Canonical Quantization of the Gowdy Model

TL;DR

This paper develops a canonical quantization of the Gowdy universe model with three-torus topology, simplifying the classical system and constructing its quantum states and observables within a reduced phase space framework.

Contribution

It introduces a gauge fixing and reduction procedure in the Ashtekar formulation, leading to a manageable quantum model of Gowdy universes with explicit state and observable construction.

Findings

Derived explicit reduced Hamiltonian and line element.

Constructed the Hilbert space of physical quantum states.

Identified the form of quantum observables.

Abstract

The family of Gowdy universes with the spatial topology of a three-torus is studied both classically and quantum mechanically. Starting with the Ashtekar formulation of Lorentzian general relativity, we introduce a gauge fixing procedure to remove almost all of the non-physical degrees of freedom. In this way, we arrive at a reduced model that is subject only to one homogeneous constraint. The phase space of this model is described by means of a canonical set of elementary variables. These are two real, homogeneous variables and the Fourier coefficients for four real fields that are periodic in the angular coordinate which does not correspond to a Killing field of the Gowdy spacetimes. We also obtain the explicit expressions for the line element and reduced Hamiltonian. We then proceed to quantize the system by representing the elementary variables as linear operators acting on a vector space of analytic functionals. The inner product on that space is selected by imposing Lorentzian reality conditions. We find the quantum states annihilated by the operator that represents the homogeneous constraint of the model and construct with them the Hilbert space of physical states. Finally, we derive the general form of the quantum observables of the model.