Helicity is a topological invariant of massless particles: C=-2h

Abstract

There is an elementary but indispensable relationship between the topology and geometry of massive particles. The geometric spin $s$ is related to the topological dimension of the internal space $V$ by $\dim V = 2s + 1$. This breaks down for massless particles, which are characterized by their helicity $h$, but all have 1D internal spaces. We show that a subtler relation exists between the topological and geometry of massless particles. Wave functions of massless particles are sections of nontrivial line bundles over the lightcone whose topology are completely characterized by their first Chern number $C$. We prove that in general $C = -2h$. In doing so, we also exhibit a method of generating all massless bundle representations via an abelian group structure of massless particles.