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arXiv:2202.13364

A reconstruction of quantum theory for spinning particles

U. Klein

Published Feb 27, 2022• 0 citations
arXivHTMLPDF

Abstract

As part of a probabilistic reconstruction of quantum theory (QT), we show that spin is not a purely quantum mechanical phenomenon, as has long been assumed. Rather, this phenomenon occurs before the transition to QT takes place, namely in the area of the quasi-classical (here better quasi-quantum) theory. This borderland between classical physics and QT can be reached within the framework of our reconstruction by the replacement $p \rightarrow M (q, t)$, where $p$ is the momentum variable of the particle and $M(q, t)$ is the momentum field in configuration space. The occurrence of spin, and its special value $1/2$ , is a consequence of the fact that $M(q,t)$ must have exactly three independent components $M_{k}(q,t)$ for a single particle because of the three-dimensionality of space. In the Schrödinger equation for a "particle with spin zero", the momentum field is usually represented as a gradient of a single function $S$. This implies dependencies between the components $M_{k}(q,t)$ for which no explanation exists. In reality, $M(q,t)$ needs to be represented by three functions, two of which are rotational degrees of freedom. The latter are responsible for the existence of spin. All massive structureless particles in nature must therefore be spin-one-half particles, simply because they have to be described by $4$ real fields, one of which has the physical meaning of a probability density, while the other three are required to represent the momentum field in three-dimensional space. We derive the Pauli-Schrödinger equation, the correct value $g=2$ of the gyromagnetic ratio, the classical limit of the Pauli-Schrödinger equation, and clarify some other open questions in the borderland between classical physics and QT.

