Possible identification of quarks with leptons in Lie-isotopic SU(3) theory

A possible identification of the six quarks (d,s,c;u,t,b) with the corresponding leptons (e-,μ-,tau-;v/sub e/,v/sub μ/,v/sub tau/) is attempted via the corrspondence principle, dapprox.(uv-bar/sub e/)e-, sapprox.(tv-bar/sub μ/)μ-, c(bv-bar/sub t/)t-,uapprox.(uv/sub e/) v/sub e/,..., and its inverse, which are formally represented by a non-unitary integral transformation (with kernel P) and its inverse or dual (with kernel Q), connecting the quark and lepton fields. It is shown that PQ and QP may be interpreted as hadronic and leptonic density matrix operators which obey the quantum mechanical analog of the Liouville equation of conservation from which a Lie-isotopic generalization of Heisenberg's equation of motion is abstracted. P and Q form iso-canonically conjugate dynamical veriables, i.e., Q is the isotpic element for the isoassociative product H*Q = HPQ in the equation of motion for Q. It is also shown that PQ and QP, being idempotent operators, have eigenvalues 0 or 1, which imply that both P and Q can be singular, leading to a further differentiation of ''hadronic mechanics'' into the conventional ''isotopic'' theory in which the isotopic element (g) in the isoassociative product A*B = AgB is non-singular and Hermitian, and a new ''homotopic'' theory in which g is singular and non-Hermitian A Lie-admissible generalization is also obained, and SU(2)-spin realizations are indicated