Oscillator model for particles underlying unitary symmetry

Abstract

1. For a unified description of elementary particles YUKAWA was early led to a relativistic harmonic-oscillator model in the framework of bi-local theory (1), which introduces, besides the usual external co-ordinates X~ and their conjugate momentum components P~, the internal space-time co-ordinates x~ and their conjugates p~. Recently he made a new development of the theory (~) by showing tha t it could imply certain internal quantum number such as isospin. On the other hand different oscillator schemes for particles have been set forth (~-5) to directly account for supermult iplet structures and mass rules for particles. In par t icular in ref. (~) i t was pointed out tha t the Ua symmetry, originally proposed from the s tandpoint of the Sakata model (6), could be interpreted in terms of an oscilla tor model for particles, and it Was remarked tha t the reason why the threedimensional uni tary symmetry should operate could be tr ivial ly understood, according to this kind of model, as ' the direct consequence of the dimensionality of the physical space itself. By combining both viewpoints we introduce in this note the basic assmnption (~) ( r : 1, 2, 3) exist, instead of a single x~ tha t three internal co-ordinate vectors x~ of Yt~KAWA, and show tha t this extended model provides a new basis underlying the Ua or SU a symmetry which has recently proved more and more successful for strongly interact ing particles (6.s). Indeed our theory presents a unified model for baryons and their excitations since all baryonic states belonging to various possible supermultiplets can be regarded as different oscillator levels of tha t same system, each of which is specified by the set of appropriate quantum numbers tha t include the usual ones. The immediate prediction of our model is the existence of specific