Generalizing noncommutative geometry moves toward integrating quantum mechanics with gravity

One method to combine the standard model of particle physics with the theory of gravity based on general relativity uses the noncommutative geometry standard model. However, the noncommutative approach operates under Riemannian manifolds, which consider three-dimensional space, as opposed to Lorentzian manifolds, which considers four-dimensional spacetime (or its higher-dimensional analogues). The geometric generalization of Lorentzian manifolds to incorporate the standard model, therefore, would greatly advance our understanding of the underlying operation of the universe.

New research in the Journal of Mathematical Physics describes a construction using noncommutative hypersurfaces to ensure that the noncommutative standard model is compatible with Lorentzian manifolds. Each hypersurface operates under the classical spacetime model and they are then combined to form a noncommutative spacetime. The authors decompose Dirac operators defining spin-dependent quantum mechanics in the Lorentzian framework into individual Riemann operators and parameterize them with a time dependence.

While noncommutative hypersurfaces have previously been considered as spectral triples, sets that analytically describe a noncommutative geometry, this work provides the mathematical rigor to describe these hypersurface collections as having a time dependence. The hypersurface time dependence ensures that the surfaces are inherently coupled, thus forming the noncommutative four-dimensional description.

The research provides mathematical proofs showing that the construction works in both Riemann and Lorentzian signatures. The full mathematical rigor developed also provides further opportunities to develop noncommutative hypersurface families to potentially couple gravity with the standard model. For example, this work enables the possibility of further elucidating the time evolution through solving Einstein’s theory of relativity for noncommutative spaces.

Source: “Families of spectral triples and foliations of space(time),” by Koen van den Dungen, Journal of Mathematical Physics (2018). The article can be accessed at https://doi.org/10.1063/1.5021305 .