A Locally Realistic Quantum Theory
Quantum Theory is Locally Realistic
(References listed at bottom)
(Apologies for awkward citation-style and quoting)
Introduction
Stochastic mechanics is a mathematical formulation rooted in the work of Edward Nelson (e.g. 1966, 1985; Nelson attributes the basis of the theory to Fenyes, 1952) that can reproduce all predictions of quantum mechanics, deriving the Schrodinger equation from some basic assumptions about a stochastic process. Like Bohmian mechanics, the quantum behavior is realized by particles that are always in definite positions and move along continuous trajectories; however, in stochastic mechanics particles zig-zag about randomly, describable in terms of a diffusion (which effectively replaces the role of the Bohmian pilot-wave). Formally-speaking, stochastic mechanics is agnostic about the cause of the randomness; however, it is commonly thought that in order to make the theory physically intelligible, the random motion should be caused by the particles interacting with some kind of background system that pervades all space (e.g. Nelson, 1985 [section 14]). This would be analogous to the classical example of a pollen particle suspended in a glass of water, being pushed about in a seemingly random manner by the background water molecules. Given this analogy, It may also be worth noting that stochastic mechanics has in recent years been applied more broadly to continuum media and fluids: e.g. Koide & Kodama (2023); de Matos, Koide & Kodama (2020).
In section 2, I go over what are arguably the main formal assumptions and structure that produces quantum behavior in the theory. Sections 3 and 4 focus on a locally realistic interpretation of some of this structure in terms of the aforementioned background system, its plausibility supported by analogy to hydrodynamic pilot-wave toy models. Section 5 focuses more on questions regarding whether the background assumption is justifiable or a fair trade for some of the overt benefits of the theory; notable here are the parallels that can be drawn between the background system and the orthodox quantum field-theoretic vacuum. Section 6 describes how all of the major historical criticisms of the theory have been addressed without fundamentally changing the theory in any profound way; albeit, this points to Levy & Krener's theory of reciprocal stochastic mechanics as perhaps being a more general and complete framework for describing stochastic mechanics. Finally, sections 7 and 8 go deeper into arguing that stochastic mechanics is completely locally realistic. I then propose how it can produce Bell violating correlations completely locally based on some observations about spin weak values, very briefly mentioning how stochastic mechanics would relate to quantum contextuality at the end.
Main Formal Assumptions
From Beyer (2021 [section 2]), other major assumptions for constructing stochastic mechanics might be described as:
a) The form of the diffusion coefficient, inversely proportional to particle mass: D = σ2/2, σ2 = ℏ/m.
b) The diffusion behaves according to Newton's second law, F = ma, with regard to its averaged motion, minimizing the classical action.
c) The diffusion is non-dissipative (frictionless) - i.e. the particle behavior conserves energy on average. Consequently, the diffusion's statistical behavior is time-reversible.
Similar sets of assumptions can be found in works by Yang (2021 [section iiib]) and Kuipers (2023a [1.4]).
Formally, the time-reversible, conservative nature of the diffusion postulated in assumption c) is enforced by what has been identified as an osmotic velocity (e.g. Beyer, 2021 [2, equation (9)]) from its appearance in the study of classical Brownian motion due to "osmotic pressure". The osmotic velocity is "the velocity acquired by a Brownian particle, in equilibrium with respect to an external force, to balance the osmotic force" (quote from Nelson, 1966 [3rd page, equations (25)-(27)]; also see Einstein, 1905 [section 3]; Bohm & Hiley, 1993 [chapter 9.6]). In other words, this velocity can emerge in classical situations where particles are subject to two opposing influences that are in equilibrium with each other:
1) The "osmotic force" which comes from the surrounding background system that is driving the diffusive motion of particles (the "osmotic pressure" in Einstein's words: i.e. a pressure due to the diffusion gradient of a solvent);
2) Another "external force" acting on particles (e.g. gravity as an example in Bohm & Hiley, 1993 [9.6]) that is directly counteracting the influence of the background.
The osmotic velocity is caused by the "external force" and so acts in opposition to the "osmotic pressure". In the quantum context, the osmotic velocity is the root of all strange quantum behaviors; and without it, the equilibrium probability distribution for the particle's behavior would be uniform as expected in a dissipative system. Instead, the non-dissipative equilibrium distribution is given by the Born rule (Bohm & Hiley, 1993 [9.6]).
Prima facie, none of the three stochastic mechanical formal assumptions above explicitly imply that quantum behavior requires more than a locally realistic ontology (i.e. where particles are in definite positions and cannot influence each other instantaneously over distance). For instance, assumption a) requires plugging in the value of the Planck constant, ℏ, but is otherwise unremarkable. Assumption b) is just invoking classical mechanical trajectories based on the variational method. I will talk about interpreting assumption c) in the next section. Anyone familiar with stochastic mechanics would point out that explicit non-locality does actually arise in stochastic mechanics' mathematical description; however, this can be comprehensively explained away and is addressed in the section on historical criticisms later - stochastic mechanics can be formulated without this non-locality.
A Locally Realistic Interpretation of Assumption c)
Assumption c) is the most radical of the assumptions given that classical diffusions, like the pollen particle being pushed about by H2O molecules in a glass of water, are generally dissipative; H2O molecules impart a drag or frictional force on diffusing pollen particles so that energy is not conserved in these interactions. The above description implies that, assuming quantum particles likewise move randomly due to interactions with some background system, assumption c) can be satisfied by just having those interactions conserve energy on average (e.g. Nelson, 1985 [section 14]; Beyer, 2023 [pg. 20-21) The background system would then be what is imparting the aforementioned "external force" that gives rise to the osmotic velocity, returning energy to the particle that would otherwise be lost to the background during diffusion: e.g.
"the ether fluid [(i.e. the background system)] produces a potential field … that imparts [an osmotic momentum / velocity] … to each particle, causing the particles to scatter through the ether constituents and thereby experience a counter-balancing osmotic impulse pressure … This leads to the equilibrium condition" (Derakshani, 2017 [pg. 73])
Albeit, I would say these quotes de-emphasizes a certain chicken-egg ambiguity in the directionality of the scenario regarding whether the osmotic pressure or the osmotic velocity "came first". Regardless, consistent with the overall interpretation, the osmotic velocity in stochastic mechanics always vanishes when the quantum diffusion coefficient mentioned in assumption a) vanishes, indicating that the osmotic velocity arises from the diffusion process itself. Importantly, there are no explicit reasons here for the interactions between particles and the background to be anything other than locally-mediated - they just have to be energetically conservative on average in order to produce assumption c).
It's worth noting a couple of examples of formulations related to stochastic mechanics that allude to this kind of interpretation. That the early "Entropic dynamics" of Caticha (2011) derives quantum mechanics using an assumption of mysterious "extra variables" whose role seems strongly analogous to the background system as just described. These "extra variables" are "… codified into the phase of a wave function [pg. 3] … can influence and in turn can be influenced by the particles … in such a way that there is a conserved ‘energy’ [section 10]"; without this type of coupling, "one obtains a fairly standard diffusion [pg. 3]". Kuipers (2023a [1.5]; 2023b [12]) similarly interprets his stochastic quantum mechanics formalism in terms of the coupling of particle behavior to the behavior of a background system, respectively corresponding to the real and imaginary components of a complex-valued stochastic process. Quantum unitarity is a direct result of the two components being maximally correlated; when the components are uncorrelated, a standard dissipative Brownian motion is produced like the example of pollen in the glass of water.
