Casimir effect at finite temperature of charged scalar field in an external magnetic field

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, 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 logZ(β). In the limit of zero temperature F (∞) e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] 1 gives the usual Casimir energy. The Casimir effect has been computed for fields other than the electromagnetic and boundary conditions different from the one implemented by conducting plates [1, 2]. In those cases also it is important to consider the finite temperature effects. Here we are interested in the case of a charged scalar field at finite temperature under the combined influence of confinement and an external magnetic field. The confinement is implemented by two impermeable parallel plates, as in the usual electromagnetic Casimir effect described above, and the external magnetic field B is constant and perpendicular to the plates. The influence of the external magnetic field on the Casimir effect has already been computed [8] and here we consider the combined effect of external magnetic field and finite temperature on the Casimir effect. In our calculation we use Schwinger’s formula for the effective action [9], which can be used to calculate the Casimir energy [10]. It can be used also to compute the partition function Z(β) [11] in the imaginary time formalism [12] for finite temperature 1/β: logZ(β) = 1 2 ∫ ∞ so ds s Tr e , (1) where so is a cutoff in the proper-time s, Tr means the total trace and H is the proper-time Hamiltonian in which the frequencies have been discretized to the values i2πn/β (n ∈ 6Z). For the charged scalar field we have H = (−i∂ − eA) + m, where e and m are the charge and mass of the field. We have for the trace in (1): Tr e = 2e 2 ∞ ∑ n=1 eBl 2π e +1) ∞ ∑