Quantum Dynamics of the Polarized Gowdy Model

Abstract

The polarized Gowdy ${\bf T}^3$ 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 non-linear constraint. The quantum Gowdy model has been defined by using a representation for $ϕ$ 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 $ϕ$ 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 $ϕ$ are well-defined, self-adjoint operators for all time, admitting the usual probability interpretation despite the lack of unitary dynamics.