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- TL;DR Summary
- Applying findings of Jung's 2020 paper (first source) to a stochastic interpretation of quantum mechanics. This results in a Bell non-local description of Bell correlations but with no FTL communication; the perfect correlations are remembered from a phase-shift at source. Taking into account that stochastic mechanical (angular) velocities will satisfy the quantum mechanical commutation relations, you can use Jung's findings to get Bell correlations with ensembles of stochastic particles.
Jung derives Bell correlations from phase-shifted circularly polarized waves, paralleling Alain Aspect's descriptions of Bell experiments in terms of circularly polarized photons (e.g. here). I just interpret the Jung paper result through stochastic mechanics.
First assumption is a spin model (section 5, figure 15) in terms of local orbital motions of particles around their current positions in an advancing circularly polarized wave (image); see the following links for model: water(very nice, detailed article), Wittig, sound1, sound2, light (Bliokh calls it "local ellipticity"). Obviously a classical beam would have many particles, but these Bell experiments deal with them one at a time. Orbits can be represented as rotating velocity vectors that carry the same superposition / interference properties as light, photons, sound or water.
Next assumption is stochastic mechanics. Like in the aforementioned spin model, spin in stochastic mechanics (Beyer) is represented by (angular) velocities; they also satisfy quantum commutation relations which would produce the desired superposition properties. Two caveats: 1) traditionally stochastic spin has been interpreted in terms of a spinning particle; however, moving to an orbiting description wouldn't prima facie require any deep conceptual changes. Yang (2024) has produced a model along the lines of circular orbits. 2) There has never been a stochastic mechanical model of photons, field-theoretic or otherwise; but, I don't think this necessarily detracts from arguments about plausibility involving quite general features like phase-shifts and commutation relations. Extremely bizarre glitch means end of paragraph 3 is in comments - post #3.
Polarizing filters would then probe the superposition properties of these ensembles, the ensemble's rotating spin vector interacting with the polarizer always at a precise angle. The ensemble can then be actualized by repeating a polarizer experiment one particle at a time given a fixed angle of interaction. For a wire-grid polarizer, electrons are allowed to oscillate along a specific plane / axis (e.g. horizontal, diagonal, vertical, etc). Photons moving in directions along that plane / axis will force electrons to oscillate, via a stochastic Newton's law (e.g. as described in this review), leading to photon absorption - the photons have been blocked. Due to the random nature of particle motion, not all particles will be moving in directions that lead to their absorption. The superposition principle implies that over many repeated experiments, we will be left with two sub-ensembles of photons that have respectively passed (i.e. are aligned with the filter) or been blocked by the filter; the average velocities of these sub-ensembles will be in perpendicular directions. The relative frequencies of particles in each sub-ensemble will depend on the angle between direction of the initial (pre-filtered) ensemble velocity (a) and the filter angle (θ) - Malus' law:
cos2(θ - a)
This is visualized in graphic 4/11 of the earlier circularly polarized wave link. The red orientation at each angle is a superposition of green and blue orientations of different sizes that are perpendicular to each other.
A circularly polarized wave can also be seen in terms of superpositions of two circularly polarized sub-waves, both rotating at a constant rate in the same direction, with possibly different amplitudes, but phase-shifted so that the orientation of one is always 90° (i.e. perpendicular like in paragraph 4) lagging behind the other in their respective rotations. Similar can then be said for the ensembles. The initial ensemble took a trajectory starting from source, its velocity vector constantly rotating along the way until it contacts the polarizer at a single angle. So too can its two sub-ensembles be seen as traveling from source, vectors rotating along the way; the particles from one of the rotating sub-ensembles are then blocked by the polarizer. The superposition principle here is saying that angular momentum is conserved for both an ensemble and its sub-ensembles. To emphasize, the traveling of these ensembles is one particle at a time from repeated experiments. An initial ensemble is also compatible with many different superpositions of sub-ensembles; but, effectively this just means carving up particle ensembles in different ways which can be orthogonally filtered by polarizers aligned at various angles.
