Feynman Rules for Any Spin. II. Massless Particles

Abstract

The Feynman rules are derived for massless particles of arbitrary spin $j$. The rules are the same as those presented in an earlier article for $m>\mathrm{}0\mathrm{}$, provided that we let $m\to 0$ in propagators and wave functions, and provided that we keep to the ($2j+1$)-component formalism [with fields of the ($j$, 0) or (0, $j$) type] or the $2(2j+1)\mathrm{}$-component formalism [with $(j, 0)\oplus (0, j)$ fields]. But there are other field types which cannot be constructed for $m=0\mathrm{}$; these include the ($\frac{j}{2}, \frac{j}{2}$) tensor fields, and in particular the vector potential for $j=1\mathrm{}$. This restriction arises from the non-semi-simple structure of the little group for $m=0\mathrm{}$. Some other subjects discussed include: T, C, and P for massless particles and fields; the extent to which chirality conservation implies zero physical mass; and the Feynman rules for massive particles in the helicity formalism. Our approach is based on the assumption that the $S$ matrix is Lorentz invariant, and makes no use of Lagrangians or the canonical formalism.