Necessity of Quantizable Geometry for Quantum Gravity

TL;DR

This paper explores the quantization of 3D gravity using a novel approach that quantizes geometric 1-forms directly, revealing the importance of quantizable geometry for consistent quantum gravity formulations.

Contribution

It introduces a Dirac quantization method for 3D gravity that quantizes connection and triad 1-forms, highlighting the role of quantizable geometry in quantum gravity.

Findings

Transition amplitude computed with this method closely matches LQG results

Inconsistencies suggest non-quantizable geometry affects quantum gravity formulations

Highlights the necessity of quantizable geometric structures for consistent quantization

Abstract

In this paper, Dirac Quantization of $3D$ gravity in the first-order formalism is attempted where instead of quantizing the connection and triad fields, the connection and the triad 1-forms themselves are quantized. The exterior derivative operator on the space of differential forms is treated as the `time' derivative to compute the momenta conjugate to these 1-forms. This manner of quantization allows one to compute the transition amplitude in $3D$ gravity which has a close, but not exact, match with the transition amplitude computed via LQG techniques. This inconsistency is interpreted as being due to the non-quantizable nature of differential geometry.