Non-Commutative Worlds

Aspects of gauge theory, Hamiltonian mechanics and quantum mechanics arise naturally in the mathematics of a non-commutative framework for calculus and differential geometry. This talk consists in a number of sections including the introduction. The introduction sketches our general results in this domain. The second section gives a derivation of a generalization of the Feynman-Dyson derivation of electromagnetism using our non-commutative context and using diagrammatic techniques. The third section discusses, in more depth, relationships with gauge theory and differential geometry. The last section discusses the structure of curvature, Bianchi identity and general relativity. We begin the talk by showing how constructing sqrt[-1] naturally leads to Clifford Algebras and the seeds of this non-commutative context.