1-loop effective action on the 4-ball

This paper applies zeta-function regularization to evaluate the 1-loop effective action for scalar field theories and Euclidean Maxwell theory in the presence of boundaries. After a comparison of two techniques developed in the recent literature, the vacuum Maxwell theory is studied and the contribution of all perturbative modes to zeta'(0) is derived: transverse, longitudinal and normal modes of the electromagnetic potential, jointly with ghost modes. The analysis is performed by imposing magnetic boundary conditions, when the Faddeev–Popov Euclidean action contains the particular gauge-averaging term which leads to a complete decoupling of all perturbative modes. It is shown that there is no cancellation of the contributions to zeta'(0) resulting from longitudinal, normal and ghost modes