Uniqueness of the Fock quantization of a free scalar field on $S^1$ with time dependent mass

TL;DR

This paper proves the unique quantum representation for a free scalar field on a circle with time-dependent mass, ensuring a consistent unitary evolution in the quantum theory.

Contribution

It establishes the uniqueness of the Fock quantization for a scalar field with time-dependent mass on $S^1$, extending previous results to more general mass functions.

Findings

The massless case provides a unitary implementation of dynamics.

The representation is uniquely determined among $S^1$-invariant complex structures.

Results extend to fields on the two-sphere with axial symmetry.

Abstract

We analyze the quantum description of a free scalar field on the circle in the presence of an explicitly time dependent potential, also interpretable as a time dependent mass. Classically, the field satisfies a linear wave equation of the form $\ddot{\xi}-\xi"+f(t)\xi=0$. We prove that the representation of the canonical commutation relations corresponding to the particular case of a massless free field ($f=0$) provides a unitary implementation of the dynamics for sufficiently general mass terms, $f(t)$. Furthermore, this representation is uniquely specified, among the class of representations determined by $S^1$-invariant complex structures, as the only one allowing a unitary dynamics. These conclusions can be extended in fact to fields on the two-sphere possessing axial symmetry. This generalizes a uniqueness result previously obtained in the context of the quantum field description of the Gowdy cosmologies, in the case of linear polarization and for any of the possible topologies of the spatial sections.