Local gauge invariant operator on isometry breaking background

TL;DR

This paper explores how to construct local gauge invariant operators in gravity with spontaneously broken isometries, addressing issues of locality, gauge invariance, and implications for black hole information paradox.

Contribution

It introduces a method using the St"uckelberg mechanism to build gauge invariant operators in backgrounds with broken isometries, linking to black hole physics and cosmology.

Findings

Gauge invariant operators can be constructed via St"uckelberg mechanism.

Spontaneous isometry breaking affects local operator definitions.

Strong isometry breaking is needed to suppress fluctuations in late-time scenarios.

Abstract

Whereas local field operators play the crucial role in reconciling quantum mechanics and special relativity, they are not trivially compatible with the diffeomorphism invariance of gravity. In order to address this issue, we consider the background geometry which breaks the isometry spontaneously. Then the local gauge invariant operator can be constructed through the St\"uckelberg mechanism, where the fluctuation of the metric in the direction of the isometry breaking combines with that of matter whose classical solution breaks the isometry. This is equivalent to introducing the clock and the rod to promote the local field operators to the gauge invariant ones. A typical example is the curvature perturbation in quasi-de Sitter space arising from the spontaneous breaking of the timelike isometry. We also discuss the features of the local gauge invariant operator when the spacelike isometry is spontaneously broken. Meanwhile, even if the local gauge invariant operators exist, it does not guarantee the reliable construction of the gauge invariant operators on the local region like the island, which is regarded as an essential ingredient to resolve the black hole information paradox. This is because the fluctuation of the spacetime point is accumulated in time, which in fact also gives rise to eternal inflation in quasi-de Sitter space. In order to suppress the fluctuation at late time, the isometry must be strongly broken by the background. In the case of the evaporating black hole, it may be achieved by the transition to the higher dimensional black hole.