Toy Models And Experiments Supporting the Plausibility of a Locally Realistic Interpretation for Assumption c)
Possible insights might also be gleaned from hydrodynamic pilot-wave experiments where oil droplets bidirectionally interact with a bath of fluid by bouncing on it, superficially analogous to the hypothetical stochastic mechanical particle-background interaction.The bath is vibrated so as to counteract and attenuate viscous dissipation in its fluid by injecting energy into the bath. The droplet can then start to evince a range of quantum-like behaviors mediated in a way that is describable analogously to the role of the energy of the osmotic velocity - also known as the quantum or Bohmian potential (Bush et al., 2024 [iiiA]). This seems loosely analogous to the role of assumption c) of stochastic mechanics in producing quantum behavior.
These behaviors may appear non-local despite all system interactions involving the fluid bath and droplet clearly being local. This is enabled because the attenuation of dissipation allows the bath to store information about distant parts of the environment and events in the past, which then influence the droplet's behavior via the droplet's interactions with the bath (e.g. Eddi et al., 2011). Without the attenuation of dissipation, the waves that would carry this information would otherwise decay.
It has even been shown in a simulation that correlated behaviors between droplets, due to locally-mediated interactions in a shared bath, can be maintained even after the droplets have been physically separated so that they can no longer communicate information to each other through the bath (Nachbin et. al, 2022); this is quite reminiscent of quantum entanglement. Another interesting example comes from a hydrodynamic pilot-wave analog of quantum interferometers, showing phenomena analogous to an Elitzur-Vaidman bomb tester experiment (Frumkin et al., 2022; 2023).
While gulfs lie between the kind of system implied by stochastic mechanics and the mechanics of these hydrodynamic pilot-wave systems, as toy models they add plausibility to the idea that strange quantum behaviors can emerge entirely due to local interactions related to a non-dissipative background system, in analogy to the dissipation-attenuated fluid baths in these toy models. See the reviews by Bush et. al (2020; 2023; 2024).
Is the Background Assumption Plausible and Justifiable?
i: Is it too Speculative?
Ofcourse, a clear caveat in all of this is that since the background system is an additional assumption over and above the formalism of the theory, it is not immediately clear exactly how or why it may have non-dissipative properties; there has not yet been a comprehensive model for such a kind of system as far as I know. Skepticism is understandable given that the usual classical diffusions are generally dissipative. However, it is not obvious either why a non-dissipative diffusion, where an osmotic velocity arises from a background system that always conserves (or returns) energy on average in its interactions, should be impossible or even implausible. Ultimately, there is no direct physical evidence for the background assumption, although any ontological claims of other quantum interpretations would similarly struggle for empirical support. Nonetheless, radical claims that introduce novel ontologies will always call out for justification
ii: A possible Relation to Quantum Fields: Pros and Cons.
Arguably, one positive is that the background assumption may not seem so radical or strange when we consider that a similar kind of background ontology is already implied in quantum field theory - the vacuum of space is not really empty, but has a zero-point energy. At the very least, we can say that a kind of pervasive background ontology across space with conservative properties is something we would have to entertain anyway, regardless of our interpretation of quantum theory. Neither do we have an underlying explanation for energy conservation in quantum field theory any more than for the hypothetical background of stochastic mechanics. Asserting a background system assumption is then arguably not as radical as, say, the ontology of Everettian Many-Worlds, which has no comparable precedent at all in any well-established physical understanding of the universe.
In stochastic mechanical field theory, the random fluctuations and osmotic velocity that are hypothetically attributable to the background system drive the behavior of the quantum vacuum state. They would also be hypothetically responsible for the conservative properties of quantum fields. At the same time, juxtaposing both the constructs of a stochastic mechanical background and the quantum vacuum state arguably begs for further elucidation about what is actually happening in the "vacuum". It risks accusation of being unparsimonious or clumsy because it looks like we have two distinct kinds of "background" that pervade space.
We would then eventually want it demonstrated that "the effective treatment of the background field [in terms of random noise + osmotic velocity] on one level arises from … [appropriately coarse-graining] … a field theory on a deeper level" (Carosso, 2024 [Discussion]). In other words, we would like a more fundamental description where we can say that what we have so far been calling the background system in stochastic mechanics really just represents influences from the very same energy-filled "vacuum" ontology that quantum field theory, at least using a coarse-grained description, already alludes to. Integrating them could turn a potentially clumsy picture into a very appealingly parsimonious one, ontologically-speaking.
Therefore, even if there is no direct evidence for a stochastic mechanical "background system", quantum theory at least already implies a kind of ontology that could conceivably take on this role; albeit, it is still just gross speculation at this point. However, Hiley & Flack (2018 [section 2.2]) interestingly do highlight a formal parallel between the behavior of the Bohmian (osmotic) potential and issues related to renormalization in quantum field theory, that would be interpretable as related to the background system.
iii: Realism as an Attractive Feature?
But, as long as they do not unveil serious contradictions or conflicts with established scientific facts, such speculations and open questions regarding the nature of a background system may still be a fair trade when choosing a quantum interpretation. Stochastic mechanics derives and gives an intelligible explanation of quantum behavior from non-quantum assumptions related to already well-established and understood subjects (e.g. stochastic processes, classical mechanics), giving a working mathematical formulation that reproduces all predictions. In contrast, alternative interpretations are completely silent on the exact reason that quantum theory behaves the way it does.
As a bonus, the assumptions of stochastic mechanics are prima-facie locally realistic and aligned with the more metaphysically conventional intuitions about reality that inhabit our everyday experiences and most other areas of science (e.g. the moon doesn't disappear when we turn around). Again, even the speculative background hypothesis is arguably not any more metaphysically radical than what is already otherwise implied by quantum field theory. Through its realism regarding particles and other ontologies, stochastic mechanics therefore completely avoids the measurement problem of other interpretations and will also have a more straightforward approach to the classical limit. Such rewards may be more than worth the epistemic risk that comes with a speculative background hypothesis.
Stochastic mechanics is then arguably the least radical of all quantum interpretations in terms of ontology, and this in itself may be a strength for those that share the intuition that we should not change our beliefs (i.e. pre-quantum beliefs) about the world any more radically than is necessary to explain the data. In fact, the idea that one should minimize how far new beliefs diverge from priors (model complexity, which is balanced with accuracy) is an inherent feature of Bayesian inference, arguably our best formal theory of rational belief updating (e.g. See Wikipedia: Evidence Lower Bound, Variational Bayesian Methods - explained at end of references below). I like to think of this kind of thing as behind my own intuition that it would seem conspiratorially strange if the underlying metaphysics of the universe were to manifest as some inherently unintuitive or even inscrutable ontology while there existed models that could explain all of the same data in the kind of locally realistic fashion that is intuitive to us, intuitive precisely because local realism is how the world generally presents itself in our everyday experience and everywhere else in science.
