Quaternion Quantum Mechanics: The Baryons, Quarks, and Their q-Potentials

The results presented here are based on the Planck-Kleinert crystal concept. The rigorous use of quaternion algebra allows postulating the scalar, vectorial, and quaternion propagators in the ideal elastic continuum. The propagators are used in constructing the proton, electron, and neutron 2nd order partial differential equation systems, PDES. The results generate the two 2nd order PDES for the \(u\) and \(d\) quarks from the \(up\) and \(down\) groups. It was verified that both the proton and the neutron obey experimental findings and are formed by three quarks. The proton and neutron are formed by \(d\)-\(u\)-\(u\) and \(d\)-\(d\)-\(u\) complexes, respectively. All particle PDES comply with the Cauchy equation of motion and can be considered as stable particles. The u and d quarks do not meet the relations of the Cauchy equation of motion. The inconsistencies of the quarks' PDES with the quaternion forms of the Cauchy equation of motion account for their lifetime and the observed Quarks Chains. That is, they explain the Wilczek phenomenological paradox: "Quarks are Born Free, but Everywhere They are in Chains".