ONE-LOOP EFFECTIVE ACTION FOR EUCLIDEAN-MAXWELL THEORY ON MANIFOLDS WITH A BOUNDARY

This paper studies the one-loop effective action for Euclidean Maxwell theory about flat four-space bounded by one three-sphere, or two conc entric three-spheres. The analysis relies on the Faddeev-Popov formali sm and zeta-function regularization, and the Lorentz gauge-averaging t erm is used with magnetic boundary conditions. The contributions of tr ansverse, longitudinal, and normal modes of the electromagnetic potent ial, jointly with ghost modes, are derived in detail. The most difficu lt part of the analysis consists in the eigenvalue condition given by the determinant of a 2X2 or a 4X4 matrix for longitudinal and normal m odes. It is shown that the former splits into a sum of Dirichlet and R obin contributions, plus a simpler term. This is the quantum-cosmologi cal case. In the latter case, however, when magnetic boundary conditio ns 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 effecti ve action in covariant gauges, on manifolds with boundary. The final r esult differs from the value obtained when only transverse modes are q uantized, or when noncovariant gauges are used.