Calculus, Gauge Theory and Noncommutative Worlds

Abstract

This paper shows how gauge theoretic structures arise in a noncommutative calculus where the derivations are generated by commutators. These patterns include Hamilton’s equations, the structure of the Levi–Civita connection, and generalizations of electromagnetism that are related to gauge theory and with the early work of Hermann Weyl. The territory here explored is self-contained mathematically. It is elementary, algebraic, and subject to possible generalizations that are discussed in the body of the paper.

Keywords

• derivative, derivation
• commutator
• curvature
• parallel translation
• covariant derivative
• connection