AI 中文总结
通过经典声子模型（Frenkel-Kontorova模型）形式上得到单自旋无粒子的薛定谔方程，从耦合谐振子晶格出发经一系列变换在非相对论极限下将克莱因-戈登方程写成薛定谔方程，还能将经典可观测量重写为量子可观测量。
AI 中文摘要
通过经典声子模型，即Frenkel-Kontorova模型，形式上获得单自旋无粒子的薛定谔方程。从一维耦合谐振子晶格开始，表明相应牛顿运动方程的连续极限产生实值场的克莱因-戈登方程。通过引入混合实值位移和速度场的复值变量变换，并分离快慢时间尺度，在非相对论极限下将克莱因-戈登方程写为薛定谔方程。这种复变量变换还允许将声子场的经典全局可观测量，如总能量或动量，重写为相应的量子可观测量。此外，表明当将摩擦力纳入经典模型时，相应的克莱因-戈登方程可以重写为具有非厄米哈密顿量的薛定谔方程。虽然这里的全局方法限于非相对论 regime 且未解决测量问题、量子化或相对论效应，但它说明了如何使用经典动力学精确再现量子代数和复值波函数。本系列的第二部分讨论了无自旋粒子的相对论 regime 以及对易子和泊松括号之间的联系。
英文摘要
The Schrodinger equation for a single spinless particle is formally obtained via a classical phonon model, namely the Frenkel-Kontorova model. Starting from a one-dimensional lattice of coupled harmonic oscillators, we show that the continuous limit of the corresponding Newtonian equation of motion yields the Klein-Gordon equation for a real-valued field. By introducing a complex-valued change of variables mixing the real-valued displacement and velocity fields, and by separating fast and slow time scales, the Klein-Gordon equation is written as the Schrodinger equation within the non-relativistic limit. This complex change of variable also allows to rewrite classical global observables of the phonon field, such as the total energy or momentum, as the corresponding quantum observables. Additionally, we show that when a friction force is incorporated into the classical model, the corresponding Klein-Gordon equation can be rewritten as a Schrodinger equation with a non-Hermitian Hamiltonian. While the global approach is limited here to the non-relativistic regime and does not address the measurement problem, quantization or relativistic effects, it nonetheless illustrates how quantum algebra and complex-valued wave functions can be exactly reproduced using classical dynamics. The relativistic regime for a spinless particle and the link between commutators and Poisson brackets is addressed in the second part of this series.
