Schrödinger and Fock representation for a field theory on curved spacetime

Abstract Linear free field theories are one of the few quantum field theories that are exactly soluble. There are, however, (at least) two very different languages to describe them, Fock space methods and the Schrodinger functional description. In this paper, the precise sense in which the two representations are related is explored. Several properties of these representations are studied, among them the well-known fact that the Schrodinger counterpart of the usual Fock representation is described by a Gaussian measure. A real scalar field theory is considered, both on Minkowski spacetime for arbitrary, non-inertial embeddings of the Cauchy surface, and for arbitrary (globally hyperbolic) curved spacetimes. We present both the Gaussian representation with a non-trivial measure and the homogeneous representation where the non-triviality lies in the vacuum state. As concrete examples, the Schrodinger representations on stationary and homogeneous cosmological spacetimes are constructed.