[论文解读] Stochastic Mechanics Without Ad Hoc Quantization: Theory And Applications To Semiclassical Gravity

Maaneli Derakhshani|arXiv (Cornell University)|Mar 31, 2018

Quantum Mechanics and Applications参考文献 213被引用 3

一句话总结

本文提出了一种无需人为引入量子化过程的随机力学框架，以zitterbewegung（快速振荡运动）作为经典基础，推导出类似量子的行为。该框架一致地推导出Schrödinger-Newton方程与半经典的引力模型，表明量子特性可从经典随机动力学中自然涌现，而无需预先假定波函数的存在。

ABSTRACT

Stochastic mechanics (SM), as proposed by Edward Nelson and others in the 20th century, aims to reconstruct quantum mechanics (QM) from a more fundamental theory of classical point particles interacting with a classical-like ether, where said interaction causes the particles to undergo a diffusion process that conserves their average total energy. However, Timothy Wallstrom and others have emphasized that SM runs into the problem that it cannot recover the Schroedinger equation of QM unless an ad hoc quantization condition is assumed. In this thesis, I reformulate SM so that the quantization condition arises as a natural consequence of the classical point particles interacting with the classical-like ether. This is done by combining SM with a proposal by Louis de Broglie and David Bohm, in which an elementary particle in its rest frame is viewed as a localized periodic phenomenon of fixed frequency, from which it follows that the phase of the periodic phenomenon in the lab frame satisfies a relation that's equivalent to the quantization condition. In doing so, I argue that SM can once again be regarded as a viable approach to reconstructing QM. In addition, I show that SM yields novel and empirically viable models of semiclassical Newtonian gravity and electrodynamics, simply by incorporating classical Newtonian gravitational and electrostatic interactions between the particles. I also show how the Schroedinger-Newton equation and the Schroedinger-Coulomb equation arise as mean-field approximations from within these SM models of semiclassical Newtonian gravity and electrodynamics, and how classical Newtonian gravity can be recovered from an appropriate center-of-mass description of many interacting SM particles. Finally, I argue that SM has distinct advantages over the standard and other heterodox approaches to combining quantum mechanics and Newtonian gravity semiclassically.

研究动机与目标

• 通过引入基于zitterbewegung的类经典模型，解决随机力学中Wallstrom对量子化过程的批评。
• 将Schrödinger-Newton方程作为具有zitterbewegung动力学的类经典随机系统的平均场极限进行推导。
• 基于随机力学构建一个一致的半经典引力理论，避免对波函数坍缩或隐变量的假设。
• 为量子现象提供一个经典基础，包括量子势的涌现与非定域关联的产生。
• 证明半经典引力模型可从随机框架的第一原理出发推导，而无需依赖正则量子化。

提出的方法

• 提出一种类经典zitterbewegung模型，其中粒子在康普顿尺度上经历快速振荡运动。
• 推导zitterbewegung粒子的随机哈密顿-雅可比方程，表明概率密度的相位满足类似Schrödinger方程的形式。
• 将zitterbewegung模型应用于多体系统，推导出质心运动的有效非线性Schrödinger方程。
• 利用Madelung变换将随机流体力学方程转化为类似Schrödinger-Newton方程的形式。
• 在平均场极限下，建立随机势与牛顿引力势之间的联系。
• 证明该理论在大N极限下重现Schrödinger-Newton方程，为半经典引力提供了经典起源。

实验结果

研究问题

• RQ1 能否通过一个类经典的随机zitterbewegung模型，在不预先假定波函数的前提下重现Schrödinger方程？
• RQ2 zitterbewegung模型如何解决随机力学中关于相位量子化的Wallstrom批评？
• RQ3 能否从类经典的随机框架中推导出半经典引力（如Schrödinger-Newton方程），而无需人为引入量子化？
• RQ4 zitterbewegung的频率与振幅在多体系统中生成类似量子行为的过程中起什么作用？
• RQ5 与其它半经典引力模型相比，该随机方法在一致性和物理诠释方面表现如何？

主要发现

• zitterbewegung模型通过为随机力学中相位的量子化提供经典起源，成功解决了Wallstrom的批评。
• zitterbewegung粒子的随机哈密顿-雅可比方程导出一个类似Schrödinger方程的概率振幅方程，从而恢复标准的量子形式体系。
• 在大N极限下，zitterbewegung粒子的质心运动遵循非线性Schrödinger-Newton方程，确立了半经典引力的经典推导。
• 该理论作为平均场近似重现Schrödinger-Newton方程，表明引力可从随机动力学中自然涌现。
• 该模型提供了一个无需波函数坍缩或隐变量假设的一致半经典引力框架。
• 该理论中的随机势在平均场极限下对应于牛顿引力势，验证了半经典方法的合理性。