It is an interpretation in the sense of Bohmian mechanics, in fact their underlying mathematical structure is very similar. — Apustimelogist

That is only true if the so-called background hypothesis, which is typically assumed to consist of a random field locally perturbing the motions of the particles, is assumed to have supplementary non-local Bohmian character as necessary to explain the statistics of quantum entanglement ... in which case your preferred interpretation becomes a variant of the Bohmian interpretation.

The stochastic interpretation provides a conception of wave-particle duality with an uncertainty principle, but without additional Bohmian mechanics it cannot explain Bells Theorem, for there is no getting around the fact that classical particle-field interactions that communicate slower than the speed of light cannot explain the 'action at a distance' of Bells inequalities; either the medium denoted by the background hypothesis is local, in which case we do not have quantum semantics, else the medium has non-local effects, in which case we have Bohmian mechanics.

I think the stochastic interpretation is pedagogically useful for providing a common-sense physical explanation for potentially classical aspects of complementarity that are often mistaken for inherently quantum phenomena, analogous to how Spekken's toy model of quantum mechanics is useful for providing common-sense epistemic intuition for understanding complementarity without assuming a physical account of the Schrodinger equation. But in neither case is there either a physical or epistemic explanation for entanglement.

But notice that embodied unstated realist assumptions about 'what the world is like'. And as Sabine Hossenfelder points out in Lost in Math, there's this tendency in today's physics to rationalise posits on the basis that they supposedly make intuitive sense and then to devise the mathematics to make them stand up. So given your realist predilections, then this approach seems natural to you. — Wayfarer

Very true, we all have different dispositions for intuition; albeit, I would say that the only reason these intuitions are open to us in physics is because of results like quantum theory. Consensus would have opted for a realistic interpretation had it been conceivable in the first place, so I would say a realistic interpretation should be preferable if available. There would be no QBism if a realistic interpretation has heen initially available to us.

And this is being borne out by experimental validation of 'Wigner's Friend'-type scenarios. — Wayfarer

Yes, but a stochastic interpretation has its own viee of Wigner's friend which isn't subjectively perspectival.

That is only true if the so-called background hypothesis, which is typically assumed to consist of a random field locally perturbing the motions of the particles, is assumed to have supplementary non-local Bohmian character as necessary to explain the statistics of quantum entanglement ... in which case your preferred interpretation becomes a variant of the Bohmian interpretation. — sime

If someone wants to call it a variation of Bohmian mechanics, I don't really see an inherent issue but you don't explicitly need Bohmian kind of non-locality for the theory to produce Bell violations. There is at least one version by Levy & Krener (1996) which is does not have Bohmian non-locality, produces all the correct predictons, and even explains that non-locality only comes when an artificially idealized assumption is used for constructing the theory.

quote="sime;983839"]The stochastic interpretation provides a conception of wave-particle duality with an uncertainty principle, but without additional Bohmian mechanicks it cannot explain Bells Theorem, for there is no getting around the fact that classical particle-field interactions that communicate slower than the speed of light cannot explain the 'action at a distance' of Bells inequalities; either the medium denoted by the background hypothesis is local, in which case we do not have quantum semantics, else the medium has non-local effects, in which case we have Bohmian mechanics.[/quote]

Well Levy & Krener's stochastic mechanics van in principle do so without explicit non-local communication.

I think the stochastic interpretation is pedagogically useful for providing a common-sense physical explanation for potentially classical aspects of complementarity that are often mistaken for inherently quantum phenomena, analogous to how Spekken's toy model of quantum mechanics is useful for providing common-sense epistemic intuition for understanding complementarity without assuming a physical account of the Schrodinger equation. But in neither case is there either a physical or epistemic explanation for entanglement. — sime

False. It can produce all quantum behavior and explains it in terms of a non-dissipative diffusion. The issue is that its depiction of QM is radically different from what people are used to so ita difficult for them to imagine. For instance, on stochastic mechanics, spin is a statistical property and doesn't belong to individual particles. An interesting possible consequence is that we no longer have to think of measurement in entanglement experiments as actively changing properties of individual particles.

If someone wants to call it a variation of Bohmian mechanics, I don't really see an inherent issue but you don't explicitly need Bohmian kind of non-locality for the theory to produce Bell violations. There is at least one version by Levy & Krener (1996) which is does not have Bohmian non-locality, produces all the correct predictons, and even explains that non-locality only comes when an artificially idealized assumption is used for constructing the theory. — Apustimelogist

I have a suspicion that the authors you mention aren't intending to address foundational questions of QM ,and are instead focusing on the technicalities of constructing laws and diffusion models that cohere with the Schrodinger equation, with potential relevance to the subject of modelling quantum decoherence, by which classical diffusion can emerge in the limit of quantum diffusion, but without relevance as to the question of the nature and ontology of quantum states and quantum measurements.

A quantum stochastic process that in principle can model non-local correlations, i.e. a random vector field of a complex Hilbert Space that is interpretable as the evolution of a quantum state vector, cannot be explicated in terms of the local interactions of a regular stochastic process such as Brownian motion. The former can serve to explicate the latter, but not conversely unless one supplements entanglement relations.

