Schrödinger representation for the polarized Gowdy model

Abstract

The polarized T3 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 R × S1. The Fock representation of the scalar field has been well studied. We construct the Schrödinger 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 Schrödinger representation is unitarily equivalent to the Fock representation, and hence all the Schrödinger 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.