Uniqueness of the Fock representation of the Gowdy $S^1\times S^2$ and $S^3$ models

TL;DR

This paper proves the uniqueness of a Fock representation for the Gowdy $S^1\times S^2$ and $S^3$ models, ensuring a consistent quantum description with unitarily implemented dynamics and symmetries.

Contribution

It establishes the uniqueness (up to equivalence) of the SO(3)-invariant Fock quantization with unitary dynamics for these cosmological models.

Findings

Existence of a unique Fock representation with unitarity

Implementation of SO(3)-symmetries unitarily

No alternative invariant complex structure yields unitarity

Abstract

After a suitable gauge fixing, the local gravitational degrees of freedom of the Gowdy $S^1\times S^2$ and $S^3$ cosmologies are encoded in an axisymmetric field on the sphere $S^2$. Recently, it has been shown that a standard field parametrization of these reduced models admits no Fock quantization with a unitary dynamics. This lack of unitarity is surpassed by a convenient redefinition of the field and the choice of an adequate complex structure. The result is a Fock quantization where both the dynamics and the SO(3)-symmetries of the field equations are unitarily implemented. The present work proves that this Fock representation is in fact unique inasmuch as, up to equivalence, there exists no other possible choice of SO(3)-invariant complex structure leading to a unitary implementation of the time evolution.