I have a suspicion that the authors you mention aren't intending to address foundational questions of QM ,and are instead focusing on the technicalities of constructing laws and diffusion models that cohere with the Schrodinger equation, with potential relevance to the subject of modelling quantum decoherence, by which classical diffusion can emerge in the limit of quantum diffusion, but without relevance as to the question of the nature and ontology of quantum states and quantum measurements. — sime

I am pretty sure this is not the case. For instance:

https://scholar.google.co.uk/scholar?cluster=856861870672922375&hl=en&as_sdt=0,5&as_vis=1

"Nelson’s stochastic mechanics formulation of quantum mechanics, which started it development with his article On the derivation of the Schrödinger equation from Newtonian mechanics in 1966 [6], is completely in accord with the requirements of Bells inequalities. By now, it has been developed to a mathematical rigor that completely parallels the formulation of classical analytical mechanics. It thus provides sufficient mathematical structure to suggest a clear physical picture of quantum phenomena."

cannot be explicated in terms of the local interactions of a regular stochastic process such as Brownian motion. — sime

Stochastic mechanics shows mathematically that entanglement follows from a non-dissipative / conservative diffusion.

I cannot access the Levy & Krener (1996) paper, so instead I asked Grok for an account of reciprocal stochastic processes, of which I am unfamiliar which gave an interesting reply.

If the papers referred to are "Dynamics and kinematics of reciprocal diffusions" and "Stochastic mechanics of reciprocal diffusions", then I can see how these papers are of relevance to foundational questions of QM, in the sense of attempting to reconstruct properties of quantum diffusion in terms of time-symmetric but otherwise classical stochastic processes that implement non-local aspects of the time-symmetric transactional interpretation of QM.

However, Grok's conclusion at the end is along the lines of my initial thoughts, namely that such processes fail to account for quantum entanglement as should be expected by the Kohen Specker Theorem.


" Reciprocal stochastic processes are compelling because they challenge the necessity of quantum mechanics’ formalism for describing quantum diffusion. If a classical-like stochastic model can replicate quantum behavior, it suggests that some quantum phenomena might be emergent from underlying probabilistic structures, aligning with interpretations like stochastic mechanics or hidden-variable theories. However, their inability to fully capture quantum non-locality (e.g., entanglement) reinforces the uniqueness of quantum mechanics, prompting deeper inquiry into what makes quantum systems distinct.
Specific foundational questions they address:
Can quantum non-locality be reduced to time-symmetric stochastic correlations?

Is the wave function a physical entity, or can it be replaced by a stochastic process with equivalent predictive power?

How does the time-symmetric nature of quantum mechanics relate to causality and the arrow of time?

Can the quantum-classical transition be fully understood as a shift from reciprocal to Markovian stochastic processes?

Conclusion
Reciprocal stochastic processes can reproduce many aspects of quantum diffusion, such as probability density evolution, interference-like patterns, and non-local effects, making them a powerful tool for modeling quantum dynamics in a probabilistic framework. Their time-symmetric and non-Markovian nature makes them particularly relevant to foundational questions about quantum non-locality, the quantum-classical transition, and the ontology of the wave function. While they do not fully explain quantum non-locality (e.g., entanglement), they offer a semi-classical perspective that challenges quantum mechanics’ uniqueness and invites exploration of alternative formulations, such as stochastic mechanics or time-symmetric interpretations."

So i stand partially corrected.

However, their inability to fully capture quantum non-locality (e.g., entanglement) reinforces the uniqueness of quantum mechanics, prompting deeper inquiry into what makes quantum systems distinct. — sime

This is absolutely false and A.I. do not reliably give you information. Often when I research something, I will read the A.I. summary thing on google just as to give me some indication if what the answer might be, but I absolutely cannot trust this. I look at the sources it says it gets the information from every time and I find that it's relatively common that the A.I. will mix things up or confabulate ideas from various sources that actually don't make sense together, ending up in wrong or misleading answers.

I think the A.I. can be used as a research aide, but it cannot be trusted to reliably give you answers to things.

For instance, I can tell that the following phrase came from an A.I. :

that implement non-local aspects of the time-symmetric transactional interpretation of QM. — sime

Because the transactional interpretation has nothing to do with stochastic mechanics and is almost a polar opposite interpretation.

I cannot access the Levy & Krener (1996) paper — sime

Btw, you can see a pdf download link by just searching something like "Levy & Krener (1996)" in google. The pdf link comes from the author's website associated with their university.

Stochastic mechanics shows mathematically that entanglement follows from a non-dissipative / conservative diffusion. — Apustimelogist

Well not according to your source "On the Stochastic Mechanics Foundation of Quantum Mechanics ".

There is no mention let alone explanation of entanglement anywhere in that paper, although there is a mention of the Bohm potential, indicating that the authors are perhaps imagining their stochastic mechanics supplemented with some other foundational interpretation, perhaps to account for the non-locality of their background hypothesis. As it stands, it is a metaphysical interpretation of the Schrodinger equation that reproduces a fragment of the least problematic parts of Quantum Mechanics with deafening silence on the most critical aspects of QM that the interpretation either fails to address, or helps itself to by appealing to unstated non-local premises.

As it stands, I view that paper, which I have admittedly scantly read, is a non-earth shattering exercise in using stochastic differential equations to simulate whatever one wishes.

This is absolutely false and A.I. do not reliably give you information. — Apustimelogist

Neither do publicity seeking authors advertising grandoise and unproven claims.

For instance, I can tell that the following phrase came from an A.I. :

that implement non-local aspects of the time-symmetric transactional interpretation of QM.
— sime — Apustimelogist

They were actually my own opinion in my own words, prompted by my understanding that the authors of the other paper you mention were reconstructing quantum diffusion out of time symmetric diffusions that is reminiscent of the symmetric casuality inherent in the transactional interpretation of QM. Personally I think that more modest paper is much more informative.

For what its worth, I'm finding vanilla ChatGPT especially helpful with regards to navigating in a sourced way the nuances of the stochastic mechanics interpretation. As an outsider to the physics research community who nevertheless has a vested interest in understanding the mathematics and logic of a wide range of theories for purposes in relation to computing and category theory, I'm generally finding LLMs particularly useful for getting to grips quickly with unfamiliar theoretical ideas and for understanding the tone and the context of research papers, without which it can be difficult to understand what authors are selling versus what they are claiming - a very common problem indeed.

