I A new realistic stochastic interpretation of Quantum Mechanics

[jbergman]

The assertion is that all quantum systems are equivalent to indivisible (also dubbed as 'generalized') stochastic systems - stochastic systems whose time-evolution cannot be arbitrarily divided up in terms of stochastic matrices for intermediate sub-intervals. Barandes' idea then is to show that any indivisible system is a subsystem of a unistochastic system, via something called the Stinespring Dilation Theorem. The unistochastic system can then be translated into a unitarily evolving quantum system in virtue of its definition. So the implication is that all of the behaviors of quantum systems should be expressible in terms of indivisible stochastic systems - or vice versa. For instance, the statistical discrepancy due to violations of divisibility  is interference, specifically corresponding to the kind of interference you would have for trajectories in the path integral formulation (I believe).

Barandes then describes a rudimentary entanglement entirely from the perspective of these indivisible stochastic systems that you can translate a quantum system into. Correlations from local interactions between different stochastic systems results in a non-factorizable composite system. The indivisibility of the composite stochastic system's transition matrix means that it cumulatively encodes statistical information so that the correlation is effectively remembered over time (even with spatial separation) until the composite system experiences a division event (i.e. decoheres and divisibility is momentarily restored) by interacting with another system. The mechanism of entanglement for the indivisibe stochastic system is then arguably due to non-Markovianity - memory - rather than some kind of overt communication.

I am in the process of skimming the paper and I for one would like to dig deeper into the math to see if what he is claiming is even correct before discussing the implications of it. Are you convinced that what he is claiming is true?

 

[iste]

The big problem in the inference picture not that the obserer distorts the system, but if you account for the systems back reaction on the observing context. This is trivial when the observing context is dominant. But when you add gravity, and want to understand unification without ad hoc fine tuning, this is the major problem; which I think suggest that we need to provide the other half of the story as well.


I get what you are saying and agree, but this is half the answer. My reactions are due to that we often seem that alot ot approached supply exactly that - only half the answer. This half of the answer represents the conventional external perspective, where "the whole macroscopic environment" observes a small subsystem (the quantum system). This conceptual framework works for small subsystems, which is where QM is corroborated as it stands.

But when the systems get big enough to cause non-trivial feedback into the "baground observer, macroscopic envirionment"; we tend to fall back to semiclassical models.

So half answer, can make us understand the action of small subsystem, embedded in a dominant context where we have perfect control; and the smaller subsystems we need to explain, the more finetuning to de need to make in the embedding, even to the point nwhere we can't handle it.

So the missing part of the answer is; what about trying to describe the situation from the inside. Then explanatory models can not be rooted in fictional background contexts. Randomness in this inside view rather does not need explanation, as somehow randomness is the "nullhypothesis", because the starting point is a simple stupid agent INSIDE a black box; not OUTSIDE the black box. On the contrary deviation from randomness is what needs to be "explained". And this is the missing part for me... and I couldn't see Barandes offering any new grips on this...

In my view, I do not think in terms of that "something" must be causing the random motion; I see this as artifacts of the external view.

I prefer to think that inside view is that the randomness is simply a manifestation of that the observes can not predict it - the obserever is indifferent to the cause because the mechanism can not be distinguished. This should even mean that the observers actions uncouple with these details; suggesting that that the phenomenology of interactions change as the observational scale does. The problem is that the explanatory power of normal renormalization flow works so that knowledge of a complex detailed microstructure of the macrostate "explains" the reduced interactions as you goto the macroscale. So it's redutionist in nature. But it offers littel insight into the emergence logic.

/Fredrik

Well its clear we have very different perspectives. I'm happy to endorse kinds of observer perspectives along the lines of those people who frame things in terms of statistical inference, Bayes, information theory... like Jaynes or Karl Friston in biology. There is even a version of stochastic quantum mechanics along these kinds of lines by Ariel Caticha. But these perspectives are not really radical like yours are of qbists or whoever else. I'm more along the lines of Barandes who think that: if you can explain stramge observer phenomena in quantum mechanics in a kind of mundane, stochastic way as a generic feature of physical interactions then there is no need to continue thinking about radical kinds of observers. But clearly these kinds of observers are a foundational part of your metaphysics.

 

[iste]

I am in the process of skimming the paper and I for one would like to dig deeper into the math to see if what he is claiming is even correct before discussing the implications of it. Are you convinced that what he is claiming is true?

