Uniqueness of the Fock quantization of Dirac fields in 2+1 dimensions

TL;DR

This paper demonstrates that in 2+1-dimensional conformally ultrastatic backgrounds, the combined conditions of symmetry invariance and unitarily implementable dynamics uniquely determine the Fock quantization of Dirac fields, resolving ambiguity.

Contribution

It establishes a criterion combining symmetry invariance and unitarity to select a unique Fock quantization for Dirac fields in non-stationary backgrounds.

Findings

Unique Fock representation is selected by combined invariance and unitarity.

The splitting of field time dependence is essentially unique.

The approach resolves quantization ambiguities in non-stationary spacetimes.

Abstract

We study the Fock quantization of a free Dirac field in 2+1-dimensional backgrounds which are conformally ultrastatic, with a time-dependent conformal factor. As it is typical for field theories, there is an infinite ambiguity in the Fock representation of the canonical anticommutation relations. Different choices may lead to unitarily inequivalent theories that describe different physics. To remove this ambiguity one usually requires that the vacuum be invariant under the unitary transformations that implement the symmetries of the equations of motion. However, in non-stationary backgrounds, where time translation is not a symmetry transformation, the requirement of vacuum invariance is in general not enough to fix completely the Fock representation. We show that this problem is overcome in the considered scenario by demanding, in addition, a unitarily implementable quantum dynamics. The combined imposition of these conditions selects a unique family of equivalent Fock representations. Moreover, one also obtains an essentially unique splitting of the time variation of the Dirac field into an explicit dependence on the background scale factor and a quantum evolution of the corresponding creation and annihilation operators.