Families of Indefinitely Rising Trajectories and Gell-Mann's Program

Abstract

Infinite-component wave equations giving rise to a linear mass spectrum and to families of parallel linear trajectories are considered. A general discussion is given of invariant equations for wave functions belonging to $\mathrm{SL}(2,C)$ representations and the mass spectra that arise are examined. The simplest possibility corresponds to a higher-derivative equation that gives a linearly rising timelike spectrum that is free of continuum spacelike solutions. Discrete spacelike solutions are absent for the simplest choices of the $\mathrm{SL}(2,C)$ representation. The currents and the commutators among the current components are calculated by setting up for the higher-derivative equation a Lagrangian formalism and a quantization procedure based on the action principle. An explicitly factorized model is considered, with respect to the internal symmetry group, and possible nonfactorized extensions are examined. A typical feature of the current commutators is the appearance of Schwinger terms which, besides satisfying known general requirements, also appear in commutators between time components of currents. An alternative interpretation of the physical system in terms of a bound-state equation is presented. The interpretation, in terms of a bound system in two space dimensions, leads to a extension to three space dimensions, again formulable as an infinite-component wave equation. The system describes a family of parallel linearly rising trajectories spaced by one unit of angular momentum. No continuum spacelike spectrum is present, and discrete spacelike solutions are absent for physical choices of the representation of the internal spin group.