On the Connection between External and Internal Symmetries of Strongly Interacting Particles

Abstract

The relation between the mass and spin-parity Jp, and internal quantum numbers of elementary particles, hints at a nontrivial connection between the external symmetry, namely the Poincaré group or its Lie algebra, and a so-called internal symmetry. Studying this connection mathematically, we find that any extension of (resp. by) the Poincaré Lie algebra 𝒫 by (resp. of) a semisimple Lie algebra 𝒳 is equivalent to the trivial one, 𝒫 ⊕ 𝒳. Moreover, if we are looking for a Lie algebra containing 𝒫 and 𝒳 in an economical and nontrivial way, namely what we call a (nontrivial) unification of 𝒫 and 𝒳, we find restrictions on the possibilities of choice of 𝒳, which exclude compact internal symmetries. An explicit treatment of SL(3, C) as internal symmetry, related to the external symmetry by the unification process of Lie algebras, gives a mass formula which is in very good accordance with the experimental data, and can be theoretically interpreted by means of a so-called ``classification principle''.