Quantum Dynamics of the Polarized Gowdy Model

TL;DR

This paper investigates the quantum dynamics of the polarized Gowdy T^3 model, revealing that time evolution cannot be unitarily implemented, yet the model's operators remain well-defined and physically interpretable.

Contribution

It derives the classical and quantum Gowdy model using covariant phase space and demonstrates non-unitary evolution while maintaining well-defined operators.

Findings

Time evolution is not unitarily implementable.

Operators for coordinates, momenta, and Hamiltonian are self-adjoint.

The model retains a consistent probability interpretation despite non-unitarity.

Abstract

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