For instance, I notice that certain physicists who are prominent members of the PhysicsForums.com were almost automatically dismissive of stochastic mechanics for the same obvious reasons that i opined earlier in this thread, but they also suspected that the authors selling stochastic mechanics were dishonest, doing pointless metaphysics, or failing to own up to the problem of entanglement.

On the other hand, ChatGPT focused on what Stochastic mechanics is and actually claims and spoke of the authors contributions in a more neutral and worthwhile tone, more or less summarizing the interpretation as a stochastic alternative to Bohmian mechanics that replaces the guiding wave with quantum diffusion, whilst also stressing the fact that stochastic mechanics cannot be an explanation for non-locality for obvious Cohen-Specker reasons, while pointing out that the model assumes non-locality in the form of the configuration space upon which the model places a quantum diffusion - namely the space describing the joint positions of all of the particles that cannot be decoupled into independent diffusions satisfying local causality if non-local entanglement is to be describable by the model.

As to the question regarding what "reciprocal processes" brings to the table, they apparently 'upgrade' the implicit and unexplained non-locality of the original model of stochastic mechanics (i.e the configuration space) , to a more explicit model of non-locality based on time-symmetry that is similar to the transactional interpretation, which in my words and understanding can presumably reconstruct at least some of the non-local unity of the configuration space in terms of the "retrocausal" effects of the future light-cone of the particles. How successful this approach is I don't know, and didn't care to ask.

There is no mention let alone explanation of entanglement anywhere in that paper — sime

I just gave you a quote earlier that says that it is in accord with Bell inequalities but they actually do have a paper where they produce the Bell violations with stochastic mechanics:

https://scholar.google.co.uk/scholar?cluster=15973777865898642687&hl=en&as_sdt=0,5&as_ylo=2024&as_vis=1

Stochastic mechanics can reproduce all behavior of quantum mechanics.

Now, the above paper has a kind of non-local behavior, but the authors interpret it epistemically. And this is completely justified imo because the Levy-Krener paper shows that this non-locality is a byproduct of using an artificial, idealized Markovian assumption. Their model alsocan reprosuce all quantum behavior without any non-locality at all in the formalism by not using the idealized Markov assumption.

So your following statement is false:

for the non-locality of their background hypothesis. As it stands, it is a metaphysical interpretation of the Schrodinger equation that reproduces a fragment of the least problematic parts of Quantum Mechanics with deafening silence on the most critical aspects of QM that the interpretation either fails to address, or helps itself to by appealing to unstated non-local premises. — sime

Because stochastic mechanics reproduces all quantum behavior under assumptions about point particles that always are in definite places at any time and move along continuous paths. The fact you can do this suggests it is metaphysically possible; and stochastic mechanics gives a very simple physical explanation: energy conservation. All quantum behavior manifests from this. You don't need a non-local background to do this. No where is it implied that explicit non-locality is needed, only this energy conservation property; it follows that non-local entanglement behavior would follow regardless of exactly how you enforce this energy conservation, aslong as it is achieved. Non-local correlations are a consequence of initial local interactions where the resulting behavior does not dissipate but maintains its initial correlations.

is a non-earth shattering exercise in using stochastic differential equations to simulate whatever one wishes. — sime

You can't just do this though. These stochastic processes work under the same physical constraints people would normally assume are impossible in regards to producing quantum behavior. Its not an arbitrary formal reconstruction, it has strong physical interpretation implied directly by the formalism.

They were actually my own opinion in my own words, prompted by my understanding that the authors of the other paper you mention were reconstructing quantum diffusion out of time symmetric diffusions that is reminiscent of the symmetric casuality inherent in the transactional interpretation of QM. Personally I think that more modest paper is much more informative. — sime

Well its completely polar opposite to what the transactional interpretation.

What do you mean by "modest"?

For what its worth, I'm finding vanilla ChatGPT especially helpful with regards to navigating in a sourced way the nuances of the stochastic mechanics interpretation. As an outsider to the physics research community who nevertheless has a vested interest in understanding the mathematics and logic of a wide range of theories for purposes in relation to computing and category theory, I'm generally finding LLMs particularly useful for getting to grips quickly with unfamiliar theoretical ideas and for understanding the tone and the context of research papers, without which it can be difficult to understand what authors are selling versus what they are claiming - a very common problem indeed. — sime

Completely disagree. I mean, what your A.I. gave me about stochastic mechanics is false in a way that can be easily checked. And I have my own experiences of sometimes using A.I. to help find some kind of an answer and finding that they were misleading or incorrect when checking sources myself.

A.I. will give you explanations that are easy to understand; it doesn't mean they are always accurate, just like in this case!

For instance, I notice that certain physicists who are prominent members of the PhysicsForums.com were almost automatically dismissive of stochastic mechanics for the same obvious reasons that i opined earlier in this thread, but they also suspected that the authors selling stochastic mechanics were dishonest, doing pointless metaphysics, or failing to own up to the problem of entanglement. — sime

These physicists may not have reliable opinions either if they are not at all familiar with and actually understand stochastic mechanics and perhaps are even invested in their own preferred interpretation. Obviously I cannot actually evaluate their opinions if what they say isn't given.

whilst also stressing the fact that stochastic mechanics cannot be an explanation for non-locality for obvious Cohen-Specker reasons — sime

False.

the model assumes non-locality in the form of the configuration space upon which the model places a quantum diffusion - namely the space describing the joint positions of all of the particles that cannot be decoupled into independent diffusions satisfying local causality if non-local entanglement is to be describable by the model. — sime

The Levy & Krener paper doesn't have this and explains why it occurs in a model which uses an artificial Markovian assumption.

