Gravitational null rays: Covariant Quantization and the Dressing Time

TL;DR

This paper develops a gauge-invariant quantization of gravitational null rays using a gravitational field-based quantum reference frame, introducing covariant normal ordering and revealing a Virasoro algebra structure.

Contribution

It introduces a novel covariant normal ordering and a quantum dressing map that captures the full diffeomorphism group, advancing gauge-invariant quantum gravity methods.

Findings

The gauge-invariant algebra forms a Virasoro crossed product.

The dressing map deforms operator products as a quantization of the Dirac bracket.

The physical Hilbert space allows a Page-Wootters reduction with non-ideal dressing time.

Abstract

We quantize the degrees of freedom on a gravitational null ray segment in a fully gauge-invariant manner by using the dressing time as a quantum reference frame (QRF). Our work goes beyond previous models in that the QRF we employ is made out of the gravitational field itself, and accounts for the full group of diffeomorphisms along the ray, not just a locally compact subgroup. The key tool we introduce is covariant normal ordering, a QRF-dependent but background-independent renormalization prescription that restores diffeomorphism covariance at the quantum level. This enables the definition of a quantum dressing map whose image is the algebra of gauge-invariant observables. We find that this algebra carries the structure of a Virasoro crossed product, and that the dressing map induces a deformed product on gauge-fixed operators which can be understood as a quantization of the Dirac bracket, with consequences for the fluctuations of observables. We explain how to cancel anomalies in the physical Hilbert space representation of the gauge-invariant algebra by including a deformation at the classical level, thereby eliminating all spurious degrees of freedom at the quantum level. The physical Hilbert space admits a Page-Wootters reduction map to the perspective of the dressing time, and we show that the dressing time is non-ideal in the sense that its distinct coherent states have non-vanishing overlaps governed by the Teo-Takhtajan energy, i.e. the K\"ahler potential for Virasoro coadjoint orbits.