Given we are talking about entanglement, every individual particle leaves source with another particle that travels to a different polarizer. We then have two initial ensembles A and B, vectors rotating as they travel off to their respective polarizers. Jung's idea is that Bell correlations require a phase shift between A and B so that one ensemble's vector rotation is lagging behind the other's. The pairs of synchronized particle orbits, represented by the rotating vectors, behave like clocks in different time-zones. A clock set to Paris-time is always 3 hours behind a clock on Moscow-time without fail because of how clock hands rotate periodically at the same rate, regardless of how far apart the clocks are, and will do so forseeably until disturbed. The same goes for rotating vectors regarding spin; a 90° phase-shift between two particle orbits is analogous to a 3 hour time-difference and results in anti-correlations: if one particle is on a Horizontal part of its orbit, the entangled partner is always in a Vertical part, and vice versa. 0° or 180° (6 hour) phase shifts give perfect correlations: HH or VV (or any other pairs of directions). Given that these experiments involve only two particles at any time, and the phase shift is engineered at source, we can say that: when the pair leave the source, one particle is always lagging behind the other with regard to their orbiting trajectories. This is maintained until contact with the polarizers, enabled by the periodicity of the orbits. Because the phase-shift is applied on every single pair leaving the source, then not only would ensembles A and B - constructed from many repetitions of these pairs - be phase-shifted; but, any sub-ensemble of A will also be phase-shifted from a specific sub-ensemble of B also, since they are too just constructed from many phase-shifted individual pairings.
We can now calculate Bell correlations from this description. The probability of a particle from one ensemble passing a filter is ½. Given that the probability of particles from an ensemble passing a fixed polarizer angle differs with the ensemble direction according to Malus' law, the probability ½ can be found by assuming that the angle at which the ensemble contacts the polarizer is completely random and fair so that the probabilities from Malus' law for each direction average out to ½ (note that ensembles with different vector directions will overlap in general as implied by Malus' law). Given that particles will maintain the same velocities after leaving the source, a natural way of ensuring the ½ probability would be to allow for random initial vector directions at source (but pairs still being phase-shifted). This is prima facie consistent with the fact that stochastic mechanical models (e.g. section 3.1.1; also see Yang (2024) above) of the Stern-Gerlach experiment have random initial spin directions that are, perhaps counterintuitively, continuous random variables (i.e. can point at any angle or direction) that subsequently lead to a probability ½ for discrete spin-up or -down measurements.
All passing sub-ensembles for polarizer A will then be aligned with the polarizer angle despite the random angle of contact for the initial ensemble, in accordance with the description in paragraph 4 earlier. Because the overall combined sub-ensemble A is just filled with particles that have traveled one-by-one from source, where they had been paired with and phase shifted from a B particle, we now know the average direction for the velocity of a paired sub-ensemble at polarizer B. In short, sub-ensemble B is just full of the phase-shifted entangled partners of the sub-ensemble that passed polarizer A. If we wanted, we could have equivalently started from the sub-ensemble that passes polarizer B instead.
How much of sub-ensemble B passed the polarizer? We just apply Malus' law to the now known direction of sub-ensemble B's velocity and the angle of polarizer B:
cos2(θb - b)
Because sub-ensemble A is always aligned with polarizer angle A and also has a fixed phase-shift from sub-ensemble B, we can alternatively express Malus' law as:
cos2(θb - (θa + phase shift))
For 0° or 180° phase shifts that produce correlations, we have:
cos2(θb - (θa + [0° or 180°])) = cos2(θb - θa)
For 90° phase shifts that produce anti-correlations:
sin2(θb - (θa + 90°)) = sin2(θb - θa)
This is how Jung produces Bell correlations / coincidences from phase shifts and Malus' law:
½ sin2(θb - θa)
½ cos2(θb - θa)
Important to emphasize is that this applies to any angles of θb and θa which will just carve up particle ensembles in different ways. Because the probability distributions given by Malus' law follow from commutation relations, there are no local hidden variables or beables present as defined by Bell. The non-locality is due to the fact that a phase-shift between particles was fixed at source and effectively remembered because of the periodicity of particle orbits due to conservation of angular momentum.
