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 $$Gamma_{kij} + Gamma_{ikj} = abla_{j}g_{ik} = partial_{j} g_{ik} + [g_{ik}, A_j].$$ 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. Weyls idea was to work with such a wider variability of the metric. The present formalism provides a new context for Weyls original idea.
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