However, one must concede that for those who do not find the prospect of a more explanatorily intelligible quantum theory with a more conventional metaphysics appealing, then speculative assumptions and open questions about unspecified background influences, as well as their non-dissipative properties and relation to the quantum vacuum state, may seem like a large cost without sufficient motivation. On the other hand, the agnosticism and vagueness of various other interpretations still elicits open questions about the universe that are not any less difficult or frustrating than the open questions regarding a background system. Those with strong tendencies toward very epistemically conservative forms of scientific anti-realism or instrumentalism may not be bothered at all by, or even recognize, the open questions put to these alternative interpretations (e.g. Copenhagen interpretation, qbism, etc). But, presumably, instrumentalism in quantum theory is largely a consequence of the perception that an explanatorily coherent metaphysical position is unavailable.
iv: A Compromise; Stochastic Mechanics without a Background
At the same time, one may also still be able to defend stochastic mechanics while being completely agnostic about how the non-dissipative and time-reversible stochastic behavior arises. It should be emphasized that the background assumption is not central to the formalism and the three major assumptions given earlier that makes stochastic mechanics work as a model. The background assumption just helps provide a deeper explanation for how stochastic mechanics could work physically, particularly assumption c); but regardless of the background assumption, the theory will still offer the same benefits of a realistic attitude to metaphysics and consequently straightforward resolution of the measurement problem. It will also still allow an intelligible explanation of quantum behavior in terms of non-dissipative, time-reversible stochastic behavior which arguably amounts to a stochastic generalization of classical Lagrangian mechanics (e.g. Goldstein, 1987; Carlen, 1984 [intro]; de Matos, Koide & Kodama, 2020); it may just lack an explanation for the exact origin of that behavior.
Addressing Historical Criticisms
It is important to also note that, historically, stochastic mechanics has had various technical criticisms; however, these all seem to have been addressed without fundamentally changing the theory. Firstly the "Wallstrom problem" (Wallstrom, 1989; 1994), which is the issue that a seemingly ad-hoc quantization condition is required to derive the Schrodinger equation, has been resolved with the finding that the desired condition actually exists within a part of the theory that is usually thrown away because it doesn't directly contribute to the equations of motion (Kuipers, 2023a [1.4, 3.8]).
A second criticism is that stochastic mechanics gives incorrect multi-time correlations. However, this is resolved when measurement is explicitly accounted for (Blanchard et al., 1986; Petroni & Morato, 2000; Derakshani & Bacciagaluppi, 2022), similarly to Bohmian mechanics (e.g. Neumaier, 2000; Marchildon, 2000; Gisin, 2018 [and references within]; Oriols & Mompart, 2019 [section vii]). In the recent formulation of stochastic mechanics by Kuipers (2023a [1.4]; 2023b [footnote pg. 6]), the issue does not even arise due to the incorporation of an imaginary position variable into the stochastic process. It is likely that this whole issue can be understood via the known correspondence of various properties in both stochastic and Bohmian mechanics to weak values in orthodox quantum theory. The real parts of weak values carry information about the unmeasured quantum state while the imaginary components have been associated with measurement back-action (e.g. Pandey et al., 2021; Leavens, 2005; Wiseman, 2007; Hiley, 2012; Lundeen et al., e.g. 2014; Dressel, 2015). Mauro & Gozzi (2002 [section 3]) give a nice description of how the correct multi-time behavior in quantum theory depends on measurement-disturbance. Therefore, the "incorrect" correlations arguably exist within orthodox quantum theory, they are just not typically of interest because they cannot be accessed by more conventional measurements that disturb the system.
A minor criticism, more concerned with pragmatics, is that even though stochastic mechanics is derived from assumptions largely independent of quantum mechanics, it typically requires prior knowledge of the wave-function in order to be used practically. Recently, the use of stochastic optimal control techniques has been used to obviate this need so that the complete spectrum of bound states can be determined (e.g. Koppe et al., 2017, 2018, 2020; Beyer et al., 2019; also applied in Beyer et al., 2021, 2023 [and other notable applicable Beyer publications listed pg. 167 of 2023]). Another perhaps also mainly pragmatic criticism is that the "modified Hamilton-jacobi equations", that formulations such as Bohmian and Stochastic mechanics are based on, are not applicable for non-differentiable fractal wave-functions due to wave-function discontinuities (Hall, 2004). However, this problem can in principle be avoided given that such wavefunctions are linear superpositions of eigenvectors that are themselves differentiable, and so can be used to produce the correct behavior. This has been shown for Bohmian mechanics (Sanz, 2005).
The final major criticism is that despite the assumptions mentioned earlier being prima facie locally realistic, a specific kind of non-locality explicitly arises in the regular stochastic mechanical mathematical description, allowing particles to seemingly instantaneously influence the behavior of others particles far away through the osmotic velocity. I will call this osmotic non-locality to distinguish it from other kinds (e.g. Bell non-locality). However, a resolution can be inferred using a distinct formulation of stochastic mechanics constructed using reciprocal (time-symmetric) stochastic processes (Levy & Krener, 1996). Interestingly, (Markovian) reciprocal stochastic processes were first constructed by Schrodinger himself, finding that their behavior could be described by an analogue of the Born rule in the context of a large deviations problem that can be solved by minimizing the relative entropy (Schrodinger, 1931; see Chetrite et al., 2021; Huang & Zambrini, 2023; Pavon, 2002 [intro]). In their most generic definition, reciprocal processes are stochastic processes with initial and final boundary conditions in time, e.g. 0, 1. Additionally, for any pair of times s ≤ u within [0, 1], events within s ≤ u are conditionally independent of those outside of s ≤ u (e.g. Leonard, Roelly & Zambrini, 2013).
From the analysis of Levy & Krener, it can be noted that the Markovian subclass of reciprocal processes is equivalent to the stochastic processes used to construct regular stochastic mechanics. Unfortunately, they cannot reproduce the Schrodinger evolution without an ad-hoc correction very closely related to the non-local Bohmian potential; this is where the osmotic non-locality comes from. However, a non-Markovian subclass of reciprocal processes also exists that is directly equivalent to the Schrodinger evolution, and so no correction is required. As a result, the stochastic mechanics formulated using this subclass does not have the osmotic non-locality that is present in regular stochastic mechanics, even when particles are statistically coupled and their diffusions non-separable.
This analysis suggests that there is more than one way to construct assumption c) from earlier; quantum theory is actually non-Markovian but it can be equivalently constructed, albeit artificially and at the cost of osmotic locality, using Markovian stochastic processes. We can see this through the fact that the Bohmian potential is just the energy of the osmotic velocity; however, the osmotic velocity only appears in regular stochastic mechanics when the Markovian forward and backward processes used to construct a non-dissipative diffusion do not coincide (e.g. Beyer, 2023 [pg. 22]). That they do not coincide is a natural consequence of Markovian irreversibility; it follows that if these processes were naturally time-reversible to begin with, the osmotic velocity and Bohmian potential would never have to be invoked explicitly. Levy & Krener's non-Markovian reciprocal stochastic mechanics then arguably gives a more veridical presentation of stochastic mechanics compared to regular Nelsonian stochastic mechanics which might be seen as constructed using an artificial idealization, in the form of a Markovian assumption, to get the same behavior. However, reciprocal stochastic mechanics is very understudied in comparison to the Nelsonian version.