First assumption is a spin model (section 5, figure 15) in terms of local orbital motions of particles around their current positions in an advancing circularly polarized wave (image); see the following links for model: water(very nice, detailed article), Wittig, sound1, sound2, light (Bliokh calls it "local ellipticity"). Obviously a classical beam would have many particles, but these Bell experiments deal with them one at a time. Orbits can be represented as rotating velocity vectors that carry the same superposition / interference properties as light, photons, sound or water.
Next assumption is stochastic mechanics. Like in the aforementioned spin model, spin in stochastic mechanics (Beyer) is represented by (angular) velocities; they also satisfy quantum commutation relations which would produce the desired superposition properties. Two caveats: 1) traditionally stochastic spin has been interpreted in terms of a spinning particle; however, moving to an orbiting description wouldn't prima facie require any deep conceptual changes. Yang (2024) has produced a model along the lines of circular orbits. 2) There has never been a stochastic mechanical model of photons, field-theoretic or otherwise; but, I don't think this necessarily detracts from arguments about plausibility involving quite general features like phase-shifts and commutation relations. Extremely bizarre glitch means end of paragraph 3 is in comments - post #3.
Polarizing filters would then probe the superposition properties of these ensembles, the ensemble's rotating spin vector interacting with the polarizer always at a precise angle. The ensemble can then be actualized by repeating a polarizer experiment one particle at a time given a fixed angle of interaction. For a wire-grid polarizer, electrons are allowed to oscillate along a specific plane / axis (e.g. horizontal, diagonal, vertical, etc). Photons moving in directions along that plane / axis will force electrons to oscillate, via a stochastic Newton's law (e.g. as described in this review), leading to photon absorption - the photons have been blocked. Due to the random nature of particle motion, not all particles will be moving in directions that lead to their absorption. The superposition principle implies that over many repeated experiments, we will be left with two sub-ensembles of photons that have respectively passed (i.e. are aligned with the filter) or been blocked by the filter; the average velocities of these sub-ensembles will be in perpendicular directions. The relative frequencies of particles in each sub-ensemble will depend on the angle between direction of the initial (pre-filtered) ensemble velocity (a) and the filter angle (θ) - Malus' law:
cos2(θ - a)
This is visualized in graphic 4/11 of the earlier circularly polarized wave link. The red orientation at each angle is a superposition of green and blue orientations of different sizes that are perpendicular to each other.
A circularly polarized wave can also be seen in terms of superpositions of two circularly polarized sub-waves, both rotating at a constant rate in the same direction, with possibly different amplitudes, but phase-shifted so that the orientation of one is always 90° (i.e. perpendicular like in paragraph 4) lagging behind the other in their respective rotations. Similar can then be said for the ensembles. The initial ensemble took a trajectory starting from source, its velocity vector constantly rotating along the way until it contacts the polarizer at a single angle. So too can its two sub-ensembles be seen as traveling from source, vectors rotating along the way; the particles from one of the rotating sub-ensembles are then blocked by the polarizer. The superposition principle here is saying that angular momentum is conserved for both an ensemble and its sub-ensembles. To emphasize, the traveling of these ensembles is one particle at a time from repeated experiments. An initial ensemble is also compatible with many different superpositions of sub-ensembles; but, effectively this just means carving up particle ensembles in different ways which can be orthogonally filtered by polarizers aligned at various angles.
