Abstract The discussion of the recently derived quantum Hamilton equations for a spinning particle is extended to spin measurement in a Stern–Gerlach experiment. We show that this theory predicts a continuously changing orientation of the particles magnetic moment over the course of its motion across the Stern–Gerlach apparatus. The final measurement results agree with experiment and with predictions of the Pauli equation. Furthermore, the Einstein–Podolsky–Rosen–Bohm thought experiment is investigated, and the violation of Bells’s inequalities is reproduced within this stochastic mechanics approach. The origin of the violation of Bell’s inequalities is traced to the the non-local nature of the velocity fields for an entangled state in the stochastic formalism, which is a result of a non-separable probability distribution of the considered particles.
Abstract The discussion of the recently derived quantum Hamilton equations for a spinning particle is extended to spin measurement in a Stern–Gerlach experiment. We show that this theory predicts a continuously changing orientation of the particles magnetic moment over the course of its motion across the Stern–Gerlach apparatus. The final measurement results agree with experiment and with predictions of the Pauli equation. Furthermore, the Einstein–Podolsky–Rosen–Bohm thought experiment is investigated, and the violation of Bells’s inequalities is reproduced within this stochastic mechanics approach. The origin of the violation of Bell’s inequalities is traced to the the non-local nature of the velocity fields for an entangled state in the stochastic formalism, which is a result of a non-separable probability distribution of the considered particles.
This paper aims at reproducing quantum mechanical (QM) spin and spin entanglement results using a realist, stochastic, and local approach, without the standard QM mathematical formulation. The concrete model proposed includes the description of Stern–Gerlach apparatuses and of Bell test experiments. Single particle trajectories are explicitly evaluated as a function of a few stochastic variables that they assumedly carry on. QM predictions are retrieved as probability distributions of similarly-prepared ensembles of particles. Notably, it is shown that the proposed model, despite being both local and realist, is able to violate the Bell–CHSH inequalities by exploiting the coincidence loophole and thus intrinsically renouncing to one of the Bell’s assumptions.
In 1985, Edward Nelson, who formulated the theory of stochastic mechanics, made an interesting remark on Bell’s theorem. Nelson analysed the latter in the light of classical fields that behave randomly. He found that if a stochastic hidden variable theory fulfils certain conditions, the inequality of Bell can be violated. Moreover, Nelson was able to prove that this may happen without any instantaneous communication between the two spatially separated measurement stations. Since Nelson’s article got almost overlooked by physicists, we try to review his comments on Bell’s theorem. We argue that a modification of stochastic mechanics published recently by Fritsche and Haugk can be extended to a theory which fulfils the requirements of Nelson’s analysis. The article proceeds to derive the quantum mechanical formalism of spinning particles and the Pauli equation from this version of stochastic mechanics. Then, we investigate Bohm’s version of the EPR experiment. Additionally, other setups, like entanglement swapping or time and position correlations, are shortly explained from the viewpoint of our local hidden-variable model. Finally, we mention that this theory could perhaps be relativistically extended.
Abstract In 1985, Edward Nelson, who formulated the theory of stochastic mechanics, made an interesting remark about Bell's theorem. Nelson analysed the latter in the light of classical fields that behave randomly. He found that if a stochastic hidden variable theory fulfils certain conditions, the inequality of Bell can be violated. Moreover, Nelson was able to prove that this may happen without any instantaneous communication between the two spatially separated measurement stations. Since Nelson's article got almost overlooked by physicists, we try to review his comments on the theorem. We argue that a modification of stochastic mechanics published recently by Fritsche and Haugk can be extended to a theory which fulfils the requirements of Nelson's analysis. The article proceeds to derive the quantum mechanical formalism of spinning particles and the Pauli equation from this version of stochastic mechanics. Then, we investigate Bohm's version of the EPR experiment. Additionally, other setups, like entanglement swapping or time and position correlations, are shortly explained from the viewpoint of our local hidden‐variable model. Finally, we mention that this theory could perhaps be relativistically extended and useful for the formulation of quantum mechanics in curved space‐times.
In 1985, Edward Nelson, who formulated the theory of stochastic mechanics, made an interesting remark on Bell's theorem. Nelson analysed the latter in the light of classical fields that behave randomly. He found that if a stochastic hidden variable theory fulfils certain conditions, the inequality of Bell can be violated. Moreover, Nelson was able to prove that this may happen without any instantaneous communication between the two spatially separated measurement stations. Since Nelson's article got almost overlooked by physicists, we try to review his comments on Bell's theorem. We argue that a modification of stochastic mechanics published recently by Fritsche and Haugk can be extended to a theory which fulfils the requirements of Nelson's analysis. The article proceeds to derive the quantum mechanical formalism of spinning particles and the Pauli equation from this version of stochastic mechanics. Then, we investigate Bohm's version of the EPR experiment. Additionally, other setups, like entanglement swapping or time and position correlations, are shortly explained from the viewpoint of our local hidden-variable model. Finally, we mention that this theory could perhaps be relativistically extended.
Quantum theory of electron spin is developed here based on the extended least action principle and assumptions of intrinsic angular momentum of an electron with random orientations. The novelty of the formulation is the introduction of relative entropy for the random orientations of intrinsic angular momentum when extremizing the total actions. Applying recursively this extended least action principle, we show that the quantization of electron spin is a mathematical consequence when the order of relative entropy approaches a limit. In addition, the formulation of the measurement probability when a second Stern-Gerlach apparatus is rotated relative to the first Stern-Gerlach apparatus, and the Schrödinger-Pauli equation, are recovered successfully. Furthermore, the principle allows us to provide an intuitive physical model and formulation to explain the entanglement phenomenon between two electron spins. In this model, spin entanglement is the consequence of the correlation between the random orientations of the intrinsic angular momenta of the two electrons. Since spin orientation is an intrinsic local property of the electron, the correlation of spin orientations can be preserved and manifested even when the two electrons are remotely separated. The entanglement of a spin singlet state is represented by two joint probability density functions that reflect the orientation correlation. Using these joint probability density functions, we prove that the Bell-CHSH inequality is violated in a Bell test. To test the validity of the spin-entanglement model, we propose a Bell test experiment with time delay. Such an experiment starts with a typical Bell test that confirms the violation of the Bell-CHSH inequality but adds an extra step that Bob’s measurement is delayed with a period of time after Alice’s measurement. As the time delay increases, the test results are expected to change from violating the Bell-CHSH inequality to not violating the inequality.
The Stern-Gerlach experiment is the fundamental experiment in order to exhibit the quantization of spin and understand the measurement problem in quantum mechanics. However, although the Stern-Gerlach experiment plays an essential role in the teaching of quantum mechanics, no complete analysis of this experiment using Pauli spinors is presented in the pedagogical literature. This paper presents such an analysis and develops implications for the theory of quantum measurement. We first propose an analytic expression of both the wave function and the probability density in the Stern-Gerlach experiment. Our explicit solution is obtained via a complete integration of the Pauli equation over time and space. The probability density evolution describes a slipping of the wave packet into two separate packets due to the measurement device, but it cannot account for impacts. We therefore calculate the de Broglie-Bohm trajectories, which not only explain impacts naturally, but also accounts for the spin quantization following the magnetic field gradient. It is then possible to propose a clear explanation of measurement effects in the Stern-Gerlach experiment.