On a Noncommutative Extension of Electrodynamics

The Maxwell vector potential and the Dirac spinor used to describe the classical theory of electrodynamics both have components which are considered to be ordinary smooth functions on space-time. We reformulate electrodynamics by adding an additional structure to the algebra of these functions in the form of the algebra $M_n$ of $n \times n$ complex matrices. This involves a generalization of the notions of geometry to include the geometry of matrices. Some rather general constraints on the reformulation are imposed which can be motivated by considering matrix geometry in the limit of very large $n$. A few of the properties of the resulting models are given for the values $n=2,3$. One of the more interesting is the existence of several distinct stable phases or vacua. The fermions can be quark-like in one and lepton-like in another.