  This paper describes recent progress in the analysis of relativistic gauge
conditions for Euclidean Maxwell theory in the presence of boundaries. The
corresponding quantum amplitudes are studied by using Faddeev-Popov formalism
and zeta-function regularization, after expanding the electromagnetic potential
in harmonics on the boundary 3-geometry. This leads to a semiclassical analysis
of quantum amplitudes, involving transverse modes, ghost modes, coupled normal
and longitudinal modes, and the decoupled normal mode of Maxwell theory. On
imposing magnetic or electric boundary conditions, flat Euclidean space bounded
by two concentric 3-spheres is found to give rise to gauge-invariant one-loop
amplitudes, at least in the cases considered so far. However, when flat
Euclidean 4-space is bounded by only one 3-sphere, one-loop amplitudes are
gauge-dependent, and the agreement with the covariant formalism is only
achieved on studying the Lorentz gauge. Moreover, the effects of gauge modes
and ghost modes do not cancel each other exactly for problems with boundaries.
Remarkably, when combined with the contribution of physical (i.e. transverse)
degrees of freedom, this lack of cancellation is exactly what one needs to
achieve agreement with the results of the Schwinger-DeWitt technique. The most
general form of coupled eigenvalue equations resulting from arbitrary
gauge-averaging functions is now under investigation.
