Quantum dynamics of the polarized Gowdy ${\mathrm{T}}^3$ model

The polarized Gowdy ${\mathbf{T}}^3$ vacuum spacetimes are characterized, modulo gauge, by a “point particle” degree of freedom and a function φ that satisfies a linear field equation and a nonlinear constraint. The quantum Gowdy model has been defined by using a representation for φ on a Fock space F. Using this quantum model, it has recently been shown that the dynamical evolution determined by the linear field equation for φ is not unitarily implemented on 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, and (3) we show that the spatially smeared canonical coordinates and momenta as well as the time-dependent Hamiltonian for φ are well-defined, self-adjoint operators for all time, admitting the usual probability interpretation despite the lack of unitary dynamics.