Dirac Monopole from Lorentz Symmetry in N-Dimensions: I. The Generator Extension

Abstract

It is by now well-known that a Lorentz force law and the homogeneous Maxwell equations can be derived from commutation relations among Euclidean coordinates and velocities, without explicit reference to momentum, action or variational principle. This result was extended to the relativistic case and shown to correspond to a Stueckelberg-type quantum theory, in which gauge transformations may depend on the invariant evolution parameter, such that the associated the five-dimensional electromagnetism becomes standard Maxwell theory in the equilibrium limit. Building on the work of Berard, Grandati, Lages and Mohrbach, we construct an extension of the Lorentz generators in N-dimensions that restores the closed commutation relations in the presence of a Maxwell field, and renders the extended generators constants of the classical motion. The algebra imposes conditions on the Maxwell field, leading to a Dirac monopole solution. The construction can be maximally satisfied in a three dimensional subspace of the full Minkowski space; this subspace can be chosen to describe either the O(3)-invariant space sector, generalizing the nonrelativistic result, or an O(2,1)-invariant restriction of spacetime, and leading to a relativistic Coulomb-like potential of the type used by Horwitz and Arshansky to obtain a covariant generalization of the hydrogen-like bound state.