A simple method for illustrating the difference between the homogeneous and inhomogeneous Lorentz groups

Abstract

A method is proposed for illustrating the difference between the Poincaré (inhomogeneous Lorentz) and homogeneous Lorentz groups. Representations of the Poincaré group are constructed from solutions of the relativistic harmonic oscillator equation whose physical wave functions have been effective in describing basic high-energy hadronic features in the relativistic quark model. It is shown that the Poincaré group can be represented by solutions of the relativistic oscillator equation in a ’’moving cylindrical’’ coordinate system in which the time-separation variable in the hadronic rest frame is treated separately. Representations which are diagonal in the Casimir operators of the homogeneous Lorentz group are also constructed from solutions of the same oscillator differential equation in a hyperbolic coordinate system. It is pointed out that the difference between the Poincaré and homogeneous Lorentz groups mainfests itself in the coordinate systems in which the oscillator differential equation in separable.