Yes, the

These physicists may not have reliable opinions either if they are not at all familiar with and actually understand stochastic mechanics and perhaps are even invested in their own preferred interpretation. Obviously I cannot actually evaluate their opinions if what they say isn't given.

whilst also stressing the fact that stochastic mechanics cannot be an explanation for non-locality for obvious Cohen-Specker reasons
— sime

False. — Apustimelogist

You need to do better than that.

The Levy & Krener paper doesn't have this and explains why it occurs in a model which uses an artificial Markovian assumption. — Apustimelogist

I'm in agreement with Levy & Krener to the extent I understand their actual claims, which is what I was pointing out; they are shifting to reciprocal processes to provide a non-classical account of non-locality , as per the explanation provided by Chat-GPT , in the hope of explicating the presupposition of non-locality in stochastic models that is hard-encoded in the latter's reliance upon configuration space.

None of this makes their revised account of stochastic mechanics a locally causal model of QM in line with the classical intuitions of Brownian motion.

Do you agree or not agree, that any theory dependent entirely on local causality cannot be a full explanation of QM? Secondly, how do you propose physically interpreting the use of time-symmetric reciprocal processes for guiding a collection of particles in a way that that is compatible with local realism?

Here is a full Grok's response.

What do reciprocal processes contribute to theories of Stochastic Mechanics that interpret Quantum Mechanics?

Reciprocal processes play a significant role in stochastic mechanics, particularly in the context of interpreting quantum mechanics through stochastic frameworks like the Nelson's stochastic quantization or the stochastic interpretation of quantum mechanics. Here's a concise explanation of their contribution:

Bridging Classical and Quantum Dynamics:
Reciprocal processes are a class of stochastic processes that generalize Markov processes by allowing conditional dependencies on both past and future states. In stochastic mechanics, they provide a framework to describe quantum systems as stochastic processes while preserving time-reversal symmetry, a key feature of quantum mechanics. This helps reconcile classical stochastic dynamics with quantum behavior.

Schrödinger Equation Connection: Reciprocal processes are closely tied to the Schrödinger equation through the concept of stochastic differential equations. In Nelson's stochastic mechanics, the quantum wave function is interpreted as encoding information about a diffusion process. The forward and backward drifts of a reciprocal process correspond to the probability currents, which are derived from the wave function, thus providing a stochastic interpretation of quantum evolution.

Path Integral Interpretation: Reciprocal processes support the path integral formulation of quantum mechanics by describing trajectories that are conditioned on both initial and final states. This aligns with the Feynman path integral approach, where quantum amplitudes are sums over all possible paths. Reciprocal processes offer a probabilistic underpinning for these paths in a stochastic framework.

Non-Markovian Behavior: Unlike standard Markov processes, reciprocal processes can model non-Markovian dynamics, which are essential for capturing quantum phenomena like entanglement and superposition. This makes them suitable for describing the stochastic analogs of quantum correlations and interference effects.

Time-Symmetric Formulation: Reciprocal processes enable a time-symmetric formulation of stochastic mechanics, mirroring the time-reversal invariance of quantum mechanics. This symmetry is crucial for interpreting quantum phenomena like wave function collapse or measurement in a stochastic context.

In summary, reciprocal processes contribute to stochastic mechanics by providing a mathematical framework that captures quantum mechanical behavior—such as wave function dynamics, path integrals, and time symmetry—within a probabilistic, stochastic setting. They offer an alternative lens to interpret quantum mechanics, emphasizing diffusion-like processes while maintaining consistency with quantum predictions.


Can reciprocal processes be understood in terms of the common-sense local causality of Brownian motion?

Yes, reciprocal processes can be understood in terms of the common-sense local causality of Brownian motion, but with important extensions that make them distinct and more suitable for capturing quantum mechanical phenomena in stochastic mechanics. Here's a concise explanation:

Brownian Motion and Local Causality

Brownian motion is a Markovian stochastic process characterized by local causality: the future evolution of a particle depends only on its current state (position and velocity) and is driven by random, uncorrelated increments (white noise). This aligns with a "common-sense" notion of causality where the past influences the future in a straightforward, memoryless way, and the dynamics are governed by local interactions (e.g., diffusion via random collisions).

Reciprocal Processes: Extending Brownian Motion

Reciprocal processes generalize Brownian motion by introducing a form of conditional dependence on both past and future states, which departs from the strict Markovian property but retains a sense of local causality in a broader sense. Here's how they relate to Brownian motion and local causality:

Local Dynamics with Boundary Conditions:
Reciprocal processes can be thought of as Brownian-like diffusions constrained by boundary conditions at both initial and final times. For example, in stochastic mechanics, a particle's trajectory is modeled as a diffusion process (akin to Brownian motion) but conditioned on starting at one point and ending at another. This introduces a "teleological" aspect, where the future influences the present, but the dynamics remain locally governed by stochastic differential equations (SDEs) similar to those of Brownian motion.

The local causality is preserved in the sense that the increments of the process are still driven by Gaussian noise, and the evolution at each time step depends on local drift and diffusion terms.

Forward and Backward Drifts:
In stochastic mechanics, reciprocal processes are described by forward and backward stochastic differential equations, where the drift terms are derived from the quantum wave function. These drifts are analogous to the drift in a biased Brownian motion but incorporate information about both initial and final states. The local causality is maintained because the drifts and noise act locally in time, even though they are influenced by global (boundary) conditions.

