Group of Relativistic Internal Motion, Its Unitary Representation and Wave Equation

Abstract

Relativistic theory of a particle involving degrees of freedom of internal is set up in a general way, in the viewpoint that it should be useful for the repre­ sentation of hadronic states, to derive various general results. The relativistic internal is described by a set of primitive kinematical variables obeying certain algebra and it immediately gives rise to not only the internal Lorentz (relativistic spin group) v· but also the of relativistic internal motion 0(-:JV). According to the type of the kinematicaJ algebra models are classified into three categories: commutation relation type (A-type), anti-commutation relation type (B·type), and hybrid type(C-type). In the A-type case the internal kinematical variables consist of continuous coordinates and their conjugate momenta, correspond­ ing to the representation of realistic internal motions, and the G 1 is real symplectic Sp(JR) ( j being the number of internal degrees of freedom). The main part of the paper concerns the treatment of this A-type case. where an L' transformation on the kinematical variables induces reducible or irreducible infinite-dimensional unitary representation of £1• on the wave function at each space-time point, resulting in a spin spectrum. Furthermore the wave function is the base of an irreducible unitary representation of the wider 0. The generators of those unitary representations are conveniently expressed in terms of oscillator variables (the creation and annihilation operators for excitation of internal modes of motion) and there the wave function takes the form of infinite-component wave function. The internal dynamics of the system is represented by a wave equation, which extracts from the wave function space a reducible unitary or pseudo-unitary representation of the physical Poincare fPpbys, to determine the mass spectrum. A specified choice of wave equation fixes the intermediary relativistic group Gi (V ~ Gl ~ 0) and the compact rest symmetry Go c G •, and the solutions are classified according to irreducible representations of G.' and Co, as well as fPphys. Within the framework of this general theory (given in Part I) several typical examples are concretely treated in Part II and in Appendix with the analysis into their physical and unphysical consequences. It includes bilocal model, spinor model, Majorana model, and the model with Dirac spinor as internal coordinates, as well as their extensions to hybrid-type models. Especially we present several new versions of wave equation with interesting properties. Examples are (i) Special second-order wave equation on the bilocal model, yielding the ascending mass spectrum of exact 4-dimensional harmonic oscillator; (ii) Hybrid wave equation on the spinor or Majorana model, yielding a set of ascending Regge trajectories and having no space­ like solutions; (iii) Particular subsidiary condition on the hybrid spinor model, which, together with the wave equation, reproduces rigid-rotator levels relativistically, without admitting space-like solutions; (iv) New treatment of relativistic rotator model (Lorentz tetrad model) by this scheme.