Wong's Equations and Charged Relativistic Particles in Non-Commutative Space

In analogy to Wong's equations describing the motion of a charged relativistic point particle in the presence of an external Yang–Mills field, we discuss the motion of such a particle in non-commutative space subject to an external $U_\star(1)$ gauge field. We conclude that the latter equations are only consistent in the case of a constant field strength. This formulation, which is based on an action written in Moyal space, provides a coarser level of description than full QED on non-commutative space. The results are compared with those obtained from the different Hamiltonian approaches. Furthermore, a continuum version for Wong's equations and for the motion of a particle in non-commutative space is derived.