Schrodinger representation for the polarized Gowdy model

TL;DR

This paper constructs the Schrödinger representation for the polarized Gowdy model's scalar field, showing equivalence to the Fock representation at fixed times and analyzing the singularity of measures over time.

Contribution

It develops the Schrödinger representation for the Gowdy model's scalar field and compares it with the Fock representation, highlighting measure singularities over time.

Findings

Schrödinger and Fock representations are unitarily equivalent at fixed times.

Typical field configurations are more singular than square-integrable functions.

Time evolution measures are mutually singular at different times.

Abstract

The polarized ${\bf T}^3$ Gowdy model is, in a standard gauge, characterized by a point particle degree of freedom and a scalar field degree of freedom obeying a linear field equation on ${\bf R}\times{\bf S}^1$. The Fock representation of the scalar field has been well-studied. Here we construct the Schrodinger representation for the scalar field at a fixed value of the Gowdy time in terms of square-integrable functions on a space of distributional fields with a Gaussian probability measure. We show that ``typical'' field configurations are slightly more singular than square-integrable functions on the circle. For each time the corresponding Schrodinger representation is unitarily equivalent to the Fock representation, and hence all the Schrodinger representations are equivalent. However, the failure of unitary implementability of time evolution in this model manifests itself in the mutual singularity of the Gaussian measures at different times.