It is generally known that linear ~free! field theories are one of the few quantum field theories that are exactly soluble. In the Schro¨dinger functional description of a scalar field on flat Minkowski spacetime and for flat embeddings, it is known that the usual Fock representation is described by a Gaussian measure. In this paper, arbitrary globally hyperbolic spacetimes and embeddings of the Cauchy surface are considered. The classical structures relevant for quantization are used for constructing the Schro¨dinger representation in the general case.
It is shown that, in this case, the measure is also Gaussian. Possible implications for the program of canonical quantization of midisuperspace models are pointed out.