Arguing Plausible Absence of Non-locality in Stochastic Mechanics
With the absence of both wave-function collapse and a Bohmian potential, reciprocal stochastic mechanics lacks arguably the two biggest culprits responsible for overt non-locality in quantum interpretations. The stochastic mechanical assumption c) from earlier suggests that entanglement can be explained without overt non-locality in terms of how a non-dissipative diffusion preserves non-separable, correlated behaviors due to an initial local interaction. Specifically, the preserved correlations can be explained as direct consequences of extremizing relative entropy or Fisher information with regard to the diffusion (e.g. Yang: 2021, 2024; Reginatto,1998; Frieden et al., 1989, 2002); additional forms of non-local causation then seem explanatorily redundant. Again, hydrodynamic pilot-wave toy models also hint at how stochastic mechanical particle behavior may be sensitive to the global environmental context (e.g. including opened / closed slits in distant screens) in a way that seems non-local but is mediated by information stored in the background system; however, this kind of explanation is still speculative given that it is coming from a toy model and not the stochastic mechanical formalism. But importantly, if a stochastic mechanical quantum system isn't initially in the correct quantum equilibrium distribution implied by some global context, it will always take a finite time to relax into the correct one given by the Born rule (Hardel, Hervieux & Manfredi, 2023; Carosso, 2024).
Arguably, the ultimate barrier to a truly locally realistic interpretation is the notorious Bell violations; but what do Bell violations actually mean? It has been demonstrated in a number of results, often referred to collectively as Fine's theorem that Bell locality precisely corresponds to the presence of a joint probability distribution (or more generally global sections in the sheaf-theoretic picture by Abramsky & Brandenburger): e.g. Fine (1982a; 1982b), Abramsky & Brandenburger (2011), Pitowsky (1989), Cabello (2023 [ivA1]). This suggests that Bell inequalities are violated primarily because of the absence of a global joint probability distribution, a generic feature in quantum theory also responsible for violations of other related inequalities such as the Leggett-Garg among various others (e.g Markiewicz et. al, 2014; Das et. al, 2014).
Bell violations then do not actually explicitly prove non-local causation; what is ruled out is simply a model where the measurement statistics are context-independent. The issue is that when it comes to the paradigmatic example of two spatially separated spin measurements, where the respective measurement settings (i.e. contexts) can be manipulated, the statistics at one spin measurement will be contextually dependent on the other distant measurement. It is then quite difficult to envision how this could be explained in any other way than non-local causation even though non-local causation is not what is explicitly being demonstrated by Bell violations. However, as described next, there is reason to believe that stochastic mechanics can provide a local explanation for Bell violations.
A Local Explanation of Bell Violations
i. A Local Description of Entanglement by Wharton et. al (2024) and Sutherland (2022)
The main insight to a plausible local explanation comes from weak values. We can envision a situation where a particle is initially prepared with some spin direction and then subject to a final measurement, final settings aligned at some arbitrary orientation. Weak values are concerned with what is happening in the particle's unmeasured intermediate trajectory given some final measurement outcome. They are most often associated with experiments that measure quantum systems under conditions of minimal disturbance (i.e. weakly); however, they are intrinsic to the orthodox quantum formalism describing the quantum state. They can be seen as coming directly from the Kirkwood-Dirac complex probability distribution (e.g. Arvidsson-Shukur et al., 2024 [e.g. section 4]), one of the various quasi-distributions one can use to give a phase-space formulation of quantum mechanics. Another novel formulation of quantum mechanics has even been proposed based on weak values and their relation to the Kirkwood-Dirac distribution (Hofmann, 2014).
Analyses in papers by Wharton et al. (2024) and Sutherland (2022) suggest a local picture of entanglement when viewed from the perspective of these weak values. Given the post-selected final outcomes, pairs of particles do not seem to influence each other's intermediate weak values and their behaviors at final measurement are also completely independent; measurement timing doesn't matter either. The final outcomes and their measurement orientations retroactively shape the intermediate weak values, which Wharton et al. (2024) and Sutherland (2022) interpret as retrocausality. Ordinarily, this kind of statistical influence need not be interpreted causally; however, the authors' interpretation comes from a desire to explain entanglement locally, which is impeded by the reasonable assumption that the measurement orientation is an independent degree of freedom that actively determines how the state of an individual particle changes upon measurement. Wharton et al. (2024) and Sutherland (2022) then suggest that future measurement settings retro-causally influence each respective particle all the way back to their shared initial preparation. Because the particles are locally connected at that point, information about future measurement settings can be shared by the particles at initial preparation without non-local communication. However, when viewed from a stochastic mechanical perspective, it will be seen that there is less reason to think retrocausality must be occurring. First I will describe exactly how spin weak values correspond to stochastic mechanical quantities.
ii. Stochastic Mechanical Velocities are Weak Values
There is a direct correspondence of the stochastic mechanical current and osmotic velocities to the real and imaginary momentum weak values, respectively (e.g. Hiley, 2012; also see Berry, 2013). Stochastic mechanical spin also has analogous current and osmotic components (e.g. Beyer, 2023 [section 5]) which would then correspond to analogous real and imaginary spin weak values. For instance, this can be seen when comparing Carrara (2019 [e.g. equation (2.6), osmotic (spin) velocity in (2.15)]; also Carrara & Caticha, 2020) to Hiley & Van Reeth (2018 [equation 9]; also Hiley, 2012).
Importantly, neither the stochastic mechanical velocities nor spin can be properties of any individual particle. Current velocities describe the average behavior of trajectories through space and can be defined using expectations of trajectories from counterfactual ensembles referring to repetitions of an experimental scenario. This is explicitly depicted in Koide (2020 [section 3.2; figure 2]); direct link to a google search image from the paper here:
https://images.app.goo.gl/cC2oj
In contrast, osmotic velocities effectively describe the tendency of particles to climb the probability gradient for position. Figures 2 and 3 of Bliokh et al. (2013) depict flow lines from measurements of both the real (current) and imaginary (osmotic) parts of the momentum weak value so as to compare their behaviors:
https://images.app.goo.gl/mnpYK (light beam with orbital angular momentum)
https://images.app.goo.gl/uGDiZ (double slit experiment)
Hiley & Flack (2018) also briefly discuss these images from Bliokh et al. (2013). Bliokh et al. (2013) actually have the osmotic velocity moving in the opposite direction to convention; nonetheless, they ascribe the correct interpretation to what is happening as described at the end of the text for figure 3; Also see Beyer (2023 [figure 2.6]) for a comparable image with the conventional osmotic direction.
Spin then refers only to the statistics of particle behavior - visualizable through the above images - described by (spin) velocity fields and only realizable in terms of entire (counterfactual) ensembles of particle trajectories. The expectations of the quantum mechanical momentum and spin operators directly correspond to expectations of current and osmotic velocities (the latter disappearing under expectation): i.e. expectations of stochastic mechanical averages as opposed to properties of particles, described in Koide (2020 [section 4.3]). This would be an expectation of weak values when expressed in the orthodox quantum formalism: e.g. (Hiley, 2012 [6th equation]; Hosoya & Shikano, 2010 [equations (3)]).
What exactly is stochastic mechanical spin though? An older view is that it literally refers to the spinning of a particle's spherical extended body, as utilized by Beyer (2023 [attributable to Dankel, 1970]); however, this view implies superluminal surface velocities and so should be impossible. An alternative view (e.g. Carrara, 2019; Yang, 2025) is that spin is actually a property explicitly describing aspects of a particle's motion through space (i.e. in its trajectories). This view avoids the aforementioned superluminal issue and has actually been experimentally verified in the case of other classical field media - e.g. sound (Shi et al., 2019) and water (Bliokh et al., 2022) - where it is associated with elliptical orbiting motions in space. Also see Mita (2000 [e.g. figure 2]) to see the analogous underlying structure in the quantum state.
iii. Stochastic Mechanical Velocities Suggest A Locally Realistic Entanglement without Retrocausality
The most important observation in Sutherland (2022) and Wharton et al. (2024) responsible for their local picture of entanglement is that intermediate weak values (both real and imaginary parts) are constant and do not change between initial preparation and final measurement. Apart from the general trigonometric rules that relate different spin components, the weak values depend only on the initially preparation and final measurement spin orientations that each respectively fix specific spin components of the intermediate weak values:
"For each possible result, the smallest vector that conforms to both of these constraints, without changing between measurements, is precisely Re(w±)" (Wharton et. al).
