Non-commutative extensions of classical theories in physics

TL;DR

This paper provides an overview of non-commutative extensions of classical physics theories, highlighting their potential role in unifying gravity with other fundamental interactions and exploring algebraic generalizations for quantum gravity.

Contribution

It offers a concise introduction to non-commutative generalizations of classical geometries and discusses algebraic structures relevant for quantum gravity.

Findings

Non-commutativity may be key to unifying gravity with other forces

Examples of non-commutative geometries are presented

Properties of algebras for quantum gravity are discussed

Abstract

We present a short introductory overview of the non-commutative extensions of several classical physical theories. After a general discussion of the reasons that suggest that the non-commutativity is a major issue that will eventually lead to the unification of gravity with other fundamental interactions, we display examples of non-commutative generalizations of known geometries. Finally we discuss the general properties of the algebras that could become generalizations of algebras of smooth functions on Minkowskian (Riemannian) manifolds, needed for the description of Quantum Gravity.