This paper shows how gauge theoretic structures arise naturally in a non-commutative calculus. Aspects of gauge theory, Hamiltonian mechanics and quantum mechanics arise naturally in the mathematics of a non-commutative framework for calculus and differential geometry. We show how a covariant version of the Levi-Civita connection arises naturally in this commutator calculus. This connection satisfies the formula $$\Gamma_{ kij} + \Gamma_{ ikj} = \nabla_{ j}g_{ ik} = \partial_{ j} g_{ ik} + [g_{ ik}, A_j].$$ and so is exactly a generalization of the connection defined by Hermann Weyl in his original gauge theory. In the non-commutative world $\cal N$ the metric indeed has a wider variability than the classical metric and its angular holonomy. Weyl's idea was to work with such a wider variability of the metric. The present formalism provides a new context for Weyl's original idea.
Categories: math.DG