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.
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.
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.
It is suggested that a thorough treatment of electron spin has advantages as a method of presenting many of the concepts and techniques of intermediate quantum mechanics. The starting point is the Stern-Gerlach experiment, from which the anticommutation relations for the spin operators may immediately be deduced. The commutation relations, which demonstrate that the spin is an angular momentum, are derived from these and it is shown that the anomalous gyromagnetic ratio is a natural consequence of the properties of the spin operators. The Pauli theory, which is the simplest example of the use of matrices in quantum mechanics, is briefly summarised, and employed in an analysis of the double Stern-Gerlach experiment. This example of a change of representation is used to introduce Dirac's transformation theory which, together with a variant of the double Stern-Gerlach experiment, is made use of in a discussion of the EPR paradox, Bell's inequality and hidden-variable theories. It is emphasised that the quantum mechanical description of these phenomena is consistent and that, from the viewpoint of quantum theory, no action-at-a-distance is required to explain them.
Thus far we have concentrated on the postulates, on their ramifications, and on the means and methods by which they are applied. Connections with classical physics have been featured, and primary notions other than those of Chapter 5 have not been needed. We deviate from this path in the present chapter, introducing a new and extremely important concept; it is based on one of the most significant and fruitful experiments of the quantal era. This experiment was performed by O. Stern and W. Gerlach (1921) and subsequently interpreted by G. Uhlenbeck and S. Goudsmit (1925). The experiment and its elucidation led to the new concept of “intrinsic spin” (in particular to spin “½”), a development that was one of the major turning points in modern physics. The consequences have been monumental, and include primary theoretical developments; the interpretation of data related to, and an understanding of, empirical rules concerning a wide variety of physical systems (e.g., the periodic table of the elements/atomic structure); and the generation of new experimental techniques and of innovative measurements. Our study of intrinsic spin and its correlative, two-state systems, introduces the reader to one of the most profound topics in quantum theory. The Stern–Gerlach experiment and its interpretation The experiment of Stern and Gerlach was designed to test the “spatial quantization” of neutral atoms, i.e., to determine whether the (orbital) angular momentum was quantized. The basic ideas are as follows. Associated with the atom's orbital angular momentum would be a magnetic moment, as in the Bohr model of hydrogen.
Bell experiments have confirmed that quantum entanglement is an inseparable correlation but there is no faster-than-light influence between two entangled particles when a local measurement is performed. However, how such an inseparable correlation is maintained and manifested when the two entangled particle are space-like separated is still not well understood. The recently proposed extended least action principle for quantum mechanics brings new insights to this question. By applying this principle, we show here that even though the inseparable correlation may be initially created by previous physical interaction between the two particles, the preservation and manifestation of such inseparable correlation are achieved through extremizing an information metric that measures the additional observable information of the bipartite system due to vacuum fluctuations. This is physically realized even though there is no further interaction when the two particles move apart, and the underlying vacuum fluctuations are local. In other words, the propagation of inseparable correlation in quantum theory is realized by an information requirement and through a local mechanism. An example of two entangled free particles described by Gaussian wave packets is provided to illustrate these results.
Quantum entanglement produces correlations between spatially separated systems that cannot be reproduced by broad classes of local hidden-variable models and cannot be used for controllable faster-than-light communication. This article develops a proposed Photony interpretation of electron entanglement based on electron structure, c-spin, x-spin, electron transduction, and shared electro-magnetic chain organization. Within Photony Theory, c-spin represents persistent internal rotational handedness, while x-spin provides an externally responsive orientation through which the electron interacts with its local electromagnetic environment. Electron transduction is proposed to organize elemental charge photons into larger chain structures whose directional organization reflects the state of the participating electron. The central hypothesis is that two entangled electrons may remain associated with the same transduced chain after becoming spatially separated. A local measurement may alter the x-spin condition of one electron and thereby reconfigure the rotational state of the shared structure, which in turn constrains the state available to the second electron. This proposed response is treated as a transition of a jointly organized entangled structure rather than as a conventional signal propagating from one particle to the other. The model must preserve local measurement randomness and the quantum no-signaling condition. It must also ultimately reproduce the spin-singlet angular correlation, Bell-inequality violations, and established decoherence behavior. The present work there-fore offers a specific structural hypothesis whose viability depends on future quantitative derivation and experimentally distinguishable predictions.