On the Schroedinger Representation for a Scalar Field on Curved Spacetime

TL;DR

This paper extends the Schroedinger representation of a scalar field to arbitrary curved spacetimes, showing that the measure remains Gaussian and discussing implications for canonical quantization in quantum gravity models.

Contribution

It generalizes the Schroedinger representation for scalar fields from flat to curved spacetimes using classical structures for quantization.

Findings

The measure in the Schroedinger representation remains Gaussian in curved spacetimes.

The construction applies to arbitrary globally hyperbolic spacetimes and embeddings.

Implications for canonical quantization of midisuperspace models are discussed.

Abstract

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.