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" $h^{\prime }$, defined by $h_{\nu \lambda \dots }^{\prime }\equiv {\gamma }^{\mu }{h}_{\mu \nu \lambda \dots }$ does not vanish, and neither does $h^{\prime \prime }\equiv {\left(h^{\prime }\right)}^{\prime }$; but $h^{\prime \prime \prime }$ does vanish. The wave equation admits a gauge group, $h\to h+\mathrm{sym}\mathrm{grad} \xi$, with ${\xi }^{\prime }=0\mathrm{}$. The most interesting feature is that the source $t$ need not be divergence free, only the traceless part of $p^{\mu }{t}_{\mu \nu \dots }$ must vanish. This weaker condition on $t$ turns out to be sufficient to guarantee that only $\mathrm{helicities}\pm s$ are transmitted between sources.