The group-theoretic origin of spin: the reality locus andcohomology of the spin spectrum

Abstract

With the sole physical input that quantum symmetries act projectively on states, this work develops a group-theoretic account of spin as the obstruction to lifting a projective unitary representation to an ordinary one, controlled by two invariants: an infinitesimal class in H²(g, bR) and a topological character in Hom(pi₁(G), bT). A lifting dichotomy determines, for every connected Lie group, whether and how its projective representations lift; it recovers the double cover from the special unitary group SU(2) to the special orthogonal group SO(3), the half-integer spins, the Frobenius--Schur values nu(rho_j) = (-1)^2j, and the trichotomy of the rotation, Poincar'e, and Galilei groups. In the topological regime, the spin sectors form the Pontryagin dual widehat{pi₁(G)}, a continuum for the planar rotation group SO(2), where it yields fractional (anyonic) spin. Two further results are established. First, the self-conjugate sectors---a necessary condition for compatibility with an antiunitary time reversal---form the 2-torsion subgroup widehat{pi₁ (G)}[2], which remains finite even when the spectrum is a continuum; for the plane, it is {0, tfrac12}, explaining why generic anyons break time reversal. Second, the spectrum and this locus are the cohomology groups H¹(G; bT) and H¹(G; bZ/2), and the Bockstein sequence of bZ to bR to bT splits spin into a free part carrying the anyons and the torsion of H² (G; bZ), an integral class whose modulo-2 reduction is, for the rotation group, the Stiefel--Whitney class obstructing spin structures on spacetime.