Fock quantization of a scalar field with time dependent mass on the three-sphere: unitarity and uniqueness

TL;DR

This paper establishes a unique, unitary Fock quantization for a scalar field with time-dependent mass on the three-sphere, ensuring consistent quantum descriptions relevant for cosmological models and other compact manifolds.

Contribution

It proves the uniqueness and unitarity of the Fock representation for a scalar field with time-dependent mass on the three-sphere, extending to other compact manifolds.

Findings

The natural Fock representation for a massless field remains unitary with time-dependent mass.

The Fock representation is unique up to unitary equivalence under SO(4) invariance and unitarity.

Results apply to cosmological perturbations and other compact spatial topologies.

Abstract

We study the Fock description of a quantum free field on the three-sphere with a mass that depends explicitly on time, also interpretable as an explicitly time dependent quadratic potential. We show that, under quite mild restrictions on the time dependence of the mass, the specific Fock representation of the canonical commutation relations which is naturally associated with a massless free field provides a unitary dynamics even when the time varying mass is present. Moreover, we demonstrate that this Fock representation is the only acceptable one, up to unitary equivalence, if the vacuum has to be SO(4)-invariant (i.e., invariant under the symmetries of the field equation) and the dynamics is required to be unitary. In particular, the analysis and uniqueness of the quantization can be applied to the treatment of cosmological perturbations around Friedmann-Robertson-Walker spacetimes with the spatial topology of the three-sphere, like e.g. for gravitational waves (tensor perturbations). In addition, we analyze the extension of our results to free fields with a time dependent mass defined on other compact spatial manifolds. We prove the uniqueness of the Fock representation in the case of a two-sphere as well, and discuss the case of a three-torus.