Non-commutative worlds

Abstract

This paper presents a mathematical view of aspects of physics, showing how the forms of gauge theory, Hamiltonian mechanics and quantum mechanics arise from a non-commutative framework for calculus and differential geometry. This work is motivated by discrete calculus, as it is shown that classical discrete calculus embeds in a non-commutative context. It is shown how various processes are modeled by non-commutative discrete calculus, and how aspects of differential geometry, such as the Levi-Civita connection, arise naturally from commutator equations and the Jacobi identity. A new and generalized version of the Feynman–Dyson derivation of electromagnetic equations is given, with corresponding discrete models.