Abstract
We revisit the path integral computation of the Casimir energy between two infinite parallel plates placed in a QED vacuum. We implement perfectly magnetic conductor boundary conditions (as a prelude to the dual superconductor picture of the QCD vacuum) via constraint fields and show how an effective gauge theory can be constructed for the constraint boundary fields, from which the Casimir energy can be simply computed, in perfect agreement with the usual more involved approaches. Gauge invariance is natural in this framework, as well as the generalization of the result to $d$ dimensions. We also pay attention to the case where the outside of the plates is not the vacuum, but a perfect magnetic (super)conductor, disallowing any dynamics outside the plates. We find perfect agreement between both setups.
Highlights
We will derive the effective boundary action, followed by a different route where the boundary conditions are imposed before the gauge fixing, leading to the very same effective action and ensuing Casimir energy/force
As usual in quantum field theory, in order to obtain the field equations, one takes the variation, in the functional sense, of the action while terms obtained at the boundary are dropped by requiring that the fields decay to zero fast enough
We will report on these results soon and compare with [34]. Another generalization will be to the non-Abelian case, where we foresee an interesting interplay between the Casimir energy and the nonperturbative effects generated by Gribov copies [35,36]
Summary
The Casimir energy and its related force per unit area [1] are among the most spectacular effects of the quantum vacuum not being “empty”: thanks to the nontrivial virtual particles swarming between two (electromagnetically uncharged) parallel (flat) plates, these can attract each other. This can be traced back to the boundary conditions the quantum electromagnetic field modes are subject to, see e.g., [2–4]. We will derive the effective boundary action, followed by a different route where the boundary conditions are imposed before the gauge fixing, leading to the very same effective action and ensuing Casimir energy/force.
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