On Quantization, the Generalized Schr\"odinger Equation and Classical Mechanics

TL;DR

This paper introduces a new deformation of quantization leading to a generalized Schr"odinger equation that unifies quantum and classical mechanics, revealing chaotic dynamics in the classical limit.

Contribution

It proposes a novel state-dependent operator deforming quantization, resulting in a nonlinear equation that reproduces both quantum and classical mechanics within a unified framework.

Findings

Reproduces linear quantum mechanics at λ=1

Recovers classical mechanics at λ=0

Demonstrates existence of functionally chaotic dynamics

Abstract

Using a new state-dependent, $\lambda$-deformable, linear functional operator, ${\cal Q}_{\psi}^{\lambda}$, which presents a natural $C^{\infty}$ deformation of quantization, we obtain a uniquely selected non--linear, integro--differential Generalized Schr\"odinger equation. The case ${\cal Q}_{\psi}^{1}$ reproduces linear quantum mechanics, whereas ${\cal Q}_{\psi}^{0}$ admits an exact dynamic, energetic and measurement theoretic {\em reproduction} of classical mechanics. All solutions to the resulting classical wave equation are given and we show that functionally chaotic dynamics exists.