Non-commutative Geometry from Perturbative Quantum Gravity

TL;DR

This paper demonstrates that perturbative quantum gravity inherently predicts non-commutative spacetime effects at the Planck scale, suggesting non-commutativity arises naturally from quantum gravitational phenomena.

Contribution

It shows that non-commutative geometrical effects are already present in perturbative quantum gravity without needing to be introduced artificially.

Findings

Non-commutativity appears at the Planck scale in perturbative quantum gravity.

Quantum theory structure clarifies the emergence of non-commutative geometry.

Non-commutativity is a natural consequence, not an additional assumption.

Abstract

Trying to connect a fundamentally non-commutative spacetime with the conservative perturbative approach to quantum gravity, we are led to the natural question: are non-commutative geometrical effects already present in the regime where perturbative quantum gravity provides a predictive framework? Moreover, is it necessary to introduce non-commutativity by hand, or does it arise through quantum-gravitational effects? We show that the first question can be answered in the affirmative, and the second one in the negative: perturbative quantum gravity predicts non-commutativity at the Planck scale, once one clarifies the structure of observables in the quantum theory.