Calculus, Gauge Theory and Noncommutative Worlds
Abstract
This paper shows how gauge theoretic structures arise naturally in a noncommutative calculus. Aspects of gauge theory, Hamiltonian mechanics and quantum mechanics arise naturally in the mathematics of a noncommutative framework for calculus and differential geometry. We show how a covariant version of the LeviCivita 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 noncommutative 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.03007
 Bibcode:
 2021arXiv210803007K
 Keywords:

 Mathematics  Differential Geometry;
 53Z05
 EPrint:
 LaTeX document, 43 pages