacuum forces acting on constraining boundaries evaluating the vacuum stresses at points on the bounding surfaces. As we will see below, in the geometry under consideration these forces are position dependent on the boundary and cannot be obtained by the global method using the total Casimir energy (on the advantages of the local method see also [36]). In the limiting case from the results of the present paper the local vacuum densities are obtained for the geometry of a rectangular waveguide (for the local analysis of quantum fields confined in rectangular cavities see [36,37,38]).…”

“…Introducing in (60) y = y ′ + L 2 /2 and taking the limit L 2 → ∞ with fixed value y ′ , from (60) the vacuum forces for two infinite parallel Dirichlet plates are obtained. Note that the local vacuum densities for a quantum field confined within rectangular cavities are investigated in [36,37,38] (for corresponding global quantities such as the total Casimir energy see [1,4] and references therein).…”

Local Casimir energies for a thin spherical shell

“…For the case of the general stress tensor (3.1), that extra term is [11] 4) where the direction of the normal is out of the region in question, which arises from the T 0i component of the stress tensor, and from ∂ µ T µν = 0. The total energy in a given region V bounded by a surface S is not, therefore, just the integral of the local energy density, but has this extra contribution [11]: 5) which is independent of ξ.…”

Surface divergences and boundary energies in the Casimir effect

“…If a nonconformal scalar stress tensor is used, a position-dependent term in the stress tensor appears, which does not contribute to either the total energy or the pressure on the plates [3,4]. Local surface divergences were first discussed for arbitrary smooth boundaries by Deutsch and Candelas [5].…”