Weaving a classical geometry with quantum threads

TL;DR

This paper explores nonperturbative quantum gravity using loop variables, revealing non-local operators with finite, diffeomorphism-invariant spectra, and demonstrating a discrete Planck-scale structure in quantum states approximating flat geometry.

Contribution

It introduces a framework where non-local geometric operators are well-defined and finite, showing the quantization of surface areas in quantum gravity.

Findings

Non-local operators like surface area are finite and diffeomorphism-invariant.

Quantum states can approximate flat geometry with a discrete Planck-scale structure.

Surface area spectra are quantized in integral units of the Planck area.

Abstract

Results that illuminate the physical interpretation of states of nonperturbative quantum gravity are obtained using the recently introduced loop variables. It is shown that: i) While local operators such as the metric at a point may not be well-defined, there do exist {\it non-local} operators, such as the area of a given 2-surface, which can be regulated diffeomorphism invariantly and which are finite {\it without} renormalization; ii)there exist quantum states which approximate a given flat geometry at large scales, but such states exhibit a discrete structure at the Planck scale; iii) these results are tied together by the fact that the spectra of the operators that measure the areas of surfaces are quantized in integral units of the Planck area.