Bundle Structure of Massless Unitary Representations of the Poincaré Group

Abstract

Reviewing the construction of induced representations of the Poincaré group of four-dimensional spacetime we find all massive representations, including the ones on interacting many-particle states. Massless momentum wavefunctions of non-vanishing helicity turn out to be more precisely sections of a U(1)-bundle over the massless shell, a property which to date was overlooked in quantum field theory and in bracket notation. Our traditional notation of states in Hilbert space enables questions about square integrability and smoothness. Their answers complete the picture of relativistic quantum physics. Frobenius reciprocity prohibits massless one-particle states with total angular momentum less than the modulus of the helicity. There is no two-photon state with \(J=1\), explaining the longevity of orthopositronium. Partial derivatives of the momentum wave functions are no operators in the space of massless states with nonvanishing helicity. They allow only for covariant, noncommuting derivatives. The massless shell has a noncommutative geometry with helicity being its topological charge.