QED on a curved background and on manifolds with boundaries: Unitarity versus covariance

Some recent results show that the covariant path integral and the integral over physical degrees of freedom give contradictory results on a curved background and on manifolds with boundaries. This looks like a conflict between unitarity and covariance. We argue that this effect is due to the use of a noncovariant measure on the space of physical degrees of freedom. Starting with the reduced phase space path integral and using the covariant measure throughout the computations we recover the standard path integral in the Lorentz gauge and the Moss and Poletti BRST-invariant boundary conditions. We also demonstrate by direct calculations that in the approach based on a Gaussian path integral on the space of physical degrees of freedom some basic symmetries are broken.