In quantum field theory with confining 'Lhard" (e.g., Dirichlet) boundaries, the latter are represented in the Schrodinger equation defining spatial quantum modes by infinite step-function potentials. One can instead introduce confining "soft" boundaries, represented in the mode equation by some smoothly increasing potential function. Here the global Casimir energy is calculated for a scalar field confined by harmonic-oscillator (HO) potentials in one, two, and three dimensions. Combinations of HO and Dirichlet boundaries are also considered. Some results differ in sign from comparable hard-wall ones.PACS number(s): 03.70.+k

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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.

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Formulae are obtained for the differential decay rate, lepton spectrum, and partial lifetime for the leptonic decays of baryons, which cover effects down to the order of one percent. A weak, linear q2-dependence of the form factors is included, which should be a sufficiently good approximation in the physical q2-range allowed in the decays. The one percent discrepancy arises as a consequence of the above-mentioned approximation to the form factors, whose value and slope at qZ~ 0 are left open in the formulae; SU(3) symmetry, CVC, and PCAC yield an estimate for these parameters.