Do I know what he is saying is correct? Can I tell you anything about his math being correct and sound? Absolutely not. Am I convinced by his presentation and walkthrough of quantum phenomena in terns of an indivisible stochastic system? Yes. One thing I am sure of is that Nelsonian stochastic mechanics has shown that quantum theory can be expressed in terms of stochastic processes, so that helps brings more confidence in what Barandes is saying.

 

[mitchell porter]

@iste #249, thanks... I won't be sure until I have seen how Barandes formulates something specific like a hydrogen atom, but this sounds like it's just quantum mechanics in which you only speak about observables? e.g. everything is expressed in terms of transition probabilities between observables. Which is fine except that I don't think it would add anything philosophically. For example, it wouldn't imply any particular position about the values of unobserved physical properties, and yet that's the issue at stake regarding the uncertainty principle, local hidden variables, and so on.

 

[iste]

@iste #249, thanks... I won't be sure until I have seen how Barandes formulates something specific like a hydrogen atom, but this sounds like it's just quantum mechanics in which you only speak about observables? e.g. everything is expressed in terms of transition probabilities between observables. Which is fine except that I don't think it would add anything philosophically. For example, it wouldn't imply any particular position about the values of unobserved physical properties, and yet that's the issue at stake regarding the uncertainty principle, local hidden variables, and so on.

Yes, it is supposed to be the same as quantum mechanics but it would mean an underlying interpretations of particles in definite positions or configurations at every point in time even when not being observed. In his perspective, you can translate those indivisible transition probabilities into divisible transition amplitudes where you can construct and sum over paths just like in the path integral formulation - Barandes very briefly mentions the connection in one paper. So basically the perspective is saying the path integral trajectories are actual paths that particles take even when they are not observed. The uncertainty principle would be manifest in the long run statistics when repeating some experiment. Local hidden variables are avoided because even though there are definite outcomes, their statistics will still preclude the kinds of joint probability distributions required for those local beables, hidden variables, due to indivisibility and the fact that non-commuting properties, uncertainty principle still applies in a statistical sense.

 

[Fra]

Well its clear we have very different perspectives. I'm happy to endorse kinds of observer perspectives along the lines of those people who frame things in terms of statistical inference, Bayes, information theory... like Jaynes or Karl Friston in biology. There is even a version of stochastic quantum mechanics along these kinds of lines by Ariel Caticha. But these perspectives are not really radical like yours are of qbists or whoever else.

Ariel's nice papers on "physics from inference" along with ET jaynes is one of the papers that inspired me way back, he also used to have an idea to infer GR from inference, but I am indeed taking things one step further. On the technical side, the point where I diverge from Ariels and Jayens "choice of measure" of the observers ignorance is the real numbers. To introduce an uncountable embedding, is genereal, but brings alot of trouble and we loose control. From there on, alot of the mathematical conclusions rests on the real numbers as a good foundation.

This what I tried to mention before that I see there are essentialy 2 ways of emergence.
(1) "entropic" emergence, or regular stochastics; this is a top-down approach to
(2) some self-organisation; this is a botton-up approach

I don't think they are in conflict thougth, but to get the full answer we need to analyse both. The second think is what saves us from fine tuning.

Som objective bayesians focus more on first case, I thikn subjective bayesian more on second. The objective approach try to explain all this by fininding a microstructure where we can identify the observers, and explain objectively the conditional probabilities. But this requires a massive structure - that comes from where?

The subjective approach, can considers the observer to self-organize. In therms of comp sci, I suppose I think of the qbist agent as a evolutionary reinforced learner. The hypothesis is that, just like in biology, that the our elementaryt particles - and their properties - are emergent.

Ie ideas like this

The self-organization of selfishness: Reinforcement Learning shows how selfish behavior can emerge from agent-environment interaction dynamics​

This would be a bottom up appraoch, but the top-down stochastic perspect may is be complementary in the case where one large observer observs a small subsystem. It doesn't have to be either or. They represent different perspectives.

"Selfishness" or "self-preservation" and emergent cooperation, are it seems the keys to emergence of type 2.

This is much more radical than the entropic style emergence of objective bayesian thinking.

So the connection to Baranders for me was this...

It seems tha this idea allows transforming from a hilbert space / hamiltonian perspective, to a stochastic perspective where the system evolution is randomg but probabilistically codified directly in transition probabilities? An for those pondering about unification, this makes us work directly with pondering how the transition probabilities emerge rather han indirectly how hamiltonians emerge?

I see this as preferred, but big questions still remains. It seems to me Baranders still assumes the existence of objectivity.

