Analysis of the Generalized Dirac Equation

It is well known, that in the limit of long-wave oscillations (transition to a continuous string) the chain of coupled identical oscillators is described by the real Klein–Gordon–Fock equation. We previously considered a mass-in-mass chain with additional harmonic interaction between internal masses. This chain can be thought of as two coupled classical chains of identical masses. In the limit of long-wave oscillations, considered chain is described by the system of two second-order partial differential equations, which is a generalization of the real Klein–Gordon–Fock equation. Based on this system in the special case of the equality of the characteristic frequencies of two considered classical chains of identical masses, we have constructed a system of two equations, which is the generalization of the complex-valued quantum-relativistic Klein–Gordon–Fock equation. Then, we constructed the generalization of the quantum-relativistic Dirac equation. The generalized Dirac equation has both optical and acoustic branches of the dispersion law, each with positive and negative energies. Two massive particles with different masses relate to the optical branch of the dispersion law that is similar on electron and massive neutrinos pair. The generalized Dirac equation can be transformed into three independent systems of equations. First one has the form of the Dirac equation, second one is the Weyl equation for massless neutrinos, and third one is the Weyl equation for massless antineutrinos. We examine a mass(-in-mass)2 chain that has two optical and one acoustic branches of the dispersion law. In this case we have six different masses in corresponded generalization of the Klein–Gordon–Fock equation. We discuss possible interpretation of this fact. In addition, we discuss how these two optical branches of the dispersion law can be related to the problem of the hierarchy of fermionic masses. In addition, we considered mass(-in-mass)3 and mass(-in-mass)N chains.