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

TL;DR

This paper proves that classical Maxwell-Lorentz electrodynamics in odd-dimensional flat spacetimes results in a trivial, pure gauge potential with no electromagnetic interaction, but adding a Chern-Simons term restores nontriviality.

Contribution

It demonstrates a no-go theorem showing the failure of classical electrodynamics in odd dimensions without Chern-Simons modifications.

Findings

Retarded vector potential is pure gauge in odd dimensions

Charge appears as zero due to Gauss's law

Adding Chern-Simons term yields nontrivial classical interactions

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.