The second idea of emergence (2) is which I see as a way to explain the emergence of the transition probability matrixes baranders mentions. Then working with transittion probabilities, that are "attached" to agents/observers would perhaps be an easier ontology for applying evolutionary reinforced learning, than doing the same in hilbert space! which indeed seems less intuitive. So this this sense I see a value to baranders ideas, but it is only half the story...

/Fredrik

 

[mitchell porter]

you can translate those indivisible transition probabilities into divisible transition amplitudes where you can construct and sum over paths just like in the path integral formulation - Barandes very briefly mentions the connection in one paper

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?

 

[PeterDonis]

an underlying interpretations of particles in definite positions or configurations at every point in time even when not being observed.

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.

 

[Demystifier]

But is it useful? There is a difference between used and useful.

It's useful, it helped me to understand and accept the pragmatic instrumental interpretation.

 

[martinbn]

It's useful, it helped me to understand and accept the pragmatic instrumental interpretation.

I meant useful in a more objective way. Anything can be useful to someone, but is it useful to the subject. Can it be used to solve problems that otherwise cannot? Or at least can it be used to provide simpler solutions?

 

[Demystifier]

I meant useful in a more objective way. Anything can be useful to someone, but is it useful to the subject. Can it be used to solve problems that otherwise cannot? Or at least can it be used to provide simpler solutions?

It depends on what one means by "problems". BM provides a simpler (compared to standard QM) solution to some conceptual problems, even if it does not help in practical, technical and quantitative problems.

 

[martinbn]

It depends on what one means by "problems". BM provides a simpler (compared to standard QM) solution to some conceptual problems, even if it does not help in practical, technical and quantitative problems.

Can you give examples?

 

[Fra]

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?

You can ask..

...but it means that you can not ask about probabilities midprocess AND presume that the answers necessarily forms a disjoint partition of sample space; so that you later can just use the law of total probability to divide the process.

This is also what IS implicitly assumed in bells ansatz wrt the hypothetical probablities for the hidden variables.

/Fredrik

 

[gentzen]

Is there any actual discussion of Barandes' papers in this thread ?

It seems to me that the entire thread is off-topic, but I haven't read every single post.

Indeed, this thread seems to go into off-topic territory again :-)

It depends on what one means by "problems". BM provides a simpler (compared to standard QM) solution to some conceptual problems, even if it does not help in practical, technical and quantitative problems.

Can you give examples?

How about
"... some Bohmians (Goldstein, Tumulka, et al) attacked that problem head on (https://arxiv.org/abs/quant-ph/0309021):

It is thus not unreasonable to ask, which mu, if any, corresponds to a given thermodynamic ensemble? To answer this question we construct, for any given density matrix rho, a natural measure on the unit sphere in H, denoted GAP(rho). We do this using a suitable projection of the Gaussian measure on H with covariance rho.

Sadly, Jozsa, Robb, and Wootters already constructed the same measure in 1994 (https://journals.aps.org/pra/abstract/10.1103/PhysRevA.49.668), as I learned from a follow up paper by the Bohmians (https://arxiv.org/abs/1104.5482)."
which I mentioned in
https://www.physicsforums.com/threads/the-interpretation-of-probability.1016226/post-6643974 ?

Back then, when I was trying to solve that problem, I googled around (hoping to find at least some "official name" of that problem) and found papers trying to also solve that problem, like:
https://www.uni-kassel.de/fb10-physik/uk003909/Papers/PRE_100_052105_2019.pdf
I wrote to one of the authors (because I still didn't even know a name, much less a solution), and he pointed me to one of their earlier publications, which investigated
A. Grid-based random phase approach
B. Eigenfunction-based random phase approach
C. Gaussian random phase wave packets / freely propagated random phase
wavepackets
https://pubs.aip.org/aip/jcp/articl...emtosecond-two-photon-photoassociation-of-hot (https://arxiv.org/abs/1201.1750)
A. and C. actually failed for them, because they were not aware that there was a way to do this "correctly" discovered by Jozsa, Robb, and Wootters in 1994, and rediscovered by some Bohmians in 2003.

 

[gentzen]

Can you give examples?

One example where the mathematical formalism of BM could turn out to useful is reciprocity and time symmetry: The mathematical formalism of BM is still time symmetric, before the Bohmians came along and destroyed that symmetry by their attempts to interpret that formalism.

Most instrumentatistic interpretations (and especially Copenhagen) on the other hand have time asymmetry already baked in. Sometimes in a very transparent manner, where the past preparation has finite entropy, but the future measurements have potentially infinite entropy. But more often than not, the time asymmetry is completely entangled with the interpretation.

