Differential geometry in non-commutative worlds

Abstract

Aspects of gauge theory, Hamiltonian mechanics and quantum mechanics arise naturally in the mathematics of a non-commutative framework for calculus and differential geometry. A variant of calculus is built by defining derivations as commutators (or more generally as Lie brackets).We embed discrete calculus into this context and use this framework to discuss the pattern of Hamilton's equations, discrete measurement, the Schrodinger equation, dynamics and gauge theory, a generalization of the Feynman-Dyson derivation of electromagnetic theory, and differential geometry in a non-commutative context. © 2006 Birkhäuser Verlag Basel/Switzerland.