  The polarized Gowdy ${\bf T}^3$ vacuum spacetimes are characterized, modulo
gauge, by a ``point particle'' degree of freedom and a function $\phi$ that
satisfies a linear field equation and a non-linear constraint. The quantum
Gowdy model has been defined by using a representation for $\phi$ on a Fock
space $\cal F$. Using this quantum model, it has recently been shown that the
dynamical evolution determined by the linear field equation for $\phi$ is not
unitarily implemented on $\cal F$. In this paper: (1) We derive the classical
and quantum model using the ``covariant phase space'' formalism. (2) We show
that time evolution is not unitarily implemented even on the physical Hilbert
space of states ${\cal H} \subset {\cal F}$ defined by the quantum constraint.
(3) We show that the spatially smeared canonical coordinates and momenta as
well as the time-dependent Hamiltonian for $\phi$ are well-defined,
self-adjoint operators for all time, admitting the usual probability
interpretation despite the lack of unitary dynamics.
