Semiquantisation Functor and Poisson-Riemannian Geometry, I

TL;DR

This paper develops a functorial approach to first-order semiclassical noncommutative Riemannian geometry, quantising classical geometric data into bimodules over a deformed algebra, with applications to black holes and quantum spacetime models.

Contribution

It introduces a functorial method to quantise classical bundles and geometric structures into noncommutative geometry, addressing metric compatibility and obstructions.

Findings

Quantisation functor maps classical bundles to bimodules with connections.

Full metric compatibility requires vanishing of a classical Ricci 2-form obstruction.

Application to Schwarzschild black hole reveals nonassociativity in quantum differential calculus.

Abstract

We study noncommutative bundles and Riemannian geometry at the semiclassical level of first order in a deformation parameter $\lambda$, using a functorial approach. The data for quantisation of the cotangent bundle is known to be a Poisson structure and Poisson preconnection and we now show that this data defines to a functor $Q$ from the monoidal category of classical vector bundles equipped with connections to the monodial category of bimodules equipped with bimodule connections over the quantised algebra. We adapt this functor to quantise the wedge product of the exterior algebra and in the Riemannian case, the metric and the Levi-Civita connection. Full metric compatibility requires vanishing of an obstruction in the classical data, expressed in terms of a generalised Ricci 2-form, without which our quantum Levi-Civita connection is still the best possible. We apply the theory to the Schwarzschild black-hole and to Riemann surfaces as examples, as well as verifying our results on the 2D bicrossproduct model quantum spacetime. The quantized Schwarzschild black-hole in particular has features similar to those encountered in $q$-deformed models, notably the necessity of nonassociativity of any rotationally invariant quantum differential calculus of classical dimensions.