On the Relation Between Fock and Schroedinger Representations for a Scalar Field

TL;DR

This paper reviews the relationship between Fock and Schrödinger representations in free scalar quantum field theories, analyzing their properties on various spacetimes and explicitly constructing the Schrödinger representation in cosmological models.

Contribution

It provides a detailed analysis of the connection between Fock and Schrödinger representations, including the Gaussian measure description and explicit constructions in curved spacetimes.

Findings

Fock and Schrödinger representations are related via Gaussian measures.

The Schrödinger representation can be explicitly constructed on stationary cosmological spacetimes.

Properties of these representations are studied on arbitrary non-inertial embeddings and curved spacetimes.

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 Schroedinger functional description. In this paper, the precise sense in which the two representations are related is reviewed. Several properties of these representations are studied, among them the well known fact that the Schroedinger 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. As a concrete example, the Schroedinger representation on stationary and homogeneous cosmological spacetimes is constructed.