  Some recent results show that the covariant path integral and the integral
over physical degrees of freedom give contradicting results on 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
non-covariant measure on the space of physical degrees of freedom. Starting
with the reduced phase space path integral and using covariant measure
throughout computations we recover 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 Gaussian path
integral on the space of physical degrees of freedom some basic symmetries are
broken.