 

[martinbn]

Indeed, this thread seems to go into off-topic territory again :-)



How about
"... some Bohmians (Goldstein, Tumulka, et al) attacked that problem head on (https://arxiv.org/abs/quant-ph/0309021):

Sadly, Jozsa, Robb, and Wootters already constructed the same measure in 1994 (https://journals.aps.org/pra/abstract/10.1103/PhysRevA.49.668), as I learned from a follow up paper by the Bohmians (https://arxiv.org/abs/1104.5482)."
which I mentioned in
https://www.physicsforums.com/threads/the-interpretation-of-probability.1016226/post-6643974 ?

Back then, when I was trying to solve that problem, I googled around (hoping to find at least some "official name" of that problem) and found papers trying to also solve that problem, like:
https://www.uni-kassel.de/fb10-physik/uk003909/Papers/PRE_100_052105_2019.pdf
I wrote to one of the authors (because I still didn't even know a name, much less a solution), and he pointed me to one of their earlier publications, which investigated
A. Grid-based random phase approach
B. Eigenfunction-based random phase approach
C. Gaussian random phase wave packets / freely propagated random phase
wavepackets
https://pubs.aip.org/aip/jcp/articl...emtosecond-two-photon-photoassociation-of-hot (https://arxiv.org/abs/1201.1750)
A. and C. actually failed for them, because they were not aware that there was a way to do this "correctly" discovered by Jozsa, Robb, and Wootters in 1994, and rediscovered by some Bohmians in 2003.

This is a lot to absorb but on first glance I cannot see anything specifically Bohmian here?

One example where the mathematical formalism of BM could turn out to useful is reciprocity and time symmetry: The mathematical formalism of BM is still time symmetric, before the Bohmians came along and destroyed that symmetry by their attempts to interpret that formalism.

Most instrumentatistic interpretations (and especially Copenhagen) on the other hand have time asymmetry already baked in. Sometimes in a very transparent manner, where the past preparation has finite entropy, but the future measurements have potentially infinite entropy. But more often than not, the time asymmetry is completely entangled with the interpretation.

What problems does this help solve?

 

[gentzen]

This is a lot to absorb but on first glance I cannot see anything specifically Bohmian here?

It is the topic of how to interact with Density Matrices in BM. There is also another approach to that topic, from the "same clique of Bohmians":
On the Role of Density Matrices in Bohmian Mechanics, with D. Dürr, R. Tumulka, and N. Zanghì, Foundations of Physics 35, 449-467 (2005), quant-ph/0311127

 

[martinbn]

It is the topic of how to interact with Density Matrices in BM. There is also another approach to that topic, from the "same clique of Bohmians":
On the Role of Density Matrices in Bohmian Mechanics, with D. Dürr, R. Tumulka, and N. Zanghì, Foundations of Physics 35, 449-467 (2005), quant-ph/0311127

Ok, but how does this answer my question? You are just listing papers by bohmians. So they define what conditional density matrix is. My question remains, what problems does this solve?

 

[gentzen]

This is a lot to absorb but on first glance I cannot see anything specifically Bohmian here?

Ok, but how does this answer my question? ... My question remains, what problems does this solve?

Which question? I quoted the question from you which I tried to answer. From my perspective, the question didn't remain the same.

Maybe it helps you, if I selectively quote parts of the paper title and its abstract:

... photoassociation of hot magnesium atoms ...​

... model the thermal ensemble of hot colliding atoms ...

One example where the mathematical formalism of BM could turn out to useful is reciprocity and time symmetry: ...

What problems does this help solve?

I could go into more concrete details, but I am unsure whether it would not just trigger again your "a lot to absorb" reaction. As soon as it gets concrete, there is always much stuff to absorb.

 

[jbergman]

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?

I don't think so, but maybe someone more informed can answer. As I understand it, indivisibility is just a condition on the evolution of the transition matrix.

For instance, he describes $\Gamma(n \delta t) = \Gamma(0)^n$ as an example of a divisible evolution of the transition matrix.

I think the typical Unitary evolution we associate with QM can be formulated as indivisble evolution of the transition matrix.

 

[Fra]

IMO the "conceptual value" added seems to mainly be that we can describe or "phrase" what makes "quantum inference" possible, simply in terms of conditional probabilites, without using the hilbert space ontology and complex wave functions.

One can view the non-commutativity in complex hilbers space, instead as a kind of "statistical dependence". I think that is good.

But all other hard problems remains and are not solved as far as I see.

