We discuss the Casimir effect for a massive bosonic field with mixed (Dirichlet-Neumann) boundary conditions. We use the ζ-function regularization prescription to obtain our physical results. Particularly, we analyse how the Casimir energy varies with the mass of the field and compare this mass dependence with those obtained for other boundary conditions. This is done graphically. Some other graphs involving a massive fermionic field are also included.

We present explicitly another example of a temperature inversion symmetry in the Casimir effect for a nonsymmetric boundary condition. We also give an interpretation for our result. PACS numbers: 11.10.Wx, 12.20.Ds, This brief report was motivated by a recent paper published by Santos et al. [1], in which they discuss the temperature inversion symmetry in the Casimir effect [2] for mixed boundary conditions (for a detailed discussion on the Casimir effect see [3,4] and references therein). In an earlier paper, Ravndal and Tollefsen [5] showed that for the usual setup of two parallel plates a simple inversion symmetry arises in the Casimir effect at finite temperature. Temperature inversion symmetry also appeared in the Brown-Maclay work[6] where they related directly the zero-temperature Casimir energy to the energy density of blackbody radiation at temperature T . A few other papers on this kind of symmetry have also been published [7,8,9,10,11]. Until the publication of Ref. [1], this kind of inversion symmetry had appeared only in calculations of Casimir energy involving symmetric boundary conditions. In 1999, Santos et al. [1] showed, for the case of a massless scalar field submitted to mixed boundary conditions (Dirichlet-Neumann), that the Helmholtz free energy per unit area could be written as a sum of two terms, each of them obeying separately a temperature inversion symmetry. Our purpose here is to present another kind of nonsymmetric boundary condition for which there exists such a symmetry. We show * Electronic address: acap@if.ufrj.br † Electronic address: britto@if.ufrj.br ‡ Electronic address: fabiopr@if.ufrj.br § Electronic address: siqueira@if.ufrj.br