Mechanics of a particle in a gauge field

Abstract

Classical motion of a charged particle in an electromagnetic field is described by the Lorentz equation in terms of the field components. The field is defined by the infinitesimal gauge group elements associated with closed curves. The Lorentz equation may be derived from the action principle with an appropriate Lagrangian. The Lagrangian may also be used to associate group elements with curves in the space-time manifold. The action principle is shown here to be an equivalence relation between the infinitesimal elements so defined for a collection of closed curves and the identity element. This suggests a natural extension to require the equivalence of global elements with the identity and by considering all curves. The resulting equation, in addition to providing an extension of the Lorentz equation, also admits a straightforward generalization to non-Abelian gauge fields. The extended equation has an infinite number of trajectories as solutions. The properties of these paths are shown to impart wavelike properties to the particles in motion. In view of these results, the motion of a particle is formulated within the framework of the path integral formalism that yields a generalized Schrödinger type equation in a general gauge field. As a further implication of the properties of the trajectories assigned to a particle, this equation is shown to reduce to a set of equations, one of them being the Klein–Gordon equation.