On the structure of the supplementary series of unitary, irreducible representations of the proper, orthochronous Lorentz group

Representations from the supplementary series of unitary, irreducible representations of the proper, orthochronous Lorentz group are classified according to the parameter z, 0 < z < 1. The representations with 0 < z < 1/2 are qualitatively different from those with 1/2 < z < 1. This is shown in the form of the following theorem: the Casimir operator of the little group of a spacelike vector has for 0 < z < 1/2 a single bound state, i.e., a single normalizable eigenstate which disappears for 1/2 < z < 1. To this end the scalar product for the supplementary series is explicitly calculated in both regions, 0 < z < 1/2 and 1/2 < z < 1 in a coordinate system provided by common eigenfunctions of the Casimir operator of the little group of a spacelike vector and the commuting generator of a parabolic rotation. The choice of this coordinate system allows to use the well established properties of the Kontorovich-Lebedev pair of integral transforms. (c) 2005 American Institute of Physics.