Construction of Schr\"odinger, Pauli and Dirac equations from Vlasov equation in case of Lorentz gauge

TL;DR

This paper derives fundamental quantum and classical equations, including Schrödinger, Pauli, Dirac, Hamilton-Jacobi, and Maxwell equations, from the probability conservation law and Helmholtz decomposition, linking classical and quantum physics.

Contribution

It introduces a novel approach to derive key physical equations from first principles, unifying classical and quantum descriptions.

Findings

Derived Schrödinger, Pauli, Dirac, Hamilton-Jacobi, and Maxwell equations from basic principles.

Established a natural connection between classical and quantum systems.

Provided a unified framework based on probability conservation and Helmholtz decomposition.

Abstract

On the basis of the first principle -- the law of probability conservation and the Helmholtz decomposition theorem the authors have succeeded to construct the Schr\"odinger, Pauli, Dirac equation, the Hamilton-Jacobi equation and the Maxwell equations. The approach described in this paper makes it possible to naturally connect the classical and quantum systems.