Non-Commutative Worlds

Abstract

This paper shows how the forms of gauge theory, Hamiltonian mechanics and quantum mechanics arise from a non-commutative framework for calculus and differential geometry. Discrete calculus is seen to fit into this pattern by reformulating it in terms of commutators. Differential geometry begins here, not with the concept of parallel translation, but with the concept of a physical trajectory and the algebra related to the Jacobi identity that governs that trajectory. The paper discusses how Poisson brackets give rise to Jacobi identity, and how Jacobi identity arises in combinatorial contexts, including graph coloring and knot theory. The paper gives a highly sharpened derivation of results of Tanimura on the consequences of commutators that generalize the Feynman-Dyson derivation of electromagnetism, and a generalization of the original Feynman-Dyson result that makes no assumptions about commutators. The latter result is a consequence of the definitions of the derivations in a particular non-commutative world. Our generalized version of electromagnetism sheds light on the orginal Feynman-Dyson derivation, and has many discrete models.