Such a picture of constant spin weak values is similarly implied by Berry (2011 [section 2]).
I will then set out a picture. The calculations presented in Sutherland (2022) imply that when the spin orientation is the same at both final measurement and initial preparation, the constant weak vector is just the initially prepared spin since the final measurement doesn't meaningfully invoke any additional constraints. Ofcourse, all particles also end up at the same final measurement outcome in this situation, and also the same intermediate counterfactual ensemble of trajectories. If you then gradually rotate the final measurement orientation (changing the constraints on the weak value), more and more particles will end up in the other final measurement outcome just because the Born rule probabilities require this. Again, in the orthodox picture spin is a property of an individual particle which changes at the point of measurement; this is difficult to reconcile with locally realism. But in stochastic mechanics, spin clearly cannot be identified with any individual particle; it is only meaningful statistically on the level of whole counterfactual ensembles of particle trajectories associated with each final measurement outcome, as described by current and osmotic spin velocity fields.
Again, to emphasize, it would just be a necessary fact of spin experiments under the stochastic mechanical formulation that different measurement orientations partition a counterfactual ensemble of trajectories into sub-ensembles whose relative size depends on the Born rule probabilities. This then leads to the possibility that simply partitioning an ensemble with the initially prepared spin statistics in different ways is what leads to the different spin statistics of the resulting subsets / sub-ensembles. In the absence of argument to the contrary, this would be the most parsimonious stochastic mechanical description of what is happening. The constant statistics of these trajectory sub-ensembles would then be constrained to be simultaneously consistent with both the initial preparation and final measurement outcomes just because of how the final measurement outcomes end up partitioning in different ways the overall ensemble implied by the initial spin statistics. This picture is no different from conventional statistical post-selection. For instance, one may examine the statistics for grades of a high school class then post-select subsets based on future outcomes like college grades, income brackets, health, etc, etc., examining the statistics of the novel subsets.
It is then interesting to note in Wharton's qubit example how:
"the weighted average of w+ and w− (using their Born-rule probabilities) is exactly (0, 0, 1), with no imaginary part surviving. This average matches ˆi [the initial spin]".
This description seems to be related to the fact that quantum mechanical spin expectations can be described as expectations of weak values, as already mentioned (e.g. Hoyosa & Shikano, 2010 [equation (3)]). Wharton et al. (2024) also earlier mentions this:
"Nevertheless, in the usual situation where the result of the final measurement is unknown and a weighted average is taken over the possible outcomes, the weak value Re(W[A])(t) can then be shown to be exactly equal to the usual expectation value ⟨A⟩(t)."
If we look at this from a stochastic mechanical perspective where we are talking explicitly about particle statistics, the initially prepared spin in the Wharton et al. (2024) qubit example then seems to be related to the statistics of its post-selected sub-ensembles by just a very conventional expectation. The initially prepared spin is combining the statistics related to both post-selected sub-ensembles, weighted by the relative size of those associated sub-ensembles and regardless of measurement orientation. Different post-selected partitions will therefore also have equivalent statistics under expectation. This seems exactly what you would expect if the final spin outcomes just came from just partitioning an ensemble of intermediate trajectories into different subsets - like any other kind of post-selection in statistics. There doesn't seem to be any overt reason to invoke retro-causality when looked at in this sense.
iv. Reproducing the Bell State Correlations
This plausibly suggests a completely local explanation of entanglement scenarios from the stochastic mechanical perspective. Because every individual particle leaves the initial preparation paired with another particle, this implies paired sub-ensembles that will both respectively maintain constant spin directions between preparation and their respective final measurements Af and Bf. Bell correlations can then be explained by just locally fixing a perfect correlation between any and all paired sub-ensemble spin directions at initial preparation.
The correlation can only actually be physically, methodologically imposed on each individual pair of particles that go through the experimental setup in any given experimental repetition. The correlation imposed on each pair would then naturally apply to every (sub-)ensemble due to the fact that any arbitrary (sub-)ensemble is constructed from groups of these individually-correlated pairs that have been produced by experimental repetition.
For photons and spin ½, the Bell states give the following perfect correlations:
For Φ+: spin orientations θa = θb at initial preparation.
For Ψ-: θa = θb ± 90° and vice versa (±180° for spin ½).
For Φ-: θa = -θb and vice versa, using 0° as a reference orientation, e.g. cos2(350 + 10) = cos2(-10 + 10) = 1; cos2(315 + 45) = cos2(-45 + 45) = 1; etc. etc.
For Ψ+: θa = -θb ± 90° and vice versa, using 0° as a reference orientation (±180° for spin ½). Equivalently, it is θa = -θb but with 45° (90° for spin ½) as the reference orientation.
The last two correlations for Φ- and Ψ+ might plausibly be conceptualized as like a mirror-image flipping of θb and θb ± 90° from the first two correlations to -θb and -θb ± 90°.
Conditioned on a sub-ensemble at a final measurement Af, we will always know the spin direction of its paired sub-ensemble that was produced at initial preparation; but, now we want to know what happens to that paired sub-ensemble when it is measured at Bf. Given any initial Bell state and the spin direction at Af, you can plug in the entailed spin direction at Bf, denoted θb, and an arbitrary measurement orientation, bf, into Malus' Law (which gives the probabilities for a final spin measurement) and find that cos2(bf - θb) will always correspond to the correct correlation for whatever Bell state was assumed. It works the same whether starting from (conditioning on) Af or Bf.
Because the sub-ensemble at Af must always be aligned with its final measurement orientation, we have:
For Φ+: θb = af ; therefore, cos2(bf - θb) = cos2(bf - af).
For Ψ-: θb = af + 90° (af + 180° for spin ½); therefore, cos2(bf - θb) = cos2(bf - af + 90) = sin2(bf - af).
For Φ-: θb = -af ; therefore, cos2(bf - θb) = cos2(bf - (-af)) = cos2(bf + af). Equivalently, cos2(θb - bf) = cos2(-af + -bf) = cos2(af + bf).
For Ψ+: θb = -af + 90° (-af + 180° for spin ½); therefore, cos2(bf - θb) = cos2(bf - (-af) + 90) = sin2(bf + af)
It must be emphasized that these equivalences are independent of interpretation.
The additional marginal probability of ½ for each Bell state then just comes from assuming that initial preparation fairly produces states (i.e. ensembles) with different spin directions or orientation in a way that the expected orientation is zero (e.g. Beyer, 2024 [section 3.1.1.]). For instance, as produced by the spontaneous parametric downconversion procedure described in Dehlinger & Mitchell (2002 [section V]) with the initial preparation resulting in HH and VV states in equal amounts prior to final measurement.