Mathematically, for a reciprocal process XtX_tX_t
, the forward SDE might look like:
dXt=bf(Xt,t)dt+2νdWt,dX_t = b_f(X_t, t) dt + \sqrt{2\nu} dW_t,dX_t = b_f(X_t, t) dt + \sqrt{2\nu} dW_t,
where bfb_fb_f
is the forward drift, ν\nu\nu
is the diffusion coefficient, and WtW_tW_t
is a Wiener process (Brownian motion). A similar backward SDE exists, ensuring compatibility with future conditions.

Non-Markovian but Locally Causal:
Unlike Brownian motion, reciprocal processes are generally non-Markovian because the probability of a trajectory depends on both past and future states. However, the local causality of Brownian motion is preserved in the sense that the process is still driven by local stochastic increments, and the non-Markovian behavior arises from the conditioning rather than nonlocal interactions.

This conditioning can be understood as a statistical constraint rather than a violation of locality, much like how a Brownian bridge (a Brownian motion conditioned to end at a specific point) remains locally causal despite its endpoint constraint.

Quantum Interpretation:
In the context of stochastic mechanics, reciprocal processes model quantum particles as undergoing Brownian-like motion with drifts determined by the wave function. The "common-sense" causality of Brownian motion is extended to account for quantum phenomena like interference and entanglement, where the conditioning on future states mimics the global correlations of quantum mechanics.

For instance, the interference patterns in the double-slit experiment can be modeled as reciprocal processes where the particle's stochastic paths are influenced by both the initial state and the final detection point, yet the paths themselves are locally diffusive, like Brownian motion.

Key Difference from Brownian Motion
While Brownian motion is purely Markovian and driven by local, memoryless noise, reciprocal processes introduce a form of global conditioning that makes them suitable for quantum mechanics. This conditioning does not violate local causality but modifies the drift terms to reflect quantum probabilities, effectively bridging the local, diffusive nature of Brownian motion with the nonlocal correlations of quantum systems.

Summary
Reciprocal processes can be understood as an extension of Brownian motion's local causality, where the random, diffusive nature of Brownian motion is preserved, but the trajectories are conditioned on both initial and final states. This conditioning introduces non-Markovian behavior while maintaining local stochastic dynamics, making reciprocal processes a natural framework for interpreting quantum mechanics in a stochastic setting. They retain the intuitive, local causality of Brownian motion but adapt it to capture the time-symmetric and globally correlated nature of quantum phenomena.

So are they locally causal at the expense of losing realism, as per the Kochen Specker theorem?

The question of whether reciprocal processes in stochastic mechanics are locally causal at the expense of losing realism, particularly in light of the Kochen-Specker (KS) theorem, is a nuanced one. To address this, we need to unpack the concepts of local causality, realism, and how the KS theorem applies in the context of stochastic interpretations of quantum mechanics (QM). Here's a concise and clear response:

Recap: Reciprocal Processes and Local Causality

Reciprocal processes, as used in stochastic mechanics (e.g., Nelson's stochastic quantization), model quantum systems as diffusion processes akin to Brownian motion, with forward and backward drifts derived from the quantum wave function. They are locally causal in the sense that:

The evolution of the process at each time step is governed by local stochastic differential equations (SDEs), driven by Gaussian noise and drift terms that depend on the local state.

The non-Markovian nature arises from conditioning on initial and final states, but this conditioning is a statistical constraint, not a nonlocal physical interaction.

This local causality aligns with the intuitive, diffusive picture of Brownian motion, extended to incorporate quantum behavior via time-symmetric drifts.

Kochen-Specker Theorem and Realism

The Kochen-Specker theorem (1967) is a no-go theorem in quantum mechanics that rules out non-contextual hidden variable theories under the assumption of realism. Realism, in this context, means that all measurable properties (observables) of a quantum system have definite values at all times, independent of measurement. The KS theorem shows that for quantum systems in dimensions d≥3
, it is impossible to assign definite values to all observables simultaneously in a non-contextual way (i.e., where the value of an observable is independent of which other compatible observables are measured alongside it).
In other words, the KS theorem challenges the idea that quantum systems possess pre-existing, measurement-independent properties (realism) while respecting the structure of quantum observables.

Reciprocal Processes and Realism
To address whether reciprocal processes in stochastic mechanics sacrifice realism to maintain local causality, we need to examine how stochastic mechanics interprets quantum mechanics and whether it assumes realism in the sense challenged by the KS theorem:

Stochastic Mechanics as a Hidden Variable Theory:
Nelson's stochastic mechanics attempts to reproduce quantum mechanics by modeling particles as undergoing stochastic trajectories governed by reciprocal processes. The wave function is interpreted as encoding the probability distribution and drift of these trajectories, not as a physical field but as a statistical descriptor.

In its original formulation, stochastic mechanics can be viewed as a hidden variable theory, where the particle's position and trajectory are the hidden variables, assumed to have definite values at all times (realism). The stochastic drifts are derived from the wave function, and the randomness mimics quantum uncertainty.

Impact of the KS Theorem:
The KS theorem applies to stochastic mechanics if it assumes non-contextual realism, i.e., that all observables (e.g., position, momentum, spin) have definite values independent of the measurement context. Since stochastic mechanics assigns definite positions to particles at all times (the trajectories are well-defined), it inherently assumes realism for position. However, other observables, like momentum or spin, are not directly represented as definite values in the stochastic framework but are derived statistically from the wave function or ensemble averages.

The KS theorem implies that stochastic mechanics cannot consistently assign definite values to all quantum observables in a non-contextual way for systems with Hilbert spaces of dimension d≥3.
. For example, attempting to define definite values for spin or momentum observables alongside position in a way that reproduces quantum predictions would lead to contextuality, contradicting non-contextual realism.

