The well-known discrepancies between covariant and non-covariant forma
lisms in quantum field theory and quantum cosmology are analyzed by fo
cusing on the Coulomb gauge for vacuum Maxwell theory. On studying a f
lat Euclidean background with boundaries, the corresponding mode-by-mo
de analysis of one-loop quantum amplitudes agrees with the results of
the Schwinger-DeWitt technique and of mode-by-mode calculations in rel
ativistic gauges.