We can now express the four Bell state correlations properly:
Φ+ = ½ cos2(bf - af)
Ψ- = ½ sin2(bf - af)
Φ- = ½ cos2(bf + af)
Ψ+ = ½ sin2(bf + af)
v. Anomalous Weak Values and Contextuality
A notable issue with weak values is that they can be difficult to understand, taking on strange, "anomalous" values and resulting in seemingly paradoxical quantum scenarios (e.g. Hoyosa & Shikano, 2010). On one hand, one might point out that difficulties in interpreting weak values may go hand-in-hand with interpretational ambiguity; and ultimately, there is a direct formal correspondence between quantum weak values and various important quantities in stochastic mechanics which have a relatively clear ontological interpretation (e.g. Hiley, 2012). It can also be noted that we now know that anomalous weak values and quantum paradoxes can be traced directly to the occurrence of negative probabilities and current densities (e.g. Pusey, 2014; Hofmann, 2015; Ji, Hofmann & Hance, 2024) which are indicators of quantum contextuality (e.g. Spekkens, 2008; Abramsky & Brandenburger, 2011; de Barros et al., 2014a, 2016, 2020; Camillo & Cervantes, 2024).
It is now apparent that contextuality can be described in a range of areas outside of quantum theory (e.g. Pothos & Busemeyer, 2022; Abramaky et al., 2015; Abramsky & Caru, 2019) and so should not be construed as a mysterious phenomena unique to the quantum realm. Contextuality just generically refers to context-dependence regarding joint probability distributions; for instance, as described by the contextuality-by-default theory (e.g. Dzhafarov, 2021; also see Khrennikov, 2024). From de Barros, Oas & Suppes (2014b):
"Instead of using negative probabilities, it is possible to simply extend the probability space such that when we talk about correlations between experimentally observable variables, as proposed by Dzhafarov and Kujala (2012, 2013a). To understand this point, imagine we start with three variables X, Y, and Z, as in the above example. Instead of thinking of them as three variables, we could think of them as six, one for each experimental context: XY, XZ, ..., ZY. It is easy to show that in some important physical examples, such as the famous Bell-EPR setup, such extension of the probability space is sufficient to grant the existence of a joint probability distribution, but at the cost of having XY ≠ XZ. Thus, the apparent inconsistencies mentioned in the previous paragraph could be argued to come from an identity assumption for the
random variables: that a random variable remains the same in different contexts
(Dzhafarov and Kujala, 2013b,a, 2014d,c)."
Quantum paradoxes then occur due to the juxtaposition of (orthogonal) contexts that could not possibly co-occur simultaneously (e.g. Sokolovski, 2018; Duprey & Matzkin, 2017 [section ivB]; also see Matzkin, 2020). This includes bizarre scenarios such as those under the umbrella of "Wigner's Friend" which, at their core, are based upon contextual statistics of joint probability distributions (e.g. Schmid, Ying & Leifer, 2023; Yang, 2022; also see Elouard, 2021). The stochastic mechanical interpretation of Wigner's Friend would then be that particles and observers are always in definite configurations at any given time, but there exists different measurement contexts which have mutually exclusive statistics. However, stochastic mechanics strongly implies that Wigner's Friend scenarios cannot occur for macroscopic observers because the stochastic mechanical diffusion coefficient vanishes for systems with large mass, resulting in classical-looking, non-contextual behavior when coarse-graining over microscopic degrees of freedom. This explanation can be seen in the orthodox formalism: e.g. Hofmann (2012 [e.g. section 4]).
Negative joint probabilities also overtly come up in the non-Markovian class of diffusions in reciprocal stochastic mechanics (Levy & Krener, 1996 [section viii]). It's tempting to suggest that the transparent occurence of these negative joint probabilities is due to the fact that reciprocal processes are formulated with both initial and final boundary conditions, analogous to the pre- and post-selection that explicitly can cause the contextual behavior of anomalous weak values. We should also be able to think of these negative joint probabilities as due to the context-dependence of statistics. Importantly, stochastic mechanics would provide a mechanism for this quantum contextuality in terms of its non-dissipative diffusion and background system, which would be directly responsible for things like the quantum uncertainty relations related to measurement-disturbance.
9. Conclusion:
Despite its relative obscurity, stochastic mechanics provides a promising way of interpreting quantum theory based on a complete mathematical formulation that reproduces all of the correct predictions. It also completely dissolves the measurement problem in a way that upholds conventional, common sense intuitions about reality: the moon does not disappear when we turn away, and there is no spooky action at a distance. The major cost of this interpretation is the invocation of a kind of background system in order to provide a more complete explanation; however, a similar kind of ontology is already alluded to in orthodox quantum theory, providing a conceivable home for this background influence. Alternatively, one could still be agnostic regarding this issue and still embrace the various positive aspects of the theory that do not explicitly require a background assumption on the formal level. Perhaps most importantly, all of the major historical criticisms have been resolved without requiring any fundamental changes to the theory.
References:
(PDF downloads in links):
Abramsky & Brandenburger, 2011:
https://scholar.google.co.uk/scholar?cluster=12086196826892314859&hl=en&as_sdt=0,5&as_vis=1
Abramsky & Caru, 2019:
https://arxiv.org/abs/1911.03521
Abramsky et al., 2015
https://arxiv.org/abs/1502.03097
Arvidsson-Shukur et al., 2024:
https://scholar.google.co.uk/scholar?cluster=2611832075931433402&hl=en&as_sdt=0,5&as_vis=1
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Berry, 2013:
https://scholar.google.co.uk/scholar?cluster=13071807045955206858&hl=en&as_sdt=0,5&as_vis=1
Beyer et al., 2019:
https://scholar.google.com/scholar?cluster=15036464767822647314&hl=en&as_sdt=0,5
Beyer, 2021:
https://www.mdpi.com/2218-1997/7/6/166
