Noncommutative geometrical origin of the energy-momentum dispersion relation

TL;DR

This paper explores how noncommutative geometry provides a foundational understanding of the energy-momentum dispersion relation by linking spectral distances with causal structures in a Lorentzian setting.

Contribution

It introduces a novel approach connecting noncommutative spectral geometry with physical dispersion relations through a Lorentzian framework.

Findings

Derived the energy-momentum dispersion relation from noncommutative geometric principles.

Linked spectral distances to causal structures in a Lorentzian noncommutative space.

Demonstrated the embedding of almost-commutative manifolds in higher-dimensional Minkowski spacetime.

Abstract

We investigate a link between the energy-momentum dispersion relation and the spectral distance in the context of a Lorentzian almost-commutative spectral geometry, defined by the product of Minkowski spacetime and an internal discrete noncommutative space. Using the causal structure, the almost-commutative manifold can be identified with a pair of four-dimensional Minkowski spacetimes embedded in a five-dimensional Minkowski geometry. Considering fermions travelling within the light cone of the ambient five-dimensional spacetime, we then derive the energy-momentum dispersion relation.