Vacua of the gravitational field

TL;DR

This paper explores the structure of vacua in quantum gravity, revealing a large degeneracy of inequivalent vacua characterized by Goldstone bosons and defects, and analyzing their symmetries and charges.

Contribution

It constructs a family of vacua interpolating between null infinities, characterized by Goldstone bosons, and analyzes their symmetry properties and charge structure.

Findings

Vacua are characterized by an arbitrary Goldstone boson on the sphere.

Defects in vacua carry superrotation charges but no Poincaré charges.

Positivity of energy constrains the stress-tensor, ruling out finite superrotations as canonical transformations.

Abstract

The Poincar\'e invariant vacuum is not unique in quantum gravity. The BMS supertranslation symmetry originally defined at null infinity is spontaneously broken and results in inequivalent Poincar\'e vacua. In this paper we construct the unique vacua which interpolate between past and future null infinity in BMS gauge and which are entirely characterized by an arbitary Goldstone boson defined on the sphere which breaks BMS invariance. We show that these vacua contain a defect which carries no Poincar\'e charges but which generically carries superrotation charges. We argue that there is a huge degeneracy of vacua with multiple defects. We also present the single defect vacua with its canonically conjugated source which can be constructed from a Liouville boson on the stereographic plane. We show that positivity of the energy forces the stress-tensor of the boson to vanish as a boundary condition. Finite superrotations, which turn on the sources, are therefore physically ruled out as canonical transformations around the vacua. Yet, infinitesimal superrotations are external symplectic symmetries which are associated with conserved charges which characterize the Goldstone boson.