Issues about Cosmological Ward Identities
  In this paper we first discuss how a Noether current corresponding to a gauge
or a global symmetry can locally be introduced in a path integral irrespective
of the boundary conditions defining the theory. We then consider quantization
of gravity plus minimally coupled scalar field system in the phase space path
integral approach. The complete gauge fixed action including the Faddeev-Popov
determinant is obtained in the so called $\zeta$-gauge. It turns out that in
this formalism while the dilatation survives as the residual symmetry of the
gauge fixed action, other diffeomorphisms which require field dependent
corrections fail to be so. The full Noether current for the dilatation is
determined and the spatial boundary conditions that yield a finite and
conserved charge are determined. The charge is shown to be expressible as a
surface integral at infinity and the corresponding Ward identity gives the
standard consistency relation of cosmological perturbations.
