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08/06/2006, 16:30 — 17:30 — Amphitheatre Pa1, Mathematics Building

Louis H. Kauffman, *Univ Illinois at Chicago*

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Non-Commutative Worlds

This talk 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. We discuss how Poisson brackets give rise to the Jacobi identity, and how the Jacobi identity arises in combinatorial contexts, including graph coloring and knot theory. We give 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. The talk is self-contained and begins with a discussion of how classical discrete calculus embeds in a non-commutative framework such that the Leibniz rule is restored, and the discrete derivatives are represented by commutators. This construction motivates the rest of the talk. See quant-ph/0403012.