  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.
