Quarks, Leptons, and Density Matrices

For an operator to be a pure density matrix it is necessary and sufficient that it be Hermitian, have trace one, and it be a projection operator. These are commonly generalized by giving up the projection operator requirement; the result are mixed density matrices convenient for modeling statistical mixtures. Other generalizations are also interesting. We solve for the Hermitian projection operators of a simple algebra based on the permutations of three elements, and show that the solutions give the weak quantum numbers, $(t_0,t_3)$, of the quarks and leptons.