One-Loop Effective Action for Euclidean Maxwell Theory on Manifolds with Boundary

TL;DR

This paper calculates the one-loop effective action for Euclidean Maxwell theory on bounded manifolds, using advanced regularization and gauge techniques, revealing differences from previous non-covariant or transverse-only approaches.

Contribution

It provides a detailed derivation of the one-loop effective action for Maxwell theory with boundary conditions, including longitudinal, normal, and ghost modes, in covariant gauges.

Findings

Explicit eigenvalue conditions for boundary modes are derived.

The effective action differs from previous results that only considered transverse modes.

The analysis enhances understanding of gauge theories on manifolds with boundary.

Abstract

This paper studies the one-loop effective action for Euclidean Maxwell theory about flat four-space bounded by one three-sphere, or two concentric three-spheres. The analysis relies on Faddeev-Popov formalism and $\zeta$-function regularization, and the Lorentz gauge-averaging term is used with magnetic boundary conditions. The contributions of transverse, longitudinal and normal modes of the electromagnetic potential, jointly with ghost modes, are derived in detail. The most difficult part of the analysis consists in the eigenvalue condition given by the determinant of a $2 \times 2$ or $4 \times 4$ matrix for longitudinal and normal modes. It is shown that the former splits into a sum of Dirichlet and Robin contributions, plus a simpler term. This is the quantum cosmological case. In the latter case, however, when magnetic boundary conditions are imposed on two bounding three-spheres, the determinant is more involved. Nevertheless, it is evaluated explicitly as well. The whole analysis provides the building block for studying the one-loop effective action in covariant gauges, on manifolds with boundary. The final result differs from the value obtained when only transverse modes are quantized, or when noncovariant gauges are used.