Beyer, 2023: https://scholar.google.co.uk/scholar?cluster=16239473886028239443&hl=en&as_sdt=0,5&as_vis=1 (third link, thesis manuscript, notable publications listed pg. 167)
Beyer, 2024:
https://scholar.google.co.uk/scholar?cluster=15973777865898642687&hl=en&as_sdt=0,5&as_vis=1
Blanchard et al., 1986: https://scholar.google.co.uk/scholar?cluster=777481222088581342&hl=en&as_sdt=0,5&as_vis=1
Bliokh et al., 2013
https://scholar.google.co.uk/scholar?cluster=2351034104038667923&hl=en&as_sdt=0,5&as_vis=1
Bliokh et al., 2022:
https://scholar.google.co.uk/scholar?cluster=17892291218685033829&hl=en&as_sdt=0,5&as_vis=1
Bohm & Hiley, 1993:
https://pierre.ag.gerard.web.ulb.be/textbooks/textbooks.html
Bush et al., 2020: https://scholar.google.co.uk/scholar?cluster=14933104262617517877&hl=en&as_sdt=0,5&as_vis=1
Bush et al., 2023: https://scholar.google.co.uk/scholar?cluster=6934967015559529279&hl=en&as_sdt=0,5&as_vis=1
Bush et al., 2024: https://scholar.google.co.uk/scholar?cluster=9134117041907264858&hl=en&as_sdt=0,5&as_vis=1
Cabello et al., 2022:
https://scholar.google.co.uk/scholar?cluster=791610487956722318&hl=en&as_sdt=0,5&as_vis=1
Camillo & Cervantes, 2024;
https://scholar.google.co.uk/scholar?cluster=1335702375506190562&hl=en&as_sdt=0,5&as_vis=1
Carlen, 1984: https://scholar.google.co.uk/scholar?cluster=10150740165970720615&hl=en&as_sdt=0,5&as_vis=1
Carosso, 2024: https://scholar.google.co.uk/scholar?cluster=14260860180761160032&hl=en&as_sdt=0,5&as_vis=1
It's also worth noting that a stochastic mechanical extension to fermionic fields has been presented by Yang (2025)): https://arxiv.org/abs/2503.02362
Carrara, 2019:
https://scholar.google.co.uk/scholar?cluster=4911527372220825232&hl=en&as_sdt=0,5&as_vis=1
Caticha & Carrara, 2020:
https://scholar.google.co.uk/scholar?cluster=13001253732217295191&hl=en&as_sdt=0,5&as_vis=1
Caticha, 2011: https://scholar.google.co.uk/scholar?cluster=10744847120076955847&hl=en&as_sdt=0,5&as_vis=1
Chetrite et al., 2021:
https://scholar.google.co.uk/scholar?cluster=503998788243586525&hl=en&as_sdt=0,5&as_vis=1 (Inside is additionally an English translation of Schrodinger, 1931)
Dankel, 1970:
https://link.springer.com/article/10.1007/BF00281477
Also:
Faris, 2005
https://link.springer.com/chapter/10.1007/3-540-11956-6_117
Wallstrom,1990:
https://scholar.google.com/scholar?cluster=14983865305492418170&hl=en&as_sdt=0,5
Das et al., 2014:
https://scholar.google.co.uk/scholar?cluster=9249630426781623355&hl=en&as_sdt=0,5&as_vis=1
Duprey & Matzkin, 2017:
https://arxiv.org/abs/1611.02780
Dzhafarov, 2021:
https://arxiv.org/abs/2103.07954
de Barros, Kujala & Oas, 2016:
https://scholar.google.co.uk/scholar?oi=bibs&hl=en&cluster=5889627159902768666
de Barros et al., 2014a:
de Barros, Oas & Suppes, 2014b:
https://arxiv.org/abs/1412.4888
de Barros, Kujala & Oas, 2016:
https://arxiv.org/abs/1511.02823
de Barros & Holik, 2020:
https://scholar.google.com/scholar?cluster=3743632890803712733&hl=en&as_sdt=0,5
Dehlinger & Mitchell, 2002:
https://scholar.google.co.uk/scholar?cluster=4106092653304822205&hl=en&as_sdt=0,5&as_vis=1
de Matos, Koide & Kodama, 2020: https://scholar.google.co.uk/scholar?cluster=1699168009039765853&hl=en&as_sdt=0,5&as_vis=1
Derakshani, 2017: https://arxiv.org/abs/1804.01394 (Thesis; published papers listed on pg. i-ii)
Derakshani & Bacciagaluppi, 2024: https://scholar.google.co.uk/scholar?cluster=16234615944899497335&hl=en&as_sdt=0,5&as_vis=1
Dressel, 2015:
https://scholar.google.co.uk/scholar?cluster=13612301325652718318&hl=en&as_sdt=0,5&as_vis=1
Eddi et al., 2011: https://scholar.google.co.uk/scholar?cluster=15913231279030590651&hl=en&as_sdt=0,5&as_vis=1
Einstein, 1905: https://scholar.google.co.uk/scholar?cluster=6462104001853464417&hl=en&as_sdt=0,5&as_vis=1
Elouard et. al, 2021:
https://scholar.google.com/scholar?cluster=201453720472237082&hl=en&as_sdt=0,5
Fine, 1982a:
https://scholar.google.co.uk/scholar?cluster=2543155278787880428&hl=en&as_sdt=0,5&as_vis=1
Fine, 1982b:
https://scholar.google.co.uk/scholar?cluster=8813695518940155915&hl=en&as_sdt=0,5&as_vis=1
Re: descriptions of Pitowsky & Fine's Theorem, one can also see with PDFs:
https://scholar.google.co.uk/scholar?cluster=3211140661517861045&hl=en&as_sdt=0,5&as_vis=1 (Kunjwal & Spekkens, 2015 [sections II, III])
https://scholar.google.co.uk/scholar?cluster=15428606165482907135&hl=en&as_sdt=0,5&as_vis=1 (Muller, 2001 [section 3])
Fenyes, 1952;
https://scholar.google.co.uk/scholar?cluster=7396251233018737422&hl=en&as_sdt=0,5&as_vis=1
Frieden, 1989:
https://scholar.google.co.uk/scholar?cluster=6726610757785075244&hl=en&as_sdt=0,5&as_vis=1
Frieden et al., 2002:
https://scholar.google.co.uk/scholar?cluster=16065972605565234781&hl=en&as_sdt=0,5&as_vis=1
Frumkin et al., 2022: https://scholar.google.co.uk/scholar?cluster=16295625758829094935&hl=en&as_sdt=0,5&as_vis=1
Frumkin et al., 2023: https://scholar.google.co.uk/scholar?cluster=7527318992667606476&hl=en&as_sdt=0,5&as_vis=1
Gisin, 2018:
https://scholar.google.co.uk/scholar?cluster=11957102981612580756&hl=en&as_sdt=0,5&as_vis=1
Goldstein, 1987:
https://scholar.google.co.uk/scholar?cluster=1903829374888411128&hl=en&as_sdt=0,5&as_vis=1
Hardel, Hervieux & Manfredi, 2023:
https://scholar.google.co.uk/scholar?cluster=7428854841934893720&hl=en&as_sdt=0,5&as_vis=1
Hall, 2004:
https://scholar.google.com/scholar?cluster=11048147743036685160&hl=en&as_sdt=0,5
Hiley, 2012:
https://scholar.google.co.uk/scholar?cluster=14552000466857545065&hl=en&as_sdt=0,5&as_vis=1
Hiley & Flack, 2018:
https://scholar.google.co.uk/scholar?cluster=18314655600428072956&hl=en&as_sdt=0,5&as_vis=1
Hiley & Van Reeth, 2018:
https://scholar.google.co.uk/scholar?cluster=15019391565502952501&hl=en&as_sdt=0,5&as_vis=1
Hofmann, 2012:
https://scholar.google.com/scholar?cluster=4191142676621310746&hl=en&as_sdt=0,5
Hofmann, 2014:
https://scholar.google.co.uk/scholar?oi=bibs&hl=en&cluster=6763592988199980049
Hofmann, 2015:
https://scholar.google.co.uk/scholar?cluster=10823798101126350454&hl=en&as_sdt=0,5&as_vis=1
Hosoya & Shikano, 2010:
https://scholar.google.co.uk/scholar?cluster=15985774068178401829&hl=en&as_sdt=0,5&as_vis=1
Huang & Zambrini, 2023:
https://scholar.google.co.uk/scholar?cluster=12392391838247803028&hl=en&as_sdt=0,5&as_vis=1
Ji, Hofmann & Hance, 2024:
https://scholar.google.co.uk/scholar?cluster=4089054583237093131&hl=en&as_sdt=0,5&as_vis=1
Koppe, 2017:
https://scholar.google.co.uk/scholar?cluster=663086951774709679&hl=en&as_sdt=0,5&as_vis=1 (PDF links to dissertation).
