Quantum fields and interactions of massless particles: the continuous spin case

Abstract

Among the unitary irreducible representations of the Poincaré group discovered by Wigner, there occurs a class of mass zero representations which are, for historical reasons, usually referred to as continuous spin representations, and are labeled by values of the Casimir invariants P 2 = 0, W μ W μ = − ϱ 2, with ϱ > 0. A corresponding particle can exist in a denumerable set of helicity states λ, with λ taking either all integer values λ=0, ±1, ±2,… or all half-odd integer values λ = ± 1 2 , ± 3 2 ,…. We construct free quantum fields for such particles and also study their interactions to lowest order in the coupling constant. The fields are of necessity infinite component fields, and their commutator is found to vanish at most in proper subsets of the spacelike region. Particular attention is paid to the behavior of the theory for small ϱ. In the limit ϱ → 0, the fields go over into those given by Bender, Frishman and Itzykson for the description of a single ordinary massless particle with fixed helicity. The limiting behavior of S-matrix elements is also studied, and it is demonstrated how predictions of conventional finite component field theory, including the helicity selection rules, can be recovered in the limit ϱ → 0. The interactions studied in this paper are at least quadratic in the fields of continuous spin particles. Currents and interaction Hamiltonians which contain them only linearly need a separate investigation. This will be carried out in a subsequent paper.