Non-Commutative Worlds and Classical Constraints

TL;DR

This paper explores the mathematical links between classical and discrete physics using non-commutative calculus, revealing how electromagnetism and general relativity can emerge from specific constraints.

Contribution

It demonstrates how first and second order constraints in non-commutative calculus lead to formulations of electromagnetism and general relativity, unifying discrete and classical physics frameworks.

Findings

Generalized non-commutative electromagnetism follows from a first order constraint.

Relationships with general relativity emerge from a second order constraint.

A second order constraint links non-commutative and classical worlds, producing Einstein-like equations.

Abstract

This paper reviews results about discrete physics and non-commutative worlds and explores further the structure and consequences of constraints linking classical calculus and discrete calculus formulated via commutators. In particular we review how the formalism of generalized non-commutative electromagnetism follows from a first order constraint and how, via the Kilmister equation, relationships with general relativity follow from a second order constraint. It is remarkable that a second order constraint, based on interlacing the commutative and non-commutative worlds, leads to an equivalent tensor equation at the pole of geodesic coordinates for general relativity.