  In this paper we show how to redress a shortcoming of the path integral
scheme for fermionic superfluids and superconductors. This approach is built
around a simultaneous calculation of electrodynamics and thermodynamics. An
important sum rule, the compressibility sum rule, fails to be satisfied in the
usual calculation of the electromagnetic and thermodynamic response at the
Gaussian fluctuation level. Here we present a path integral scheme to address
this inconsistency. Specifically, at the leading order we argue that the
superconducting gap should be calculated using a different saddle point
condition modified by the presence of an external vector potential. This leads
to the well known gauge-invariant BCS electrodynamic response and is associated
with the usual (mean field) expression for thermodynamics. In this way the
compressibility sum rule is satisfied at the BCS level. Moreover, this scheme
can be readily extended to address arbitrary higher order fluctuation theories.
At any level this approach will lead to a gauge invariant and compressibility
sum rule consistent treatment of electrodynamics and thermodynamics.
