The nontrivial topologies of photons, gravitons, and electromagnetic plasma waves and their implications for splitting angular momentum into spin and orbital parts*

Many physical systems, such as insulators, fluid, and plasmas, support topologically nontrivially waves. Here, we show that even the vacuum admits topologically nontrivial modes. Massless particles such as the R and L photons and gravitons form nontrivial vector bundles with Chern numbers C= [1] and C= [2]. This nontrivial topology originates from a singularity in the momentum space at k=0. By considering representations of the Poincaré group on vector bundles (rather than on vector spaces) we obtain a more rigorous version of Wigner’s little group method, showing that photons and gravitons are irreducible bundle representations of the Poincaré group with helicities h=±1 and h=±2. It is known that massless particles are characterized by helicity rather than spin. We show that this transition from spin to helicity in the massless limit occurs due to a topological singularity. A surprising implication is that this singularity obstructs the splitting of the angular momentum of massless particles into spin (SAM) and orbital (OAM) parts. We also show that EM waves in unmagnetized plasma are topologically nontrivial, and that this precludes splitting the wave angular momentum into SAM and OAM. Indeed, topological nontriviality corresponds to a twisting together of the internal and external degrees of freedom. This twisting obstructs splitting the SO(3) symmetry of the wave modes into internal (SAM) and external (OAM) SO(3) symmetries.