A new realistic stochastic interpretation of Quantum Mechanics

[iste]

We already know what such an interpretation of QM looks like: it looks like Bohmian mechanics. Which is about as far from stochastic as you can get.

Not at all; as I already said in that post, the trajectories of stochastic particles that take up definite positions in a stochastic interpretation are quite literally the physical realization of the trajectories in the path integral formulation. The fact that the path integral trajectories are generally non-differentiable is something also shared with Brownian motion - i.e. they look like particles traversing continuous paths but where their direction of motion is being constantly disturbed.

 

[iste]

So suppose there's a gap of two seconds between observations, and I want to think about what was happening one second after the first observation. Am I just not allowed to ask about probabilities for unobserved properties? Is that what indivisibility means?

Well indivisibility would apply to both unobserved and observed scenarios - observation consists in adding an additional stochastic subsystem whose role is as a measurement device. But yes, indivisibility means you would not be able to talk about probabilities for the intermediate time (one second) conditioned on the initial time. More specifically, it means there is no unique joint probability distribution that includes the intermediate 'one second' time from which you can construct transition probabilities for the 'two seconds' time.

The following paper which Barandes cites talks about this pg. 12 - 15 (where divisibility is also mentioned) and pg. 35-38. So the ability to construct marginal probabilities from unique joint probability distributions in stochastic processes is talked about in terms of Kolmogorov conaistency conditions / extension theorem here. Divisibility can be seen as a special case of that which breaks down in quantum mechanics.

https://scholar.google.co.uk/scholar?cluster=8084926175613713567&hl=en&as_sdt=0,5

I am not entirely sure what the unobserved case means but I think this paper, even just reading the abstract, gives I think a nice picture of what indivisibility means in quantum mechanics with measurements:

https://scholar.google.co.uk/scholar?cluster=251868338556687708&hl=en&as_sdt=0,5

And you see there that it is intimately related to contextuality because contextuality is characterized by similar kinds of joint probability violations to indivisibility.

 

[PeterDonis]

the trajectories of stochastic particles that take up definite positions in a stochastic interpretation are quite literally the physical realization of the trajectories in the path integral formulation.

But each individual particle only has one trajectory. It doesn't have all of them. And each individual trajectory is still deterministic, since, as you say, it's just one of the paths in the path integral. Which is exactly the same as Bohmian mechanics.

 

[iste]

But each individual particle only has one trajectory. It doesn't have all of them.

Yes, a particle would not go along all trajectories simultaneously, it can only take one. But then if you repeat some experiment or situation ad infinitum then you will see that eventually all possible trajectories will be taken over the course if repetition.That all possible paths will be taken exactly exemplifies the fact that it is not deterministic.

Which is exactly the same as Bohmian mechanics.

Not sure what you mean. Bohmian trajectoried are not the same as the ones that show up in the path integral formulation.

 

[PeterDonis]

That all possible paths will be taken exactly exemplifies the fact that it is not deterministic.

No, it does nothing of the sort. The same thing occurs in Bohmian mechanics. It has nothing to do with non-determinism in the particle trajectories. It only has to do with the randomness of the initial conditions: the initial positions of the particles are randomly assigned, so that over a large number of experiments, a random distribution of the possible initial positions, and hence of the possible paths, will be sampled.

 

[PeterDonis]

Bohmian trajectoried are not the same as the ones that show up in the path integral formulation.

Why not?

 

[iste]

It only has to do with the randomness of the initial conditions: the initial positions of the particles are randomly assigned, so that over a large number of experiments, a random distribution of the possible initial positions, and hence of the possible paths, will be sampled.

This may be the case in Bohmian mechanics with smooth deterministic trajectories but it is not the case for Path integral trajectories which zig-zag is around randomly and constantly. The fact that path integral trajectories are non-differentiable is inconsistent with the guiding equation of Bohmian trajectories. At the same time, the average velocity of path integral trajectories is the same as the velocity that deterministically guides and shapes Bohmian trajectories (i.e. the two formulation's trajectoried relate to the same velocity in very different ways). They cannot be the same object, and path integral trajectories are fundamentally stochastic as mentioned in passing in the paper below:

https://www.mdpi.com/1099-4300/20/5/367

There are some nice images in the following that depict how they look very different. Its comparing Bohmian and stochastic mechanics trajectories but stochastic mechanics trajectories are identical to path integral trajectories:

https://arxiv.org/html/2405.06324v1

 

[martinbn]

I don't know such an example, but I also don't know an opposite example, for which standard QM is better than BM. Do you?

No, I don't. My guess would be any example where in addition to the Schrodinger equations you need to work with the equations for the position, and they add difficulty to the solution. Or any problem that is better in any other basis than the position basis.

But why do you ask for such an example?

