Mathematical Formulation of the Gell-Mann-Nishijima Scheme for New Particles
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It is shown that the theory of d'Espagnat and Prentki(henceforth referred to as DP)on a mathematical formulation of the Gell-Mann-Nishijima scheme can be generalized so as to accomodate particles of higher isofermion number. Modifications made for this purpose are : (i)the isospinor formalism is adopted for all kinds of particles instead of DP's isospinor-isotensor mixed formalism ; and(ii)the invariance under inversion of isobaric spin space, imposed by DP as a postulate on strong and electromagnetic interactions, is replaced by the invariance under a continuous phase transformation group G_U generated by the operator U representing isofermion number. These modifications lead not only to extension of the framework but also to refinement of the theory as follows. In our formalism, (a)the transformation properties of field variables with respect to G_U are unambiguously determined without resort to the assumption, made by DP, that only the Yukawa interactions appear in the interaction Lagrangian ; (b)the isofermion number U proves to be the number of suffices of the first kind minus that of the second kind, with respect to inversion, of the isospinor describing a particle ; and(c)DP's identity Q=I_3+1 / 2U among the electric charge Q, the third component of is obaric spin I_3, and U as well as Racah's identity P=exp(i1 / 2πU)between U and the isobaric parity P follows directly from an identical relation among the transformation groups concerned.
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