Unifying Quantum and Classical Dynamics

TL;DR

This paper demonstrates that quantum and classical dynamics can be described by identical equations, showing a formal unification where quantum observables follow the same equations as classical variables without explicit quantum parameters.

Contribution

It provides a novel formulation of Heisenberg equations that are mathematically identical to classical Newton and Hamilton equations, unifying the two frameworks without deriving classical behavior from quantum mechanics.

Findings

Heisenberg equations can be written in a form identical to Newton's equations.

Heisenberg equations can be generalized to match classical Hamilton's equations.

Quantum and classical dynamics are governed by the same fundamental equations.

Abstract

Classical and quantum physics represent two distinct theories; however, quantum physics is regarded as the more fundamental of the two. It is posited that classical mechanics should arise from quantum mechanics under certain limiting conditions. Nevertheless, this remains a challenging objective. In this work, we explore the potential for unifying the dynamics of classical and quantum physics. This discussion does not suggest that classical behavior emerges from quantum mechanics; rather, it demonstrates the exact equivalence between the dynamics of quantum observables and their classical counterparts. It is shown that the Heisenberg equations of motion can be cast in a form that is identical to Newton's equations of motion, with $\hbar$ being absent from the formulation. In a generalized analysis, the Heisenberg equations are cast in a form that is identical to the classical Hamilton's equations of motion. This implies that both quantum and classical dynamics are governed by the same equations, with the Heisenberg operators substituting the classical observables.