To rejuvenate the plausibility of stochastic mechanics

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Three points for the plausibility of a stochastic intepretation of quantum mechanics based on:

1. A recent demonstration of a stochastic-quantum correspondence.

2. Khrennikov's (Kolmogorovian) Vaxjo contextual probability model.

3. The universality of non-commutativity and uncertainty relations in stochastic systems.
In the stochastic interpretation, quantum mechanics describes the behavior of a stochastic process. Particles have real, conventionally definite properties but their dynamics are stochastic. The wave function's only role is to generate probabilities concerning those dynamics and it does not physically collapse because it does not represent an actual physical state.

Three points for its plausibility:

1. Stochastic-quantum correspondence

Some recent pre-prints have shown a formal correspondence between generalized stochastic systems and quantum ones:

https://arxiv.org/abs/2309.03085

https://arxiv.org/abs/2302.10778

These papers go on to show that when these stochastic systems have indivisible dynamics, archetypal quantum phenomena like interference, superposition, decoherence and entanglement can all arise naturally. This suggests that in principle, it is possible that quantum phenomena can be instantiated entirely in the long-run statistical behavior of individual particles which possess definite properties - so long as those properties are subject to stochastic fluctuations and that the resultant dynamics are characterized by indivisibility. In these papers, indivisibility is denoted by dropping marginalization rules and non-negativity assumptions for the transition probabilities of trajectories.

We can note that violations of marginalization rules are actually recurring features in quantum mechanics. Considering that the crucial variables for delineating trajectories are position and momentum, it seems clear that the marginalization violations in quantum mechanics should go hand-in-hand with those that characterize indivisible trajectories. We might then consider these as different ways of looking at the same thing, which becomes more obvious when quantum mechanics talks paths and dynamics.. e.g.,

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.013603

https://journals.aps.org/pra/abstract/10.1103/PhysRevA.91.0621232. The quantum formalism plausibly has a Kolmogorovian interpretation.

Marginalization violations entail the absence of a unique joint probability distribution. Such absences suggest that the statistics of observables can only be fully described in a context-sensitive manner. Motivated by this, a natural line of enquiry is to look at the implications that context-sensitive random variables have for marginalization violations generally.

Khrennikov's Vaxjo contextual probability model does this: e.g.,

https://arxiv.org/abs/quant-ph/0105059

https://arxiv.org/abs/quant-ph/0302194

https://arxiv.org/abs/hep-th/0112076

https://www.mdpi.com/1099-4300/24/10/1380/htm

Khrennikov starts off by assuming pairs of generic Kolmogorovian random variables that are context-dependent and can be disturbed in the sense that transitions between different mutually exclusive contexts can be induced. Given certain assumptions, he then shows that this purely Kolmogorovian scheme can naturally produce violations of marginalization rules and the associated interference terms found in quantum mechanics that preclude joint distributions. He goes on to show that various other features central to quantum mechanics seem to be implicit within this scheme: e.g., complex amplitudes, the born rule, superposition and non-commutativity. This scheme is clearly more general than quantum mechanics; however, it alludes that much of the quantum formalism, and indivisibility, could be a by-product of representing context-dependent, Kolmogorovian statistics.3. Commutation and uncertainty relations are generic consequences of stochasticity.

In a quantum context, we can attribute the context-dependence of Khrennikov's model to Heisenberg's uncertainty principle. If position and momentum both have extremely concentrated probability distributions, it is necessarily entailed that a disturbance must have occurred, inducing a transition between two different, mutually-exclusive contexts. The crucial question is: Why does Heisenberg uncertainty exist?

In fact, non-commutative properties and Heisenberg-type uncertainty relations seem to be generically derivable from the continuous but inherently non-differentiable nature of stochastic trajectories. In the path integral formulation, commutation relations follow from the non-differentiability of Feynman's paths. Uncertainty relations therefore seem to be natural properties of probability distributions in stochastic systems and they exist in quantum mechanics for the same reasons: e.g.,

https://arxiv.org/abs/1208.0258

Also:

https://www.mdpi.com/1099-4300/24/10/1502/htm

https://www.sciencedirect.com/science/article/pii/S0304414910000256

At the root of these uncertainty relations is time-reversibility in the stochastic dynamics; however, even for irreversible stochastic processes, these conditions can be met in the averaged behavior at statistical equilibrium: e.g.,

https://link.springer.com/article/10.1140/epjh/s13129-023-00052-5

arXiv:quant-ph/0412132

It only then takes Fine's theorem to show a straightforward route from stochasticity-induced non-commutativity to contextuality and Bell violations:

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.48.291

https://iopscience.iop.org/article/10.1088/1367-2630/13/11/113036
 
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Khrennikov ocasionally writes here so I'll invite him: @Vaxjo
 
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