Local Causality vs. Realism:
Local causality in reciprocal processes is preserved because the dynamics are governed by local SDEs, and the conditioning on future states (non-Markovianity) is a statistical effect, not a nonlocal physical interaction. This avoids issues like Bell nonlocality, as stochastic mechanics can be constructed to respect Einstein locality for spatially separated systems.

However, to maintain consistency with quantum mechanics (and avoid contradicting the KS theorem), stochastic mechanics must abandon non-contextual realism for all observables. Instead, it may adopt contextual realism, where the values of observables depend on the measurement context (e.g., which compatible observables are measured). For instance, the momentum in stochastic mechanics is not a definite value but a statistical quantity derived from the drift, which depends on the wave function and measurement setup.

Alternatively, stochastic mechanics could abandon realism entirely for observables other than position, treating them as inherently probabilistic or emergent from the stochastic dynamics. This would align with quantum mechanics' rejection of pre-existing values but might undermine the "realist" motivation of stochastic mechanics as a classical-like interpretation.

Does Stochastic Mechanics Lose Realism?:
If stochastic mechanics insists on non-contextual realism (definite values for all observables, independent of measurement), it is incompatible with the KS theorem and fails to reproduce quantum mechanics fully. In this sense, maintaining local causality via reciprocal processes comes at the expense of non-contextual realism, as the KS theorem forbids such a model.

In practice, stochastic mechanics often adopts a weaker form of realism, where only the particle's position is real and definite (the stochastic trajectory), while other observables are contextual or probabilistic. This allows it to maintain local causality and reproduce quantum predictions (e.g., for position measurements) but sacrifices the strong realist picture where all observables have pre-existing values.

Some formulations of stochastic mechanics explicitly acknowledge this by focusing on position as the primary observable and treating other properties as statistical or context-dependent, sidestepping the KS theorem's constraints by not assigning definite values to incompatible observables.

Summary

Reciprocal processes in stochastic mechanics are locally causal, with dynamics resembling Brownian motion but conditioned on initial and final states. However, to be consistent with quantum mechanics and the KS theorem, stochastic mechanics cannot maintain non-contextual realism for all observables. It either adopts contextual realism (where observable values depend on measurement context) or restricts realism to position alone, treating other observables probabilistically. Thus, local causality is preserved, but at the expense of a strong realist interpretation where all quantum observables have definite, measurement-independent values.
In essence, the KS theorem forces stochastic mechanics to compromise on realism to remain viable, aligning its reciprocal processes with quantum mechanics' contextual or non-realist nature while retaining a locally causal framework.

So are particle velocities interpreted non-locally by this model, if they are considered to be real properties?

To address whether particle velocities in stochastic mechanics, when interpreted as real properties, are non-local in the context of reciprocal processes, we need to examine how velocities are defined in this framework, the role of realism, and the implications of non-locality. This response builds on our prior discussion, particularly the interplay between local causality, realism, and the Kochen-Specker (KS) theorem, and provides a concise answer tailored to your question.

Stochastic Mechanics and Reciprocal Processes: Overview

In stochastic mechanics (e.g., Nelson’s formulation), quantum systems are modeled as particles undergoing diffusion processes akin to Brownian motion, described by reciprocal processes. These processes are governed by stochastic differential equations (SDEs) with forward and backward drifts derived from the quantum wave function. The particle’s position is treated as a real, definite property (realism), and its stochastic trajectory is well-defined. Reciprocal processes are locally causal because the dynamics at each time step depend on local drift and noise terms, with non-Markovian behavior arising from statistical conditioning on initial and final states, not nonlocal interactions.

Velocities in Stochastic Mechanics
In stochastic mechanics, the concept of velocity is more nuanced than in classical mechanics due to the stochastic nature of the trajectories. Let’s clarify how velocities are defined and interpreted:
Forward and Backward Velocities:
The particle’s motion is described by a stochastic differential equation, such as:
dXt=bf(Xt,t)dt+2νdWt,dX_t = b_f(X_t, t) dt + \sqrt{2\nu} dW_t,dX_t = b_f(X_t, t) dt + \sqrt{2\nu} dW_t,
where XtX_tX_t
is the particle’s position, bf(Xt,t)b_f(X_t, t)b_f(X_t, t)
is the forward drift, ν\nu\nu
is the diffusion coefficient (related to ℏ/2m\hbar/2m\hbar/2m
), and WtW_tW_t
is a Wiener process (Brownian noise).

Similarly, a backward SDE exists with a backward drift bb(Xt,t)b_b(X_t, t)b_b(X_t, t)
. These drifts are derived from the wave function ψ (I snipped Grok's unprintable unicode description)

The osmotic velocity u=(bf−bb)/2u = (b_f - b_b)/2u = (b_f - b_b)/2
and current velocity v=(bf+bb)/2v = (b_f + b_b)/2v = (b_f + b_b)/2
are introduced to describe the particle’s motion. The current velocity ( v ) is analogous to the Bohmian velocity in pilot-wave theory and is often interpreted as the “physical” velocity of the particle, while the osmotic velocity accounts for the diffusive component.

Realism of Velocities:
If velocities (e.g., the current velocity ( v )) are considered real properties, they are assumed to have definite values at each point along the particle’s trajectory, consistent with the realist assumption that the particle has a well-defined position and motion.

In stochastic mechanics, the current velocity v=ℏmIm(∇ψψ)v = \frac{\hbar}{m} \text{Im} \left( \frac{\nabla \psi}{\psi} \right)v = \frac{\hbar}{m} \text{Im} \left( \frac{\nabla \psi}{\psi} \right)
depends on the wave function, which encodes global information about the quantum system. This raises the question of whether such a velocity, if real, implies non-locality.

