Quantum Dynamics of the Polarized Gowdy Model

The polarized Gowdy ${\mathbf{T}}^3$ 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 $\mathcal{H}\subset \mathcal{F}$ defined by the quantum constraint. (3) We show that the spatially smeared canonical coordinates and momenta as well as the time-dependent Hamiltonian for $\varphi$ are well-defined, self-adjoint operators for all time, admitting the usual probability interpretation despite the lack of unitary dynamics.