Finite, diffeomorphism invariant observables in quantum gravity

TL;DR

This paper constructs finite, background-independent, diffeomorphism-invariant operators in loop quantum gravity that measure surface areas and Wilson loops, with discrete spectra, enabling a correspondence between classical geometries and quantum states.

Contribution

It introduces new surface and loop observables in loop quantum gravity that are finite, background independent, and have discrete spectra, advancing the understanding of quantum geometry.

Findings

Operators are finite and background independent with proper regularization.

Area spectra are discrete, including multiples of half the Planck area.

Establishes a link between classical geometries and quantum states via Regge manifolds.

Abstract

Two sets of spatially diffeomorphism invariant operators are constructed in the loop representation formulation of quantum gravity. This is done by coupling general relativity to an anti- symmetric tensor gauge field and using that field to pick out sets of surfaces, with boundaries, in the spatial three manifold. The two sets of observables then measure the areas of these surfaces and the Wilson loops for the self-dual connection around their boundaries. The operators that represent these observables are finite and background independent when constructed through a proper regularization procedure. Furthermore, the spectra of the area operators are discrete so that the possible values that one can obtain by a measurement of the area of a physical surface in quantum gravity are valued in a discrete set that includes integral multiples of half the Planck area. These results make possible the construction of a correspondence between any three geometry whose curvature is small in Planck units and a diffeomorphism invariant state of the gravitational and matter fields. This correspondence relies on the approximation of the classical geometry by a piecewise flat Regge manifold, which is then put in correspondence with a diffeomorphism invariant state of the gravity-matter system in which the matter fields specify the faces of the triangulation and the gravitational field is in an eigenstate of the operators that measure their areas.