  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.
