The quantum structure of spacetime at the Planck scale and quantum fields

TL;DR

This paper introduces uncertainty relations for spacetime coordinates based on quantum and gravitational principles, proposes a model of Quantum Spacetime with specific commutation relations, and explores implications for quantum field theory and ultraviolet divergence smoothing.

Contribution

It develops a model of Quantum Spacetime with exact uncertainty relations and initial steps towards formulating quantum fields and interactions on this noncommutative geometry.

Findings

Quantum Spacetime replaces local interactions with nonlocal effective interactions.

Classical limit recovers Minkowski space times a bundle over S^2.

Framework sets the stage for addressing ultraviolet divergences in QFT.

Abstract

We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg's principle and by Einstein's theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations. We outline the definition of free fields and interactions over QST and take the first steps to adapting the usual perturbation theory. The quantum nature of the underlying spacetime replaces a local interaction by a specific nonlocal effective interaction in the ordinary Minkowski space. A detailed study of interacting QFT and of the smoothing of ultraviolet divergences is deferred to a subsequent paper. In the classical limit where the Planck length goes to zero, our Quantum Spacetime reduces to the ordinary Minkowski space times a two component space whose components are homeomorphic to the tangent bundle TS^2 of the 2-sphere. The relations with Connes' theory of the standard model will be studied elsewhere.