Quaternionic Quantum Mechanics: the Particles, Their q-Potentials and Mathematical Electron Model

Abstract

In this work we show the quaternionic quantum descriptions of physical processes from the Planck to macro scale. The results presented here are based on the concepts of the Cauchy continuum and the elementary cell at the Planck scale. The structurally symmetric quaternion relations and the postulate of the quaternion velocity have been important in the present development. The momentum of the expansion and compression $u̇_{0}(t, x)$ is the consequence of the scalar term $\sigma_{0}(t, x)$ in the quaternionic deformation potential. The quaternionic $G_{0}(m)(σ_{0} + \hat{\textstyle \phi} )$, vectorial $G_{0}(m) \hat{\textstyle \phi}$ and scalar $G_{0}(m)\sigma_{0}$ propagators are used to generate the second order PDE systems for the proton, electron and neutron. A mathematical model of an electron is formulated. It is described by the hyperbolic-elliptic partial differential system of quaternion equations with the initial-boundary conditions. The boundary conditions are generated by the quaternion energy flux that is found with the use of the Gauss theorem, the Cauchy–Riemann derivative and other mathematical formulas. The rigorous assessment of the second order PDE systems allows the proposal of two second order PDE systems 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 the $d$-$u$-$u$ and $d$-$d$-$u$ complexes, respectively. The u and d quarks do not comply with the Cauchy equation of motion. The inconsistencies of the quarks’ PDE with the quaternion forms of the Cauchy equation of motion account for their short lifetime and the observed Quarks Chains. That is, they explain the Wilczek phenomenological paradox: Quarks are Born Free, but everywhere they are in Chains.