Mass splitting as a covariant concept in the generalized Poincaré group

Abstract

Various consequences of the two thirty-six component momentum operators occurring in the generalized inhomogeneous Poincaré group ISL(6, C) are examined and discussed, in an effort to understand their physical significance and to discover whether they can be consistently dealt with. After a brief comparison of various properties and consequences of the recently proposed U(12) group and ISL(6, C), the properties of the multicomponent momentum operator are examined in more detail. It is shown that corrdinate frames exist in which the covariant and contragredient momenta have the same values, if certain subsidiary conditions (expressible as polynomial equations for the momenta convariant in ISL(6, C)) are imposed. Hypercharge, charge, total isotopic spin, and three four-momenta can be simultaneous observables with nonzero eigenvalues. The remaining six four-momenta have zero eigen-values. The little group, corresponding to the situation where two four-momenta are not zero, is also obtained. As an illustration of this case, new covariant free quark equations are next obtained. Their solutions are shown to give unitary representations in Hilbert space and, in the physical world, to reduce to Dirac equations with different masses for the quarks of different strangeness. Equations are also postulated for the meson representation. They lead to the prediction of four scalar K mesons, in addition to the octet of pseudoscalar and nonet of vector mesons. It is next shown that the so-called “kinematic” mass operator, proposed earlier by the authors, is naturally arrived at from the discussion of free particle equations. A typical “dynamical” mass term responsible for the spin split of the masses is briefly discussed. The general form of the S matrix for quark-singlet scattering, its univtarity and electromagnetic form factors are also considered. Symmetry breaking is an essential concomitant of almost all groups which are relativistic extensions of SU (6). However, it is shown that it can be carried out at such a late stage of the analysis that the symmetry properties are reflected in various features of the results, such as number of particles in a multiplet, and the number and structure of invariants. Symmetry breaking is required to discuss the case of multiplets with nondegenerate mass in exactly the same sense as for those with degenerate masses. Surprisingly, the subsidiary conditions in the former are less stringent than those in the latter case.