Quantum dynamics of the polarized GowdyT3model

Abstract

The polarized Gowdy ${\mathbf{T}}^{3}$ vacuum spacetimes are characterized, modulo gauge, by a ``point particle'' degree of freedom and a function \ensuremath{\varphi} that satisfies a linear field equation and a nonlinear constraint. The quantum Gowdy model has been defined by using a representation for \ensuremath{\varphi} 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 \ensuremath{\varphi} 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 $\mathcal{H}\ensuremath{\subset}\mathcal{F}$ 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 \ensuremath{\varphi} are well-defined, self-adjoint operators for all time, admitting the usual probability interpretation despite the lack of unitary dynamics.