Vacuum confinement at finite temperature for scalar QED in magnetic field and deformed boundary condition

We investigate the Casimir effect at finite temperature for a charged scalar field in the presence of an external uniform and constant magnetic field, perpendicular to the Casimir plates. We have used a boundary condition characterized by a deformation parameter θ; for θ = 0 we have a periodic condition and for θ = π, an antiperiodic one, for intermediate values, we have a deformation. The temperature was introduced using the imaginary time formalism and both the lagrangian and free energy were obtained from Schwinger proper time method for computing the effective action. We also computed the permeability and its asymptotic expressions for low and high temperatures. The electromagnetic field in classical vacuum is described by Maxwell’s lagrangian −FFμν/4. When only the field B is present, the lagrangian is given by L(0)(B) = −B2/2. The effective lagrangian, that takes into account the free electromagnetic field and the contribution from the medium is described by Leffective = L(0)(B) + L(1)(B), where L(1)(B) is the one loop lagrangian of the system. Expanding L(1)(B), the term proportional to B2 is associated to the linear polarization, while higher order terms represent nonlinear polarizations. In 1935, Euler and Heisenberg [1] obtained the corrections to the effective Lagrangian under an electromagnetic field; the so called Euler-Heisenberg lagrangian. Their result gives the exact expression of the Lagrangian and the [email protected] [email protected] [email protected] [email protected] 1 first correction is given by a term in B4 [1]. The correspondent Lagrangian for scalar QED is referred to as the Weisskopf-Schwinger effective lagrangian[2, 3]. Consider now a constant and uniform magnetic field directed to OZ axis and perpendicular to the Casimir plates. The plates are plane, perfectly conductive, parallel and a is the distance between them. The plates may also be considered squares of side l ≫ a. The boundary condition used here establishes that the scalar field experiences a recoil in phase, at each displacement a along the direction perpendicular to the Casimir plates [5]. This condition is characterized by φ(r + az) = e−iθφ(r). For θ = 0 or, in the limit 2π, this condition is reduced to the periodic one and for θ = π it is antiperiodic. For intermediate values we have a deformed boundary condition. We can consider a circular compactified dimension where the deformed condition occurs, denoted by S1 θ . This condition presents the peculiarity of recoiling the phase of the field in θ at each complete turn; naturally, for θ = 0 we have S1 0 = S 1, that is, the common compactification of R in S1. In other words, the subjacent space to the scalar field is compactified from R3 to R2 × S1 θ , It can be shown that this boundary condition can be generated from the minimum coupling of the charged scalar field in the spacetime given by R2 ×S 1×R, by the use of a constant potential θ/ea along the compactified dimension S1 [5]. We use here the imaginary time formalism [4] to write down Schwinger’s proper-time formalism [3] where the partition function for the bosonic field is logZ = 1 2 ∫ ∞ s0 ds s Tre , (1) where s is the Schwinger’s proper-time, H is the proper-time hamiltonian and s0 is a cut off in the proper time s. The operator H for such charged scalar field is given by H = (P − eA)2 +m2, where P 2 = P2 − (P 0)2. The eigenvalues px and py are restricted to Landau levels and given by px +py 2(2n+1), where n ∈ N. The imposition of the boundary condition on the z component of the momentum gives: pz = (2n1π − θ)/a, n1 ∈ Z; finally, the Matsubara frequencies are p0 = (2πn2i/β), where β = 1/T and n2 ∈ Z. Then the trace reads: Tr e = 2 eBl2 2π ∞ ∑