Algebraic description of external and internal attributes of fundamental fermions

Abstract

To describe external and internal attributes of fundamental fermions, a theory of multi-spinor fields is developed on an algebra, a {\it triplet algebra}, which consists of all the triple-direct-products of Dirac \gamma-matrices. The triplet algebra is decomposed into the product of two subalgebras, an external algebra and an internal algebra, which are exclusively related with external and internal characteristic of the multi-spinor field named {\it triplet fields}. All elements of the external algebra which is isomorphic to the original Dirac algebra $A_\gamma$ are invariant under the action of permutation group $S_3$ which works to exchange the order of the $A_\gamma$ elements in the triple-direct-product. The internal algebra is decomposed into the product of two $4^2$ dimensional algebras, called the family and color algebras, which describe the family and color degrees of freedom. The family and color algebras have fine substructures with "trio plus solo" conformations which are irreducible under the action of $S_3$. The triplet field has trio plus solo family modes with ordinary tricolor quark and colorless solo lepton components. To incorporate the Weinberg-Salam mechanism, it is required to introduce two types of triplet fields, a left-handed doublet and right-handed singlets of electroweak iso-spin. It is possible to qualify the Yukawa interaction and to make a new interpretation of its coupling constants naturally in an intrinsic mechanism of the triplet field formalism. The ordinary Higgs mechanism leads to a new type of the Dirac mass matrices which can explain all data of quark sector within experimental accuracy.