Photon topology

Following the discovery of topological insulators and the topology inherent in wave physics, researchers have begun studying the topology of plasma waves. Here, we explore the topology of a more fundamental system, that of Maxwell’s equations in vacuum. The topology of this system is interesting because there are no photons with k=0, creating a hole in the momentum space. We show that while the set of all photons forms a trivial vector bundle g over this momentum space, the R- and L-photons form topologically nontrivial subbundles g_± with Chern numbers ±2. By considering representations of the Poincare group on vector bundles (rather than on vector spaces) we obtain a more rigorous version of Wigner’s little group method, and show that g_± are irreducible bundle representations of the Poincare group with helicities ±1. This formalism also offers a method of quantizing the EM field without invoking discontinuous polarization vectors as in the traditional scheme. We also demonstrate that unlike the ordinary Chern number, the spin-Chern number of photons is not a purely topological quantity. Lastly, there has been an extended debate on whether photon angular momentum can be split into spin and orbital parts. Our work helps explain the precise issues that prevent this splitting.