Necessity of Quantizable Geometry for Quantum Gravity

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.