The topology and geometry of photons, gravitons, and waves in unmagnetized plasma

Abstract

Topology has been at the heart of many breakthroughs in modern physics including the discovery of topological insulators, the bulk-edge correspondence (BEC), and the spin Hall effects. It has been recently found that topological effects can emerge not only in lattices, but also in fluids and plasmas where topologically protected edge-modes have been identified via the BEC. While this is a major discovery on its own, it also reveals an underlying and largely unstudied topological vector bundle structure of continuum waves. We show that topologically nontrivial waves are not rare, emerging even in simple continuous media such as the vacuum—in the form of photons and gravitons—and cold unmagnetized plasmas. In this thesis we use this structure to study novel topological applications beyond the BEC. Examples include constructing nonsingular representations of massless particles, constructing globally smooth bases of polarization vectors for photons and electromagnetic plasma waves (despite claims that this would violate the hairy ball theorem), and characterizing the relationship between the geometry and topology of massless particles.Particularly interesting applications pertain to issues surrounding the spin (SAM) and orbital angular momentum (OAM) of electromagnetic waves. OAM beams have applications in a diverse range of fields including wakefield acceleration and laser-induced magnetic field generation in plasmas. However, there has been a long controversy about whether the angular momentum (AM) of massless particles can be legitimately split into SAM and OAM. We identify geometric and topological singularities that develop in the massless limit which obstruct such an SAM-OAM decomposition for photons and gravitons; a similar obstruction occurs in plasmas. The geometric singularity is related to a nonzero curvature of the polarization space. We also show that the AM eigenstates of massless particles and unmagnetized plasma waves can be expressed as spin-weighted spherical harmonics—these harmonics concretely illustrate the lack of a genuine SAM-OAM decomposition. However, we demonstrate that the AM admits a natural decomposition, induced by rotational symmetry, into helicity quasi-AM and orbital quasi-AM. Although these operators do not generate rotations and are therefore not true AM operators, they are gauge invariant and appear to be experimentally relevant.