Electrodynamics On Matrix Space: Non-Abelian By Coordinates

TL;DR

This paper explores the dynamics of particles with matrix-valued coordinates in a non-commutative space, connecting matrix space physics with non-Abelian gauge theories and D0-brane phenomenology.

Contribution

It derives Lorentz equations for matrix coordinates and discusses the non-Abelian transformation properties of gauge potentials and field strengths.

Findings

Matrix coordinates induce non-Abelian gauge transformations.

Derived Lorentz equations for matrix-valued particles.

Discussed phenomenological implications for D0-brane bound states.

Abstract

We consider the dynamics of a charged particle in a space whose coordinates are $N\times N$ hermitian matrices. Putting things in the framework of D0-branes of String Theory, we mention that the transformations of the matrix coordinates induce non-Abelian transformations on the gauge potentials. The Lorentz equations of motion for matrix coordinates are derived, and it is observed that the field strengths also transform like their non-Abelian counterparts. The issue of the map between theory on matrix space and ordinary non-Abelian gauge theory is discussed. The phenomenological aspect of "finite-N non-commutativity" for the bound states of D0-branes appears to be very attractive.