Massless fields with half-integral spin

Abstract

The Fierz-Pauli Lagrangian for massive particles with spin $s=n+\frac{1}{2}$, $n$ integer, is examined in the limit of vanishing mass. A considerable simplification occurs. The potential $h$ is a Rarita-Schwinger spinor-tensor of tensorial rank $n$. The "spinor-trace" ${h}^{\ensuremath{'}}$, defined by ${h}_{\ensuremath{\nu}\ensuremath{\lambda}\dots{}}^{\ensuremath{'}}\ensuremath{\equiv}{\ensuremath{\gamma}}^{\ensuremath{\mu}}{h}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\lambda}\dots{}}$ does not vanish, and neither does ${h}^{\ensuremath{'}\ensuremath{'}}\ensuremath{\equiv}{({h}^{\ensuremath{'}})}^{\ensuremath{'}}$; but ${h}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}}$ does vanish. The wave equation admits a gauge group, $h\ensuremath{\rightarrow}h+\mathrm{sym}\mathrm{grad} \ensuremath{\xi}$, with ${\ensuremath{\xi}}^{\ensuremath{'}}=0$. The most interesting feature is that the source $t$ need not be divergence free, only the traceless part of ${p}^{\ensuremath{\mu}}{t}_{\ensuremath{\mu}\ensuremath{\nu}\dots{}}$ must vanish. This weaker condition on $t$ turns out to be sufficient to guarantee that only $\mathrm{helicities}\ifmmode\pm\else\textpm\fi{}s$ are transmitted between sources.