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On the Schroedinger Representation for a Scalar Field on Curved Spacetime

It is generally known that linear (free) field theories are one of the few QFT that are exactly soluble. In the Schroedinger 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 space-times and embeddings of the Cauchy surface are considered. The classical structures relevant for quantization are used for constructing the Schroedinger 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.