Dispersion chain of quantum mechanics equations

TL;DR

This paper introduces a novel dispersion chain for quantum mechanics equations based on the Vlasov equations, extending classical and quantum systems analysis to generalized phase spaces with new theorems and applications.

Contribution

It proposes a new chain of quantum mechanics equations for high kinematic values, extending Hamiltonian and Lagrangian formalisms to generalized phase spaces.

Findings

Reduction to quantum phase space (Wigner function) and pilot wave theory in special cases

Derivation of theorems on Hamilton and Maxwell equations extensions

Solution of second-rank Schrödinger equation with positive distribution density

Abstract

Based on the dispersion chain of the Vlasov equations, the paper considers the construction of a new chain of equations of quantum mechanics of high kinematical values. The proposed approach can be applied to consideration of classical and quantum systems with radiation. A number of theorems are proved on the form of extensions of the Hamilton operators, Lagrange functions, Hamilton-Jacobi equations, and Maxwell equations to the case of a generalized phase space. In some special cases of lower dimensions, the dispersion chain of quantum mechanics is reduced to quantum mechanics in phase space (the Wigner function) and the de Broglie-Bohm {\guillemotleft}pilot wave{\guillemotright} theory. An example of solving the Schr\"odinger equation of the second rank (for the phase space) is analyzed, which, in contrast to the Wigner function, gives a positive distribution density function.