Representations of the Poincaré group for relativistic extended hadrons

Abstract

Representations of the Poincaré group are constructed from the relativistic harmonic oscillator wave functions which have been effective in describing the physics of internal quark motions in the relativistic quark model. These wave functions are solutions of the Lorentz-invariant harmonic oscillator differential equation in the “cylindrical” coordinate system moving with the hadronic velocity in which the time separation variable is treated separately. This result enables us to assert that the hadronic mass spectrum is generated by the internal quark level excitation, and that the hadronic spin is due to the internal orbital angular momentum. An addendum relegated to PAPS contains discussions of detailed calculational aspects of the Lorentz transformation, and of solutions of the oscillator equation which are diagonal in the Casimir operators of the homogeneous Lorentz group. It is shown there that the representation of the homogeneous Lorentz group consists of solutions of the oscillator partial differential equation in a “spherical” coordinate system in which the Lorentz-invariant Minkowskian distance between the constituent quarks is the radial variable.