Koppe et al., 2018:
https://scholar.google.co.uk/scholar?cluster=6227316334896325474&hl=en&as_sdt=0,5&as_vis=1
Koppe, 2020:
https://www.worldscientific.com/doi/abs/10.1142/9789811209796_0005
Koide & Kodama, 2023: https://scholar.google.co.uk/scholar?cluster=3213082132606119978&hl=en&as_sdt=0,5&as_vis=1
Kuipers, 2023a: https://arxiv.org/abs/2301.05467
Kuipers, 2023b: https://arxiv.org/abs/2304.07524
Khrennikov, 2024:
https://royalsocietypublishing.org/doi/full/10.1098/rsos.231953
Leavens, 2005:
https://scholar.google.co.uk/scholar?cluster=6646277109282008630&hl=en&as_sdt=0,5&as_vis=1
Leonard, Roelly & Zambrini, 2013:
https://arxiv.org/abs/1308.0576
Levy and Krener, 1996:
https://www.math.ucdavis.edu/~krener/ (PDF given at reference 68 at end; also, 72 and 61 relevant)
Lundeen et al., 2014:
https://scholar.google.co.uk/scholar?cluster=13685429957802398639&hl=en&as_sdt=0,5&as_vis=1 (also see various other papers by Lundeen on measuring the quantum state / wavefunction, plus https://scholar.google.co.uk/scholar?cluster=2611832075931433402&hl=en&as_sdt=0,5&as_vis=1 by Arvidsson-Shukur et al.)
Marchildon, 2000: https://scholar.google.co.uk/scholar?cluster=1415529095046305978&hl=en&as_sdt=0,5&as_vis=1
Markiewicz et al., 2014:
https://scholar.google.co.uk/scholar?cluster=4261036749838004631&hl=en&as_sdt=0,5&as_vis=1
Matzkin, 2020:
https://arxiv.org/abs/2002.00832
Mauro & Gozzi, 2004:
https://scholar.google.com/scholar?cluster=11196281894751885989&hl=en&as_sdt=0,5&as_yhi=2006
Nachbin, 2022: https://scholar.google.co.uk/scholar?cluster=11815274735010691195&hl=en&as_sdt=0,5&as_vis=1 (Also referenced in other Bush reviews)
Nelson, 1966: https://scholar.google.co.uk/scholar?cluster=10928480749452078078&hl=en&as_sdt=0,5&as_vis=1 (Other Nelson papers and books found here: https://web.math.princeton.edu/~nelson/papers.html)
Nelson, 1985:
PDF found here (Quantum Fluctuations book) -
https://web.math.princeton.edu/~nelson/papers.html
Neumaier, 2000: https://scholar.google.co.uk/scholar?cluster=18082693999119350419&hl=en&as_sdt=0,5&as_vis=1
Oriols & Mompart, 2019:
https://scholar.google.co.uk/scholar?cluster=13436694591113361330&hl=en&as_sdt=0,5&as_vis=1
Pandey et al., 2021: https://scholar.google.co.uk/scholar?cluster=17453749043004093198&hl=en&as_sdt=0,5&as_vis=1 (also see the supplementary documents; also the preprint: https://scholar.google.co.uk/citations?view_op=view_citation&hl=en&user=swfZnh8AAAAJ&citation_for_view=swfZnh8AAAAJ:OBSaB-F7qqsC)
Pavon, 2002:
https://scholar.google.co.uk/scholar?cluster=16349118895536726893&hl=en&as_sdt=0,5&as_vis=1
Petroni & Morato, 2000: https://scholar.google.co.uk/scholar?cluster=15066155520566221879&hl=en&as_sdt=0,5&as_vis=1 (also described in Derakshani & Bacciagaluppi, 2024 and Beyer, 2023 [pg. 31-32])
Pitowsky, 1989:
https://scholar.google.co.uk/scholar?cluster=7459227212319851370&hl=en&as_sdt=0,5&as_vis=1 Also see: https://scholar.google.co.uk/scholar?cluster=16366977415123888164&hl=en&as_sdt=0,5&as_vis=1 (Pitowsky, 1994 - PDF)
https://scholar.google.co.uk/scholar?cluster=17313080888273101986&hl=en&as_sdt=0,5&as_vis=1 (Abramsky, 2020)
Pothos & Busemeyer, 2022:
https://pubmed.ncbi.nlm.nih.gov/34546804/
Pusey, 2014:
https://scholar.google.co.uk/scholar?cluster=4581362650618351707&hl=en&as_sdt=0,5&as_vis=1
Reginatto, 1998:
https://journals.aps.org/pra/abstract/10.1103/PhysRevA.58.1775
Sanz, 2005:
https://scholar.google.com/scholar?cluster=2064940648404733431&hl=en&as_sdt=0,5
Schmid, Ying & Leifer, 2023:
https://scholar.google.com/scholar?cluster=12611254179628854230&hl=en&as_sdt=0,5
Schrodinger, 1931 (Inside Chetrite et al., 2021):
https://scholar.google.co.uk/scholar?cluster=503998788243586525&hl=en&as_sdt=0,5&as_vis=1
Shi et al., 2019:
https://scholar.google.co.uk/scholar?cluster=8920367932635165523&hl=en&as_sdt=0,5&as_vis=1
Sokolovski, 2018:
https://arxiv.org/abs/1803.02303
Spekkens, 2008:
https://scholar.google.co.uk/scholar?cluster=3111305811603149998&hl=en&as_sdt=0,5&as_vis=1
Sutherland, 2022:
https://scholar.google.co.uk/scholar?cluster=4724692290714986852&hl=en&as_sdt=0,5&as_vis=1
Yang, 2021: https://scholar.google.co.uk/scholar?cluster=5106354696323260707&hl=en&as_sdt=0,5&as_vis=1
Yang, 2022:
https://arxiv.org/abs/2204.12285
Yang, 2024:
https://arxiv.org/abs/2311.08420
Yang, 2025:
https://scholar.google.co.uk/scholar?cluster=9835259018458496839&hl=en&as_sdt=0,5&as_vis=1
Wallstrom, 1989:
https://scholar.google.com/scholar?cluster=17220464949127373559&hl=en&as_sdt=0,5
Wallstrom, 1994:
https://scholar.google.com/scholar?cluster=14207102363905249618&hl=en&as_sdt=0,5
Wharton et al., 2024
https://arxiv.org/abs/2412.05456
Wiseman, 2007: https://scholar.google.co.uk/scholar?cluster=5017634327174548872&hl=en&as_sdt=0,5&as_vis=1
Wikipedia (Bayes):
https://en.wikipedia.org/wiki/Variational_Bayesian_methods - Minimizing the variational free energy to 0 indicates the correct posterior probability distribution in Bayesian inference. "ELBO" below is an example of this (but described as maximizing the negative free energy instead).
https://en.wikipedia.org/wiki/Evidence_lower_bound - First set of equations under the "Definition" section can be rearranged as -ln p(x | z) - Dkl q(z | x) || p(z) which describes an accuracy-complexity trade-off, complexity being the expected difference (Dkl) between a prior, p(z), and a hypothetical posterior q(z | x). This kind of complexity has also been described in David Mackay's Information theory, inference and learning algorithms textbook, and is an inherent part of Bayes theorem [section iv, 28: Model Comparison and Occam's Razor]:
http://www.inference.org.uk/mackay/itila/book.html (free PDF, author's website)
Also re-described by Rougier & Priebe, 2021:
https://scholar.google.co.uk/scholar?cluster=5177100000297736827&hl=en&as_sdt=0,5&as_vis=1
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