Algebraic approach to quantum gravity III: noncommmutative Riemannian geometry

TL;DR

This paper introduces a quantum Riemannian geometry framework using quantum groups, exploring noncommutative structures and their potential role in quantum gravity, including a model on finite sets.

Contribution

It develops a noncommutative Riemannian geometry approach based on quantum groups, extending classical geometry with nonsymmetric metrics and quantized differential structures for quantum gravity.

Findings

Generalization of classical Riemannian geometry with nonsymmetric metrics

Proposal that spacetime differential structure is a quantized degree of freedom

Illustration of quantum gravity scheme on finite sets

Abstract

This is a self-contained introduction to quantum Riemannian geometry based on quantum groups as frame groups, and its proposed role in quantum gravity. Much of the article is about the generalisation of classical Riemannian geometry that arises naturally as the classical limit; a theory with nonsymmetric metric and a skew version of metric compatibilty. Meanwhile, in quantum gravity a key ingredient of our approach is the proposal that the differential structure of spacetime is something that itself must be summed over or `quantised' as a physical degree of freedom. We illustrate such a scheme for quantum gravity on small finite sets.