Quantum Spacetimes from General Relativity?

TL;DR

This paper develops a new non-commutative product for curved spacetimes, generalizing existing models, and explores its mathematical properties and implications for different physical spacetime geometries.

Contribution

It introduces a generalized non-commutative product using the exponential map and Poisson tensor, extending previous models to curved spacetimes with specific associativity conditions.

Findings

Derived conditions for associativity in curved spacetimes

Constructed non-commutative structures for various physical geometries

Identified properties of the new non-commutative product

Abstract

We introduce a non-commutative product for curved spacetimes, that can be regarded as a generalization of the Rieffel (or Moyal-Weyl) product. This product employs the exponential map and a Poisson tensor, and the deformed product maintains associativity under the condition that the Poisson tensor $\Theta$ satisfies $\Theta^{\mu\nu}\nabla_{\nu}\Theta^{\rho\sigma}=0$, in relation to a Levi-Cevita connection. We proceed to solve the associativity condition for various physical spacetimes, uncovering non-commutative structures with compelling properties.