Helicity is a topological invariant of massless particles

There is an elementary but indispensable relationship between the topology and geometry of massive particles. Namely, the spin s characterizing the particles Poincaré geometry, is related to the dimension of the internal space V (a topological invariant) by dim(V) = 2s + 1. However, this relationship breaks down for massless particles. Such particles are geometrically characterized by their helicity h, but all have one-dimensional internal spaces. We show that a more subtle and less elementary relationship exists between the topology and geometry of massless particles. Unlike massive particles, whose wave functions are sections of topologically trivial bundles, wave functions of massless particles are sections of topologically nontrivial line bundles over the forward lightcone. The topologies of such line bundles are completely characterized by their first Chern number C. We recently showed that R and L photons have helicities h=±1 and Chern numbers C =∓2 [1] while gravitons have h=±2 and C =∓4 [2]. Here we prove in general that C = -2h, giving the fundamental relationship between the geometry and topology of massless particles. In doing so, we also exhibit a method of generating all massless bundle representations via an abelian group structure of massless particles.