No-go theorem for the classical Maxwell-Lorentz electrodynamics in odd-dimensional worlds

Abstract

If the conventional Maxwell-Lorentz formulation of classical electrodynamics is adopted in a flat spacetime of arbitrary odd dimension, then the retarded vector potential Aμ generated by a point charge turns out to be pure gauge, Aμ = μ . By Gauss’ law, the charge shows up as zero. The classical electromagnetic coupling is thus missing from odd-dimensional worlds. If the action is augmented by the addition of the Chern-Simons term, then the classical interaction picture in the three-dimensional world becomes nontrivial.