arXiv:hep-th/9810033v1 5 Oct 1998 Casimir effect at finite temperature of charged scalar field in an external magnetic field M. V. Cougo-Pinto ∗ , C. Farina † , M. R. Negr˜ao ‡ and A. Tort § Instituto de F´ ısica, Universidade Federal do Rio de Janeiro CP 68528, Rio de Janeiro, RJ 21945-970, Brazil February 1, 2008 Abstract The Casimir effect for Dirac as well as for scalar charged particles is influenced by external magnetic fields. It is also influenced by finite temperature. Here we consider the Casimir effect for a charged scalar field under the combined influence of an external magnetic field and finite temperature. The free energy for such a system is computed using Schwinger’s method for the calculation of the effective action in the imaginary time formalism. We consider both the limits of strong and weak magnetic field in which we compute the Casimir free energy and pressure. The Casimir effect can be generally defined as the effect of non-trivial space topology on the vacuum fluctuations of relativistic quantum fields [1, 2]. The corresponding change in the vacuum fluctuations appears as a shift in the vacuum energy and a resulting vacuum pressure. In the original Casimir effect [3] two parallel closely spaced conducting plates confine the electromagnetic field vacuum in the region between the plates. We may consider the plates as squares of side ℓ separated by a distance a; the close spacing is implemented by the condition a ≪ ℓ. A shift in the zero point energy, known as the Casimir energy, is produced in passing from the trivial space topology of lR 3 to the topology of lR 2 × [0, a]. As a consequence, the plates are attracted towards each other, albeit being uncharged. This force of attraction has been measured by Sparnaay [4] and recently with high precision by Lamoreaux [5] and by Mohideen and Roy [6]. The slab of vacuum between the plates can be seen as a system with large volume aℓ 2 , energy given by the Casimir energy and pressure given by minus the derivative of the Casimir energy with respect to the spacing a. It is then natural to look for the thermodynamical properties of such a system by considering its behavior at finite temperature [7]. The system at temperature 1/β can be described by its partition function Z (β ) in terms of which we obtain the free energy F (β )= −β −1 log Z (β ). In the limit of zero temperature F (∞) * e-mail: marcus@if.ufrj.br † e-mail: farina@if.ufrj.br ‡ e-mail: guida@if.ufrj.br § e-mail: tort@if.ufrj.br 1