DOI
10.48550/arxiv.2202.13364

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As part of a probabilistic reconstruction of quantum theory (QT), we show that spin is not a purely quantum mechanical phenomenon, as has long been assumed. Rather, this phenomenon occurs before the transition to QT takes place, namely in the area of the quasi-classical (here better quasi-quantum) theory. This borderland between classical physics and QT can be reached within the framework of our reconstruction by the replacement p→M(q,t)p \rightarrow M (q, t)p→M(q,t), where ppp is the momentum variable of the particle and M(q,t)M(q, t)M(q,t) is the momentum field in configuration space. The occurrence of spin, and its special value 1/21/21/2 , is a consequence of the fact that M(q,t)M(q,t)M(q,t) must have exactly three independent components Mk(q,t)M_{k}(q,t)Mk​(q,t) for a single particle because of the three-dimensionality of space. In the Schrödinger equation for a "particle with spin zero", the momentum field is usually represented as a gradient of a single function SSS. This implies dependencies between the components Mk(q,t)M_{k}(q,t)Mk​(q,t) for which no explanation exists. In reality, M(q,t)M(q,t)M(q,t) needs to be represented by three functions, two of which are rotational degrees of freedom. The latter are responsible for the existence of spin. All massive structureless particles in nature must therefore be spin-one-half particles, simply because they have to be described by 444 real fields, one of which has the physical meaning of a probability density, while the other three are required to represent the momentum field in three-dimensional space. We derive the Pauli-Schrödinger equation, the correct value g=2g=2g=2 of the gyromagnetic ratio, the classical limit of the Pauli-Schrödinger equation, and clarify some other open questions in the borderland between classical physics and QT.
Within the framework of the individuality interpretation of quantum theory (QT), the basic equations of QT cannot be derived from the basic equations of classical mechanics (CM). The unbridgeable gap between CM and QT is given by the fact that a certain system which is described in CM by a finite number of degrees of freedom requires an infinite number in QT. The standard quantization method, which is conceptually closely linked to the individuality interpretation, is limited to finding structural similarities between observables and operators. The fundamental question \emph{why} one must move from a finite number to an infinite number of degrees of freedom, remains unanswered. This gap can only be closed if probabilistic aspects are already taken into account in the classical area. This may be done by taking the uncertainty in initial conditions into account. In this probabilistic version of mechanics (PM), a system is mathematically described as an ensemble, with an infinite number of degrees of freedom, thus bridging the gap mentioned above. This step then enables the reconstruction of QT, in particular the derivation of the Schrödinger equation, from PM. This work is the third in a series of works in which this program is carried out. The method used here differs from the previous one and allows a better understanding of the structural differences between classical physics and QT. The derivation of the Schrödinger equation essentially takes place in two steps: a projection from phase space to configuration space and a linearization. Some contradictions of the individuality interpretation are analyzed and eliminated from the point of view of the ensemble interpretation.
We show that quantum theory (QT) is a substructure of classical probabilistic physics. The central quantity of the classical theory is Hamilton’s function, which determines canonical equations, a corresponding flow, and a Liouville equation for a probability density. We extend this theory in two respects: (1) The same structure is defined for arbitrary observables. Thus, we have all of the above entities generated not only by Hamilton’s function but also by every observable. (2) We introduce for each observable a phase space function representing the classical action. This is a redundant quantity in a classical context but indispensable for the transition to QT. The basic equations of the resulting theory take a “quantum-like” form, which allows for a simple derivation of QT by means of a projection to configuration space reported previously [Quantum Stud Math Found 5:219–227, 2018]. We obtain the most important relations of QT, namely the form of operators, Schrödinger’s equation, eigenvalue equations, commutation relations, expectation values, and Born’s rule. Implications for the interpretation of QT are discussed, as well as an alternative projection method allowing for a derivation of spin.
Abstract Conventional methods for teaching spin at a university level generally follow four main approaches. The first introduces spin into the Schrödinger equation artificially, leaving its physical meaning unclear. The second connects spin to the representation of the rotation group, which clarifies its nature but requires knowledge of group theory. The third approach derives from field theory, separating angular momentum into orbital and intrinsic parts, the latter corresponding to spin; however, this cannot be done within quantum mechanics alone. The fourth derives electron spin from Dirac’s 1928 relativistic equation, which, though correct, has led to the misconception that spin is a purely relativistic effect. A detailed analysis of Dirac’s 1928 papers shows that the concept of spin arises from the need to construct an equation linear in both energy and momentum to preserve Born’s probabilistic interpretation. This linearization of the Klein–Gordon equation reveals that spin is conceptually independent of relativity. Historical and pedagogical perspectives can together clarify conceptual difficulties and promote a deeper understanding of fundamental ideas in physics. Drawing on selected passages from Dirac’s original work, the present discussion offers broader conceptual and cultural considerations that introduce the notion of spin with limited mathematical requirements and can be applied independently of the didactic approach adopted.
It is a scientific truism nowadays that quantum mechanical predictions are of statistical or rather of probabilistic nature: a quantum theory assigns a probability distribution to each measurable quantity (observable) of a physical system prepared according to a given preparation prescription (state). Such sets of probability distributions represent the most general quantitative predictions of physical theories. The aim of this study is to determine all theoretical frameworks which yield such a kind of output taking the probabilistic nature of quantitative physical predictions as initial hypothesis. Astonishingly, this project succeeds with a minimal number of additional assumptions, which arise quite naturally in the course of the analysis, and which essentially amount to regularity conditions. The result is that there exist only two classes of physical theories providing quantitative probabilistic predictions: classical theories where observables are represented by real functions over a phase space and states are represented by probability measures upon the phase space, and quantum theories where observables are represented by selfadjoint operators acting on a complex Hilbert space and states are represented by state operators. On empirical grounds, classical theories cannot provide the all-encompassing, fundamental frame for the mathematical description of physical systems. Quantum theories remain, and there is no more complete formal frame for quantitative physical theories. As an additional result, it is shown that there exist physical systems which possess a quantum mechanical but no classical description; a prominent example is the spin system of two spin-1/2 particles. The classical hidden variables model, which J. Bell relates to this system in order to prove his theorem, cannot be a true description of it because no such classical model exists, and Bell's theorem loses its foundation.
Spin, still a concept or property not fully exploited, has already impacted the world in many ways with its many applications. The history of how spin came to light and how it was interpreted and developed as investigations were conducted is of particular interest with the many odd outcomes of experiments concerning this intrinsic property. This article will sort out the historical outline of how spin was first indicated in theories and how it was eventually embraced as a fundamental property of quantum particles in an attempt to holistically review the process of recognition of this remarkable concept. Through this review, it becomes apparent how the recognition and development of a concept involves the intellect of a broad range of physicists who are willing to view things from both classical and quantum perspectives particularly when it comes to new concepts as spin. How elements like luck and chance play a role will also be revealed. As it turns out, our intuitions about the world around us may not be the ultimate truth, especially when it comes to Quantum Mechanics. Space Quantization was once thought of as no more than a theoretical outcome, however, experiments anyway impose this reality on physicists requiring them to abandon stubborn biases. Admittedly, not every speculation was correct, the analogy of “Spin” was an unfortunate one. But this article shows that this is just how understanding the mysteries of the quantum world really unfolds.