Noncommutativity and Physics: A non-technical review

TL;DR

This paper provides a conceptual overview of how noncommutative geometry influences physics, highlighting its role in unification theories, quantum mechanics, and the emergence of gauge theories without delving into technical details.

Contribution

It offers a non-technical, conceptual review of noncommutative geometry's applications in physics, connecting spectral geometry with physical theories and phenomena.

Findings

Noncommutativity induces canonical time evolution.

Spectral geometry links line elements and algebras.

Applications include unification models and quantum phenomena.

Abstract

We give an overview of the applications of noncommutative geometry to physics. Our focus is entirely on the conceptual ideas, rather than on the underlying technicalities. Starting historically from the Heisenberg relations, we will explain how in general noncommutativity yields a canonical time evolution, while at the same time allowing for the coexistence of discrete and continuous variables. The spectral approach to geometry is then explained to encompass two natural ingredients: the line element and the algebra. The relation between these two is dictated by so-called higher Heisenberg relations, from which both spin geometry and non-abelian gauge theory emerges. Our exposition indicates some of the applications in physics, including Pati--Salam unification beyond the Standard Model, the criticality of dimension 4, second quantization and entropy.