Off-mass-shell massless particles and the Weyl group in light-core coordinates

Abstract

Extends the author's previous work on the description of virtual relativistic particles by unitary irreducible representations of the Weyl group to the case of particles with zero on-mass-shell mass. The structure of the d-dimensional Weyl group Lie algebra on the light-cone is that of the group of inhomogeneous Galilei transformations and non-relativistic dilatations on a Euclidean space plus time of (d-2) space dimensions. The 'dilatation' generators of the two non-relativistic algebras are D+or-M0(d-1), and an infinite transformation of these operators takes the momentum operator Pmu into (P+/ square root 2, O, +or-P+or-/ square root 2) respectively, where P+or-=(P0+or-P(d-1))/ square root 2. The transformations therefore take a state of an off-mass-shell massless particle, of arbitrary momentum, transformation of these operators takes the momentum, onto the mass-shell. It is calculated how the spin and position operators transform under the infinite transformations, and express the Weyl group generators in terms of the transformed operators. A canonical form is also constructed for the transformed operators and generators, and a discussion given of the transformation properties of the single-particle states under a unitary operator of the Weyl group. The case d=4 is discussed in detail and, in particular, Weinberg's theorem on massless irreducible representations of the d=4 Poincare group is extended to the case of the Weyl group.