Quadrilocal Model of Baryons and Unitary Symmetry

Abstract

A unified theory of baryons is proposed based on a spinor wave equation that depends on four space-time points or equivalently on the center of mass and three relative-coordinate vectors. The associated subsidiary condition and the structure of the mass operator are such that the four-point association is maintained within a small region of Minkowski space-time with characteristic length and that the theory has $U(9)$ symmetry in the full symmetry limit. By the couplings of internal motions this symmetry is reduced to the direct product of the usual unitary-spin group $U(3)$ and the other unitary group $U{(3)}^{\ensuremath{'}}$ characteristic of spherical-oscillator-type motions, and then this latter is further reduced to simple rotational invariance. Baryonic states are assigned to the 165-dimensional irreducible representation (IR) of the $U(9)$ corresponding to the first excited shell with respect to the oscillatory motions of relative coordinates. These states are subgrouped according to the IR of the usual $\mathrm{SU}(3)$ and to the eigenvalue of the relative angular momentum. Identifications with known levels are then made. The whole treatment is carried out covariantly, and minimum violation of causality is implied inside the particle.