Quantum Spacetimes in the Year 1

TL;DR

This paper reviews emergent concepts of quantum spacetime from noncommutative geometry, highlighting models like fuzzy spheres, new uncertainty principles, and implications for fundamental symmetries, all within quantum physics without gravity input.

Contribution

It introduces models of quantum spacetime such as fuzzy $S^4$ and explores their radical implications, including nonlocality, modified uncertainty principles, and potential violations of fundamental symmetries.

Findings

Models like fuzzy $S^4$ exhibit nonlocal states.

Quantum spacetime may violate $CPT$ symmetry.

Planck's constant could be an operator, not a fixed value.

Abstract

We review certain emergent notions on the nature of spacetime from noncommutative geometry and their radical implications. These ideas of spacetime are suggested from developments in fuzzy physics, string theory, and deformation quantisation. The review focuses on the ideas coming from fuzzy physics. We find models of quantum spacetime like fuzzy $S^4$ on which states cannot be localised, but which fluctuate into other manifolds like $ CP^3$ . New uncertainty principles concerning such lack of localisability on quantum spacetimes are formulated.Such investigations show the possibility of formulating and answering questions like the probabilty of finding a point of a quantum manifold in a state localised on another one. Additional striking possibilities indicated by these developments is the (generic) failure of $CPT$ theorem and the conventional spin-statistics connection. They even suggest that Planck's `` constant '' may not be a constant, but an operator which does not commute with all observables. All these novel possibilities arise within the rules of conventional quantum physics,and with no serious input from gravity physics.