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https://github.com/rudschuck/stochastic-mechanics-and-quantum-mechanics

Compendium on Nelson's stochastic mechanics as an interpretation of quantum mechanics.
https://github.com/rudschuck/stochastic-mechanics-and-quantum-mechanics

nelson quantum quantum-mechanics quantum-physics skript stochastic stochastic-differential-equations stochastic-processes

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Compendium on Nelson's stochastic mechanics as an interpretation of quantum mechanics.

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# Stochastic Mechanics and Quantum Mechanics

Compendium on Nelson's stochastic mechanics as an interpretation of quantum mechanics.



This compendium arose from the necessity of having to deal with the subject of quantum mechanics and quantum computing in my job. In this context, I revised my old lecture notes.



For this reason, the compendium has two blemishes:



1. The original language was German and I got stuck with that.

2. It is primarily a personal compilation. For this reason, it is far from being ready for publication. - But hey, I'm working on it and there will be a regular update.



I would like to take this opportunity to express my sincere thanks to all those who are contributing to the development of this compendium. Yes, the presence is intentional. This compendium is not finished but will be continuously developed by me. The aim is not to produce an to create an all-encompassing textbook, but to bring together my ideas, comments and criticisms on stochastic mechanics. 



(c) Michael Rudschuck; [email protected]



### Foreword



This compendium on stochastic mechanics and quantum mechanics is based on a lecture transcript of the lectures Quantenmechanik I and II, held in WS 95 and SS 96 at the TU Clausthal by Prof. Dr. Lothar Fritsche. In this lecture, quantum mechanics was derived from classical mechanics.



The corresponding publications are



* [(Link)](https://arxiv.org/abs/0912.3442)

* [(Link)](https://www.researchgate.net/figure/Spin-effective-components-of-the-quivering-motion_fig3_45890773)

* [(Link)](https://www.karlsruher-physikkurs.de/download/qm_wo_obs_2.pdf)



(The third publication is very detailed and an absolute must read.)



The compendium shows a value-free derivation of quantum mechanics from classical mechanics as one of many approaches. 

Stochastic mechanics is an interpretation of quantum mechanics. 

Interestingly, however, only one assumption is required for its derivation. This is the existence of fluctuations of the electromagnetic field in a vacuum. If such fluctuations are assumed, the results of quantum theory can be reproduced, without using quantum theory. For example, stochastic mechanics can be used to explain particle spin. 

as well as the position of the energy levels of the shell electrons in atomic physics. Furthermore, the long-known problem of radiation emission from moving electrons is also solved by this theory. All statements known from quantum mechanics can be applied directly within the framework of 

stochastic mechanics without having to use the formalism of quantum mechanics. 



To be fair, however, it must also be said that stochastic mechanics is only one of many interpretations of quantum mechanics. quantum mechanics. However, it is characterised by a very intuitive approach via classical mechanics.



### Introduction 



Quantum mechanics is undoubtedly one of the most fascinating and challenging theories in physics. Since its development at the beginning of the 20th century, it has changed our understanding of nature in the most fundamental and enabled numerous technological innovations that characterise our daily lives. From electronics, medicine and cryptography, quantum mechanics has found applications in almost every area of science and technology. 



### Stochastic Mechanics



Stochastic mechanics can be seen as a kind of generalisation of the theoretical scheme of classical mechanics. The classical deterministic trajectories are replaced by random trajectories of well-defined stochastic processes. Under suitable conditions and for a large class of dynamical systems, the basic equations of stochastic mechanics show a surprising connection to the basic equations of quantum mechanics. equations of quantum mechanics. It is then possible to develop a suitable scheme of physical interpretation so that the phenomenological content of stochastic mechanics coincides with the phenomenological content of quantum mechanics as far as all experimentally observable effects are concerned. observable effects are affected. Stochastic mechanics thus offers an approach to the quantisation of dynamic systems that is based on methods of probability theory and stochastic processes. This approach is completely different from the traditional approach, which is based on operator methods. From a physical point of view, however, it is completely equivalent if the correct physical interpretation scheme is used.



The foundations of stochastic mechanics were laid by Edward Nelson in 1966, based on earlier ideas by Fenyes, Weizel and others. Nelson's original formulation is based on two basic hypotheses. The first assumes that the trajectories of the dynamical system are system are disturbed by an underlying Brownian motion. movement. The second is a special form of the second principle of dynamics, in which the classical acceleration is replaced by a suitable form of stochastic acceleration. Further developments of the theory show that the basic equation of stochastic mechanics can be derived from variational principles, in complete analogy to classical mechanics, the basic equation of stochastic mechanics can be derived from variational principles based on the same classical action. which are based on the same classical action but use stochastically perturbed trajectories as experimental trajectories. From this view, stochastic mechanics is closely related to classical mechanics as a kind of stochastic generalisation.



Nelson's original formulation from 1966 also contains the basic features of the equivalence between stochastic mechanics and quantum mechanics, including the effects of physical interpretation. This equivalence has been confirmed by research in recent years. been confirmed. It should be noted, however, that stochastic mechanics has some features that, in principle, can potentially be used within the proposed theoretical scheme, but which have no correspondence in the structure of effective physical observability within the conventional interpretation. observability within the conventional interpretation of quantum mechanics. Therefore, it seems completely open the problem of deciding whether the particular structure of stochastic mechanics has an effective value, in the sense of a physical theory that is more detailed than the usual formulation. more detailed than the usual formulation of quantum mechanics. Otherwise, the richer structure of observability in stochastic mechanics would have to be based on purely mathematical terms, without an empirically observable physical counterpart. From this point of view, it would be extremely important to find some kind of physical motivation for the underlying Brownian movement that gives all trajectories a stochastic character.



## Editorial Notes



### Revisions



     - 2024-09-13 - v0.1 - First version released on GitHub