/Fredrik

 

[physika]

Baranders still assumes the existence of objectivity

...So ?

 

[Fra]

...So ?

Most do, and use it like a constraint, so not unusual of course.

But for those having an observer centered starting point, usually try to explain effective objectivity in the sense of observer equivalence, via some emergent asymptotic process.

I had hoped Barandes had something of this up his sleves, but seems not.

/Fredrik

 

[Demystifier]

Can you give examples?

After decoherence, why does only one branch of the wave function becomes physically relevant? In other words, where does the collapse rule come from? The Bohmian answer to this conceptual question is much clearer than the standard one.

 

[Demystifier]

@gentzen concerning papers about foundations of statistical physics by Bohmians that you mentioned, have you seen my https://arxiv.org/abs/2308.10500 ?

 

[martinbn]

After decoherence, why does only one branch of the wave function becomes physically relevant? In other words, where does the collapse rule come from? The Bohmian answer to this conceptual question is much clearer than the standard one.

This is a question about interpretations. I wanted to see an example of a problem that BM solves and QM, not some interpretation of QM, doesn't solve. Or at least gives an easier/better solution. More along the examples of @gentzen , but his examples are not clear to me.

 

[martinbn]

Maybe it helps you, if I selectively quote parts of the paper title and its abstract:

... photoassociation of hot magnesium atoms ...​

... model the thermal ensemble of hot colliding atoms ...

Is this the example? Is this something that is handled in BM better than in QM?

 

[gentzen]

Is this the example? Is this something that is handled in BM better than in QM?

Yes, it is the concrete example that I alluded to. The concrete problem is not easy, so just using the solution suggested by BM would fix A. and C., but this doesn't mean that it will be the best solution for this problem. The considered approaches were

A. Grid-based random phase approach
B. Eigenfunction-based random phase approach
C. Gaussian random phase wave packets / freely propagated random phase
wavepackets

I guess the best solution also depends on the gaps between the Eigenvalues, i.e. if there are some dominant Eigenvalues clearly separated from the rest, then it is better to handle those explicitly, and only use the canonical measure on the rest of the spectrum of Eigenvalues.

So the role of BM here is to add a specific perspective to your arsenal available when tackling concrete problems. For the concrete problem, there is no clear cut right or wrong, best or optimal. But for BM, there was an important question, where already the fact that it had a concrete answer was important, and in addition it turned out that the concrete answer was even practical to a certain extent. (If you look at the details, there is a rejection samping part caused by the required normalization of the Gaussian measure, i.e. the "adjusted" in "Gaussian adjusted projected measure". It would have been nice if that could have been avoided, but I guess it cannot.)
Another part of BM was that it made it easier to search for that solution, because it is a "canonical" problem in that context, so you expect that other people might have tried to tackle it already. For the concrete problem on the other hand, there was rather the expectation that even if other people had struggled with similar problems, it would be hard for you to find their publications.

 

[Demystifier]

This is a question about interpretations. I wanted to see an example of a problem that BM solves and QM, not some interpretation of QM, doesn't solve. Or at least gives an easier/better solution. More along the examples of @gentzen , but his examples are not clear to me.

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?

 

[Lord Jestocost]

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).”

 

[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.

 

[iste]

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.

Yes, it is very general and minimalist but it gives interpretation at least in the following sense. Because we are talking about quantum mechanics as being stochastic systems, just this very fact, if it were true, would imply that the system is always in a definite configuration at any point in time. So at the very least you would have a picture of the universe full of particles that are always in one place at a time but they just move randomly. I would say that is definitely interpretational in a minimal way.

In two of Barandes' papers he mentions the mechanism for entanglement being the fact that correlations induced by local interactions between two different stochastic systems are remembered over time until the system is later disturbed (e.g. by measurement devices), after which it basically forgets what had happened in the past at the original local interaction. This is purely because the indivisible transition matrix is non-Markovian - divisibility or division events means it no longer has these memory properties. There is no superdeterminism because the correlation is solely due to the local interaction. Any correlations in the measurement devices are solely due to the fact that the correlation from the original local interaction is remembered; the devices do not causally influence each other over distances independently of this. It is very general though. His entanglement examples I don't think give strong insight to entangled polarization / spin experiments.

 

[Demystifier]

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

They are only constrained by their initial and final points. It is in fact somewhat wrong to think about them as paths. They are functions $x(t)$, a function can be totally weird, like $x(t)=0$ for rational $t$ and $x(t)=1$ for irrational $t$. The "path" integral is really the functional integral, i.e. the integral over all functions.