Bundle Structure of Massless Unitary Representations of the Poincar\'e Group

TL;DR

This paper analyzes the structure of massless and massive unitary representations of the Poincaré group, revealing geometric and topological properties of massless states, and clarifying constraints on angular momentum and derivatives in relativistic quantum physics.

Contribution

It uncovers the bundle structure of massless helicity states, applies Frobenius' reciprocity to angular momentum constraints, and explores the noncommutative geometry of the massless shell.

Findings

Massless helicity states are sections of a U(1)-bundle over the massless shell.

Frobenius' reciprocity restricts massless states with low angular momentum.

Partial derivatives are non-operator covariant derivatives, indicating noncommutative geometry.

Abstract

Reviewing the construction of induced representations of the Poincar\'e group of four-dimensional spacetime we find all massive representations, including the ones acting 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 bracket notation. Our traditional notation enables questions about square integrability and smoothness. Their answers complete the picture of relativistic quantum physics. Frobenius' reciprocity theorem 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 which can be applied to massless states Psi with nonvanishing helicity. They allow only for covariant, noncommuting derivatives. The massless shell has a noncommutative geometry with helicity being its topological charge. A spatial position operator for Psi which constitutes Heisenberg pairs with the spatial momentum, is excluded by the smoothness requirement of the domain of the Lorentz generators.