Radiative corrections to the Casimir energy in the λ|φ| model under quasi-periodic boundary conditions

We compute the first radiative correction to the Casimir energy in the (d + 1)dimensional λ|φ|4 model submitted to quasi-periodic boundary conditions in one spatial direction. Our results agree with the ones found in the literature for periodic and antiperiodic boundary conditions, special cases of the quasi-periodic boundary conditions. The idea of introducing an arbitrary parameter in order to interpolate continuously distinct theories is not new in the literature. For instance, fermionic and bosonic partition functions can be obtained as particular cases of more general ones which are computed assuming that these fields satisfy a more general boundary conditions (BC) in the imaginary time, where the fields acquire a phase e whenever τ → τ + β (β = 1/T ) [1] (for nonrelativistic partition functions see Ref. [2]). Periodic and antiperiodic BC (for bosons and fermions, respectively) correspond to θ = 0 and θ = π. It can be shown that the same effects can be obtained if instead of introducing the parameter θ, we couple the charged field (bosonic or fermionic) appropriately with a constant gauge potential of the form (A0, 0), which cannot be gauged away due to the compactification in the x0-direction, introduced to take into account the thermal effects. Similarly, we can consider that the field under study is submitted to quasi-periodic BC in a space dimension, which interpolates the periodic and antiperiodic ones. Analogously, the introduction of the interpolating parameter is equivalent to coupling the charged field with a constant gauge field with a non-vanishing component along the space-dimension which is assumed to be compactified [3]. In this work we discuss the effects of an interpolating BC in the vacuum energy of a complex scalar field. More precisely, we compute the O(λ) correction to the Casimir energy Email: [email protected] Email: [email protected] Email: [email protected] CO02-6 2 of a complex scalar field whose dynamics is described by the (Euclidean) lagrangian density LE = |∂μφ| +m|φ| + λ|φ| + Lct, (1) where Lct contains the renormalization counterterms, and subject to quasi-periodic boundary conditions in the xd-direction, i.e., φ(x0, x1, . . . , xd + a) = e φ(x0, x1, . . . , xd), 0 ≤ θ < 2π. (2) In previous works we performed similar calculations for Dirichlet-Dirichlet, Neumann-Neumann [4], and Dirichlet-Neumann [5] boundary conditions. The Casimir energy (per unit area) of a free field subject to those boundary conditions was computed in [6] in the (3 + 1)-dimensional case; the result is E (0) θ (a) ∣