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 discusses our general results in this domain. We give a derivation of a generalization of the Feynman-Dyson derivation of electromagnetism using the non-commutative context and diagrammatic techniques. We then discuss, in more depth, relationships with gauge theory and differential geometry.  The key aspect of this approach is the representation of derivatives as commutators. This creates the context of a non-commutative world and allows a synoptic view of patterns in mathematical physics.

Finally, we explore the relationship of general relativity and non-commutative algebra via the expression of covariant derivatives in terms of commutators and articulate the Bianchi Identity in terms of the Jacobi Identity. In this way we can formulate curvature in terms of commutators and formulate algebraic constraints on curvature so that our curvature tensors have the requisite symmetries to produce a divergence free Einstein tensor. The talk will discuss these structures and their relationship with classical general relativity.