Quantum Gowdy $T^3$ model: A uniqueness result

TL;DR

This paper proves the uniqueness of a specific Fock quantisation for the Gowdy $T^3$ cosmology model, ensuring consistent quantum dynamics under symmetry and unitarity constraints, and extends the result to more general field dynamics.

Contribution

It establishes the uniqueness (up to unitary equivalence) of the Fock quantisation for the Gowdy $T^3$ model under symmetry and unitarity requirements, and generalizes the proof to broader field dynamics.

Findings

The chosen Fock quantisation is unique up to unitary equivalence.

The proof applies to more general field dynamics beyond Gowdy cosmologies.

The quantisation respects the global $S^1$ translation symmetry.

Abstract

Modulo a homogeneous degree of freedom and a global constraint, the linearly polarised Gowdy $T^3$ cosmologies are equivalent to a free scalar field propagating in a fixed nonstationary background. Recently, a new field parameterisation was proposed for the metric of the Gowdy spacetimes such that the associated scalar field evolves in a flat background in 1+1 dimensions with the spatial topology of $S^1$, although subject to a time dependent potential. Introducing a suitable Fock quantisation for this scalar field, a quantum theory was constructed for the Gowdy model in which the dynamics is implemented as a unitary transformation. A question that was left open is whether one might adopt a different, nonequivalent Fock representation by selecting a distinct complex structure. The present work proves that the chosen Fock quantisation is in fact unique (up to unitary equivalence) if one demands unitary implementation of the dynamics and invariance under the group of constant $S^1$ translations. These translations are precisely those generated by the global constraint that remains on the Gowdy model. It is also shown that the proof of uniqueness in the choice of complex structure can be applied to more general field dynamics than that corresponding to the Gowdy cosmologies.