The Casimir energy in Maxwell-Chern-Simons electrodynamics with various perfect electric or magnetic boundary condition combinations
We investigate the Casimir effect between two parallel wires in $(2+1)D$ Maxwell-Chern-Simons (MCS) electrodynamics under different boundary conditions and separated by a distance $L$. Using a functional approach in which the boundary constraints are implemented by auxiliary fields localized on the wires, we integrate out the gauge field and derive an effective boundary theory whose functional determinant allows us to compute the Casimir energy. We consider perfect electric conductor (PEC), perfect magnetic conductor (PMC), and infinitely permeable (PMC2) boundary conditions. For two PEC wires, we recover the known MCS result and show that it coincides with the result obtained in the Abelian limit of the Curci-Ferrari (CF) model, implying that the Casimir effect in this case probes the photon mass $μ$ but not its origin. The same energy is obtained for two PMC2 wires. In sharp contrast, the Casimir energy vanishes identically for the covariant PMC condition in MCS theory, both for PMC-PMC and PEC-PMC. Since this cancellation does not occur in the CF model, it could provide a signature of the topological origin of the photon mass in $(2+1)D$. Finally, the PEC-PMC2 case yields a non-vanishing Casimir force and reproduces the known result for a perfectly conducting wire facing an infinitely permeable one. Interestingly enough, we show that for PEC-PMC2, the Casimir force in the MCS theory is smaller than the one of the CF model for any $L,μ>0$. Our results thus show that, depending on the boundary conditions, the Casimir effect could probe not only the photon mass but also reveal signatures of its origin.