Magnetic properties of confined bosonic vacuum at finite temperature

We compute the combined effect of confinement, an external magnetic field and temperature on the vacuum of the charged scalar field using Schwinger’s formula for the effective action in the imaginary time formalism. The final result reproduces an effective Lagrangian similar to the Heisenberg-Euler one in the limit of no confinement, in the case of confinement it provides the necessary corrections to this Lagrangian at each order of magnitude of the magnetic field. The results show a finite temperature contribution to the vacum permeability constant apart from the one due to confinement alone. [email protected] [email protected] [email protected] 1 In a previous work [1] we computed the combined effect of confinement and an external magnetic field on the constitutive relations of the vacuum of the charged scalar field. Knowing that both the confinement and the applied electromagnetic field affect the quantum vacuum of the charged scalar field [1] it is natural to ask what is the effect of considering the whole system at finite temperature. Let us consider the vacuum of the complex scalar field of mass m and charge e confined between two large parallel plates of side l and separation a, under the influence of an external uniform constant magnetic field B with direction perpendicular to the plates. The confinement is described by the Dirichlet boundary conditions, which demand a vanishing field within the plates. The system at temperature 1/β can be described by its partition functiom Z(β). From the Schwinger’s formulas for the partition function [2] and for the effective action at zero temperature[3] it is straightforward that they obey the relation: W = ilogZ(β). (1) The expression for logZ(β) [4] is given by: logZ(β) = 1 2 ∫ ∞ so ds s Tr e , (2) 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 (2): Tr e = 2e 2 ∞ ∑ n=1 eBl 2π e +1) ∞ ∑