Quantum Dynamics of the Polarized Gowdy Model

Abstract

The polarized Gowdy ${\mathbf{T}}^3$ vacuum spacetimes are characterized, modulo gauge, by a ``point particle'' degree of freedom and a function $\varphi$ that satisfies a linear field equation and a non-linear constraint. The quantum Gowdy model has been defined by using a representation for $\varphi$ on a Fock space $\mathcal{F}$. Using this quantum model, it has recently been shown that the dynamical evolution determined by the linear field equation for $\varphi$ is not unitarily implemented on $\mathcal{F}$. 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.

24 pages, some typos corrected

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