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.