 

[martinbn]

There is indeed no physical problem that BM solves and QM not. However, David Bohm and Jeffrey Bub pointed out to their uneasy feeling with regard to the orthodox interpretation of QM. In their paper “A Proposed Solution of the Measurement Problem in Quantum Mechanics by a Hidden Variable Theory“ (Rev. Mod. Phys. 38, 453, 1966), they wrote:

“It is not easy to avoid the feeling that such a sudden break in the theory (i.e., the replacement, unaccounted for in the theory, of one wave function by another when an individual system undergoes a measurement) is rather arbitrary. Of course, this means the renunciation of a deterministic treatment of physical processes, so that the statistics of quantum mechanics becomes irreducible (whereas in classical statistical mechanics it is a simplification – in principle more detailed predictions are possible with more information).”

To me that shows that they had a problem accepting that nature can be probabilistic. But nature could be that way, and all we know so far suggests it is that way. So this is more their problem than a problem of QM.

 

[Demystifier]

But why do you ask for such an example?

Because you think that standard QM is better than BM, so I wondered if you could back that up with an example.

 

[pines-demon]

It is frustrating how Barandes can be so eloquent and at the same time not give straight answer to what is the mental picture to have here. If he cannot do that, then it is just another obscure rewritting of QM that does not allow for interpretations. Like ok, you are allowed to violate Bell's inequalities but how should I think of it? He ask us to exchange the wavefunction and collapse for an all permeating fluctuating force. Does this force updates faster than light to produce entanglement results? Or is there a memory effect from the past over large regions of space that allows us to measure QM-like effects (conspiracy, superdeterminism)?

It would be really helpful if Barandes just gave a course solving an example with a single qubit and then and two entangled qubit example. His formalism seems very general, it does not depend on fundamental objects being qubit, particles or fields, and the non-classicality is encoded in his indivisibility. So it also does not give any new insight on the fundamental nature either.

Edit: thinking more about it, I think Barandes just stumbled into duality, it could be helpful if it can be used to to solve non-Markovian problems with quantum mechanics and viceversa. Nevertheless, calling it an interpretation is flawed at this stage, it is like calling a Wick rotation an interpretation.

 

[martinbn]

Because you think that standard QM is better than BM, so I wondered if you could back that up with an example.

This is a question for someone else, most physicists may be. But it is a strange question! QM was developed first, then came BM. So it is BM that need to show what new it has to offer. Any way I gave you two reasons why QM is better. 1) It doesn't have huge number of additional equations and 2) It ca work in any basis not just the position basis. Don't you think these are advantages?

 

[PeterDonis]

path integral trajectories are non-differentiable

Are they? Don't they still have to be solutions of a differential equation?

 

[Demystifier]

This is a question for someone else, most physicists may be. But it is a strange question! QM was developed first, then came BM. So it is BM that need to show what new it has to offer.

I cannot beat that argument, so I wrote this:
https://arxiv.org/abs/physics/0702069

Any way I gave you two reasons why QM is better. 1) It doesn't have huge number of additional equations and 2) It ca work in any basis not just the position basis. Don't you think these are advantages?

Yes, these are advantages. But BM has corresponding counter-advantages:
1) It doesn't have huge number of additional collapses (one collapse whenever a measurement happens).
2) It can derive the Born rule in any basis from Born rule in the position basis.
Don't you think these are advantages too?

 

[Demystifier]

Are they? Don't they still have to be solutions of a differential equation?

No, path integral trajectories are not solutions of a differential equation.

 

[PeterDonis]

No, path integral trajectories are not solutions of a differential equation.

Then what constraints do they have to obey? Do they just have to be connected?

 

[renormalize]

Don't they still have to be solutions of a differential equation?

No, since for a Feynman path integral, a general path is nowhere differentiable. Here's a recent reference that discusses this:
https://link.springer.com/article/10.21136/CMJ.2024.0493-22
Non-differentiability of Feynman paths by Pat Muldowney
Abstract:
A well-known mathematical property of the particle paths of Brownian motion is that such paths are, with probability one, everywhere continuous and nowhere differentiable. R. Feynman (1965) and elsewhere asserted without proof that an analogous property holds for the sample paths (or possible paths) of a non-relativistic quantum mechanical particle to which a conservative force is applied. Using the non-absolute integration theory of Kurzweil and Henstock, this article provides an introductory proof of Feynman’s assertion.

 

[PeterDonis]

for a Feynman path integral, a general path is nowhere differentiable.

Thanks for the reference--another thing to add to my already overloaded reading list.

This property of general paths is not one I have seen discussed in what I have read previously; I suspect that is because, in practice, the contributions of such paths to actual amplitudes is negligible, so they are mostly ignored. I don't know how that practical issue affects the interpretation under discussion in this thread.