  Free scalar field theory on a flat spacetime can be cast into a generally
covariant form known as parametrised field theory in which the action is a
functional of the scalar field as well as the embedding variables which
describe arbitrary, in general curved, foliations of the flat spacetime.
  We construct the path integral quantization of parametrised field theory in
order to analyse issues at the interface of quantum field theory and general
covariance in a path integral context. We show that the measure in the
Lorentzian path integral is non-trivial and is the analog of the Fradkin-
Vilkovisky measure for quantum gravity. We construct Euclidean functional
integrals in the generally covariant setting of parametrised field theory using
key ideas of Schleich and show that our constructions imply the existence of
non-standard `Wick rotations' of the standard free scalar field 2 point
function. We develop a framework to study the problem of time through
computations of scalar field 2 point functions. We illustrate our ideas through
explicit computation for a time independent 1+1 dimensional foliation. Although
the problem of time seems to be absent in this simple example, the general case
is still open. We discuss our results in the contexts of the path integral
formulation of quantum gravity and the canonical quantization of parametrised
field theory.
