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 $A^{\mu }$ generated by a point charge turns out to be pure gauge, $A^{\mu }={\partial }^{\mu }\chi$. 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.

arXiv commentV1 LaTeX, 4 pages, V2. The title is changed, two new authors appear for the reason explained at the end of the text, the main result remains unchanged, new comments are added, references are renewed