Are Velocities Non-Local?
Non-locality in quantum mechanics typically refers to correlations or influences that violate Bell’s inequalities or Einstein locality, where the state of one system instantaneously affects another at a distance without a local mediating mechanism. To determine if velocities in stochastic mechanics are non-local when treated as real properties, we consider the following:

Dependence on the Wave Function:
The current velocity ( v ) is determined by the gradient of the phase of the wave function ψ\psi\psi
. In quantum mechanics, the wave function is a global object that describes the entire system, including entangled or spatially extended states. For example, in an entangled two-particle system, the wave function ψ(x1,x2)\psi(x_1, x_2)\psi(x_1, x_2)
depends on the positions of both particles, and the velocity of particle 1, v1=ℏm1Im(∇1ψψ)v_1 = \frac{\hbar}{m_1} \text{Im} \left( \frac{\nabla_1 \psi}{\psi} \right)v_1 = \frac{\hbar}{m_1} \text{Im} \left( \frac{\nabla_1 \psi}{\psi} \right), may depend on the position of particle 2, even if they are far apart.

If ( v ) is a real property, this dependence suggests non-locality, as the velocity of one particle is instantaneously influenced by the state or position of another, without a local physical mechanism. This is analogous to the non-locality in Bohmian mechanics, where the velocity of a particle is guided by the non-local quantum potential or wave function.

Reciprocal Processes and Local Causality:

Reciprocal processes themselves are locally causal in their dynamics: the SDEs governing the particle’s motion depend only on the local drift bfb_fb_f or bbb_bb_b and noise at the current position XtX_tX_t
. The non-Markovian conditioning (dependence on initial and final states) is a statistical constraint, not a dynamical non-locality.

However, the drifts (and thus the velocities) are derived from the wave function, which can encode non-local correlations. For a single particle in a non-entangled state, the velocity ( v ) depends only on the local gradient of ψ\psi\psi , and the dynamics appear local. But in entangled or multi-particle systems, the wave function’s global nature introduces non-local dependencies, even though the stochastic evolution of each particle’s position is locally governed.

Comparison to Bohmian Mechanics:
Stochastic mechanics shares similarities with Bohmian mechanics, where the particle’s velocity is explicitly non-local due to its dependence on the wave function. In Bohmian mechanics, the velocity v=ℏmIm(∇ψψ)v = \frac{\hbar}{m} \text{Im} \left( \frac{\nabla \psi}{\psi} \right)v = \frac{\hbar}{m} \text{Im} \left( \frac{\nabla \psi}{\psi} \right) is a real property, and its non-locality is evident in entangled systems (e.g., EPR pairs), where measuring one particle’s position affects the other’s velocity instantaneously.

In stochastic mechanics, the current velocity plays a similar role, and if treated as a real property, it inherits the same non-local character. The stochastic noise adds randomness to the trajectories, but the drift (and thus velocity) is still tied to the non-local wave function.

Kochen-Specker Theorem and Contextuality:
As discussed previously, the KS theorem implies that stochastic mechanics cannot assign definite, non-contextual values to all observables (including velocity or momentum) while reproducing quantum mechanics. If velocity is treated as a real, definite property, it must be contextual—its value depends on the measurement context or the state of the system (e.g., the wave function).

This contextuality reinforces the potential non-locality: the velocity’s dependence on the global wave function means it cannot be determined solely by local properties of the particle, especially in entangled systems.

Non-Locality in Entangled Systems:
For a single particle or non-entangled system, the velocity ( v ) depends on the local gradient of ψ\psi\psi
, and the dynamics may appear local. However, in entangled systems, the wave function’s non-separable nature means that the velocity of one particle depends on the state of others, regardless of distance. This is a hallmark of quantum non-locality, as demonstrated by violations of Bell inequalities.

If velocities are real properties in stochastic mechanics, this dependence implies dynamical non-locality, where the motion of one particle is influenced by distant events (e.g., the position or measurement of another particle).

Conclusion
If particle velocities (e.g., the current velocity ( v )) are interpreted as real properties in stochastic mechanics, they are non-local in the sense that they depend on the global quantum wave function, which can encode non-local correlations, particularly in entangled systems. This non-locality arises because the velocity is derived from the wave function’s phase gradient, which may involve the states of distant particles, mirroring the non-locality in Bohmian mechanics. While reciprocal processes themselves are locally causal in their stochastic dynamics (local SDEs with noise), the realism of velocities introduces non-locality through their dependence on the non-local wave function.
Thus, treating velocities as real properties in stochastic mechanics sacrifices locality in entangled or multi-particle systems, consistent with quantum mechanics’ non-local correlations. This does not conflict with the KS theorem, which addresses non-contextual realism, but it highlights that realism for velocities comes at the cost of non-locality, especially in quantum systems exhibiting entanglement.

You need to do better than that. — sime

I linked you the EPR paper and the whole centra idea in stochastic mechanics is that you can derive the Schrodinger equation which means that all predictions can be reproduced. The diffusion process in Levy & Krener has conservation laws identical to the Schrodinger equation.

they are shifting to reciprocal processes to provide a non-classical account of non-locality , as per the explanation provided by Chat-GPT , in the hope of explicating the presupposition of non-locality in stochastic models that is hard-encoded in the latter's reliance upon configuration space. — sime

I don't really understand where you are Chatgp is getting this which just suggests to me that this based on the aforementioned unreliability of these A.I. explanations.

What is interesting about Levy & Krener's theory is that it is based on realistic point particles but does not have the non-locality that Bohmian mechanics has, whilst fulfilling all predictions.