Given we are talking about entanglement, every individual particle leaves source with another particle that travels to a different polarizer. We then have two initial ensembles A and B, vectors rotating as they travel off to their respective polarizers. Jung's idea is that Bell correlations require a phase shift between A and B so that one ensemble's vector rotation is lagging behind the other's. The pairs of synchronized particle orbits, represented by the rotating vectors, behave like clocks in different time-zones. A clock set to Paris-time is always 3 hours behind a clock on Moscow-time without fail because of how clock hands rotate periodically at the same rate, regardless of how far apart the clocks are, and will do so forseeably until disturbed. The same goes for rotating vectors regarding spin; a 90° phase-shift between two particle orbits is analogous to a 3 hour time-difference and results in anti-correlations: if one particle is on a Horizontal part of its orbit, the entangled partner is always in a Vertical part, and vice versa. 0° or 180° (6 hour) phase shifts give perfect correlations: HH or VV (or any other pairs of directions). Given that these experiments involve only two particles at any time, and the phase shift is engineered at source, we can say that: when the pair leave the source, one particle is always lagging behind the other with regard to their orbiting trajectories. This is maintained until contact with the polarizers, enabled by the periodicity of the orbits. Because the phase-shift is applied on every single pair leaving the source, then not only would ensembles A and B - constructed from many repetitions of these pairs - be phase-shifted; but, any sub-ensemble of A will also be phase-shifted from a specific sub-ensemble of B also, since they are too just constructed from many phase-shifted individual pairings.
We can now calculate Bell correlations from this description. The probability of a particle from one ensemble passing a filter is ½. Given that the probability of particles from an ensemble passing a fixed polarizer angle differs with the ensemble direction according to Malus' law, the probability ½ can be found by assuming that the angle at which the ensemble contacts the polarizer is completely random and fair so that the probabilities from Malus' law for each direction average out to ½ (note that ensembles with different vector directions will overlap in general as implied by Malus' law). Given that particles will maintain the same velocities after leaving the source, a natural way of ensuring the ½ probability would be to allow for random initial vector directions at source (but pairs still being phase-shifted). This is prima facie consistent with the fact that stochastic mechanical models (e.g. section 3.1.1; also see Yang (2024) above) of the Stern-Gerlach experiment have random initial spin directions that are, perhaps counterintuitively, continuous random variables (i.e. can point at any angle or direction) that subsequently lead to a probability ½ for discrete spin-up or -down measurements.
All passing sub-ensembles for polarizer A will then be aligned with the polarizer angle despite the random angle of contact for the initial ensemble, in accordance with the description in paragraph 4 earlier. Because the overall combined sub-ensemble A is just filled with particles that have traveled one-by-one from source, where they had been paired with and phase shifted from a B particle, we now know the average direction for the velocity of a paired sub-ensemble at polarizer B. In short, sub-ensemble B is just full of the phase-shifted entangled partners of the sub-ensemble that passed polarizer A. If we wanted, we could have equivalently started from the sub-ensemble that passes polarizer B instead.
How much of sub-ensemble B passed the polarizer? We just apply Malus' law to the now known direction of sub-ensemble B's velocity and the angle of polarizer B:
cos2(θb - b)
Because sub-ensemble A is always aligned with polarizer angle A and also has a fixed phase-shift from sub-ensemble B, we can alternatively express Malus' law as:
cos2(θb - (θa + phase shift))
For 0° or 180° phase shifts that produce correlations, we have:
cos2(θb - (θa + [0° or 180°])) = cos2(θb - θa)
For 90° phase shifts that produce anti-correlations:
sin2(θb - (θa + 90°)) = sin2(θb - θa)
This is how Jung produces Bell correlations / coincidences from phase shifts and Malus' law:
½ sin2(θb - θa)
½ cos2(θb - θa)
Important to emphasize is that this applies to any angles of θb and θa which will just carve up particle ensembles in different ways. Because the probability distributions given by Malus' law follow from commutation relations, there are no local hidden variables or beables present as defined by Bell. The non-locality is due to the fact that a phase-shift between particles was fixed at source and effectively remembered because of the periodicity of particle orbits due to conservation of angular momentum.
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