  The well-known discrepancies between covariant and non-covariant formalisms
in quantum field theory and quantum cosmology are analyzed by focusing on the
Coulomb gauge for vacuum Maxwell theory. On studying a flat Euclidean
background with boundaries, the corresponding mode-by-mode analysis of one-loop
quantum amplitudes agrees with the results of the Schwinger-DeWitt technique
and of mode-by-mode calculations in relativistic gauges.