It is still Bell non-local which is different. It is widely believed that a kind of Bohmian non-locality is required to explain Bell non-locality in these kinds of models, but Levy & Krener direcrly refutes this thought while also explaining ehy Bohmian non-locality emerges in Markovian stochastic models.

Do you agree or not agree, that any theory dependent entirely on local causality cannot be a full explanation of QM? Secondly, how do you propose physically interpreting the use of time-symmetric reciprocal processes for guiding a collection of particles in a way that that is compatible with local realism? — sime

A full explanation of QM need to violate Bell inequalities whoch falsify non-contextual hidden variable models. But stochastic mechanics is not a non-contextual hidden variable model. The subtlety is that the fact that the model is contextual doesn't necessarily have to mean non-local causation, but such an explanation is vwry intuitive in a scenario like spin measurements.

I think that what sets apart stochastic mechanics from other interpretations of QM is that in all other interpretations, spin is a property of individual particles. In stochastic mechanics, spin is a property of particle statistics that only describe the behavior of particles counterfactuslly under infinite repetition of an experiment.

You can then imagine an infinite ensemble of particle trajectories between some initial preparation and final spin measurement. The final spin measurement just divides the infinite ensembles into pairs of sub-ensembles with different statistics. It can be shown that the spin statistics of these sub-ensembles would have to remain constant between initial preparation and final spin measurement which means that if you introduce another spin measurement on a different ensemble of particles, you could correlate their respective final measurements by allowing them to share an initial preparation which fixed a correlation between them.

Overall, especially closer to the beginning, this was fairly good albeit I would say it saidsome of its claims very overconfidently imo.

But there is some nuance regarding descriptions like the following:

Alternatively, stochastic mechanics could abandon realism entirely for observables other than position, treating them as inherently probabilistic or emergent from the stochastic dynamics. This would align with quantum mechanics' rejection of pre-existing values but might undermine the "realist" motivation of stochastic mechanics as a classical-like interpretation. — sime

In essence, the KS theorem forces stochastic mechanics to compromise on realism to remain viable, aligning its reciprocal processes with quantum mechanics' contextual or non-realist nature while retaining a locally causal framework. — sime

Realism of Velocities:
If velocities (e.g., the current velocity ( v )) are considered real properties, they are assumed to have definite values at each point along the particle’s trajectory, consistent with the realist assumption that the particle has a well-defined position and motion. — sime

For example, attempting to define definite values for spin or momentum observables alongside position in a way that reproduces quantum predictions would lead to contextuality, contradicting non-contextual realism. — sime

The realism of particle configurations in stochastic mechanics has nothing to do with the realism related to the KS theorem. From the stochastic mechanical perspective, KS theorem and similar are about statistics. Particles can then always be in definite positions but their statistics respect the KS non-realism.

I actually used to refer to realism completely in this statistical sense until someone suggested to me it is more intuitive to talk about realism in QM interpretation in realistic fundamental ontology. So I started changing my use of the word more in line with this intuitive way of thinking about what realism is. Stochastic mechanics has realistic ontology of particles but is statistically non-realistic.

The A.I. is right though in line with what I said in my previous post that in stochastic mechanics, momentum and spin are defined statistically so they are not actual properties of individual particles but statistics that could only be related to lots of particles under repetition. You actually can have definite spin or momentum at points in space or for measurement results, just it doesn't apply to any specific particle and there may be a statistical spread over all results. The inclination to call these exact same statistics in other interpretations as "indefinite" in some scenarios (e.g. measurement results indicate equal superposition of spin up and down) comes from the assumption that all this stuff is talking about the property of a single particle. If you do that you cannot help but say that momentum and spin or momentum in these situations are indefinite. But under a statistical or stochastic interpretation, no claims are being made about a single particle so the idea of "indefiniteness" doesn't necessarily hold up except for in the kind of trivial notion that statistics have a spread, which is incontroversial and mundane. Conversely, particle ontology may always take definite positions, but their statistics can have a spread which is what in conventional interpretations would seemingly look like "indefinite" position. But again, this is not about a specific particle but statistics. When you take this into account, the A.I.'s claim that stochastic mechanical definite position is linked to KS realism is false; particle positions in stochastic mechanics can be indefinite in the statistical KS sense (i.e a statistical spread or uncertainty related to measurement interactions) while being definite ontologically for each particle.

Then there is also the following:

In stochastic mechanics, the current velocity v=ℏmIm(∇ψψ)v = \frac{\hbar}{m} \text{Im} \left( \frac{\nabla \psi}{\psi} \right)v = \frac{\hbar}{m} \text{Im} \left( \frac{\nabla \psi}{\psi} \right) depends on the wave function, which encodes global information about the quantum system. This raises the question of whether such a velocity, if real, implies non-locality. — sime

If ( v ) is a real property, this dependence suggests non-locality, as the velocity of one particle is instantaneously influenced by the state or position of another, without a local physical mechanism. This is analogous to the non-locality in Bohmian mechanics, where the velocity of a particle is guided by the non-local quantum potential or wave function — sime

But in entangled or multi-particle systems, the wave function’s global nature introduces non-local dependencies, even though the stochastic evolution of each particle’s position is locally governed. — sime

Yes, wavefunction and velocity fields are global and carry global information but they are not physical things. They are epistemic descriptions of statistics.

It is right Markovian stochastic mechanics is non-local as described in terms of instantaneous influences. But the non-Markovian reciprocal process version by Levy & Krener does not have this property at all, and explains it away as an artifact of an Markovian idealization. It then reproduces the correct behaviors without explicit Bohmian non-locality also responsible for configuration space descriptions where distant particles depend on each other.