Casimir force for the maxwell-chern-simons-proca model

Casimir Force for the Maxwell-Chern-Simons-Proca Model
Carlos Rafael M. S., J. F. de Medeiros Neto - Universidade Federal do Par´ a Rudnei O. Ramos -Departamento de F´sica Te´ rica, Universidade do Estado do Rio de Janeiro, 20550-013, Rio de Janeiro, RJ, Brazil ı o

Junho de 2011, Bel´ m, Pa, Brasil e

Introducao ¸˜
In a recent work, two of us considered the vortex condensation in the CSH model [1], specialized to the case of the self-dual potential for the scalar field [2, 3], in which case vortices can be considered as noninteracting. The strategy used to study the vortex condensation problem in the CSH model was to make explicit the vortex excitations in the functional action, by making use of a series of dual transformations for the original Lagrangian fields, obtaining an equivalent action, in which it became clear the vortex contributions, represented by a complex scalar vortex field ψ coupled to a gauge field. In a (2+1)-dimensional Euclidean space-time, the dual action can be written as S= +
2 d3x c1α2Hµν + ic2α2ǫµνγ hµ∂ν hγ |∂µψ + ic3αhµψ|2 (1) (∂µhµ)2 2 2 , M |ψ| + 2α

2

The Boundary Conditions

NWe must investigate if the boundary conditions considered for gauge field can be mapped properly in well defined boundary conditions for the scalar fields {φ, ϕ} (and viceversa), in order to obtain the Casimir force for the MPCS associated to the initial vortex model. For this purpose, we need to invert the equations (given in [8]), in order to obtain Ai in terms of {φ, ϕ}. Initially, we have 1 √ 1 √ 2φ − 2ϕ A1 = 2θµ1 2θν1 (5)

which implies that A1 must also obey Neumann boundary conditions: ∂1A1(0) = ∂1A1(L) = 0. (19) In order to find the behavior of A2 (or a function of of A2), we proceed inverting the equations of Ref. [8] that lead from A1, A2, π 1, π 2 to {φ, ϕ, πϕ, πφ}. We find √ √ 1 1 A2 = √ µ1πφ − M 2µ1∂2∂1φ − θν1πϕ + θ 2N ν1∂2∂1ϕ. 2 2 (20) with 1 1 µ µ µ µ M= , N =− . 1+ + 1+ + 2 2 2 2 2θ 4m 4m 2θ 4m 4m (21) For the geometry that we are