Radiation reaction in various dimensions

We discuss the radiation reaction problem for an electric charge moving in flat space-time of arbitrary dimensions. It is shown that four is the unique dimension where a local differential equation exists accounting for the radiation reaction and admitting a consistent mass renormalization (the Lorentz-Dirac equation). In odd dimensions Huygens’s principle does not hold, and, as a result, the radiation reaction force depends on the whole past history of a charge (radiative tail). We show that the divergence in the tail integral can be removed by the mass renormalization only in the $2+1$ theory. In even dimensions higher than four, divergences cannot be removed by the mass renormalization.