A Understanding Barandes' microscopic theory of causality

[iste]

Beables have diagonal matrices wrt configurations, as they can be read off from the existing configuration (see equation 19 in the correspondence paper). Emergeables don't, and hence are given meaning by a measurement context.

But it doesn't seem to me that there is any preferred basis of configurations. Surely, Barandes formulation doesn't stop you from creating beables with a configuration space for any quantum observable? Moreover, the diagonal vs. non-diagonal aspect I am not sure is relevant because under Barandes' formulation, beables and emergeables act similarly with regard to the measurement device and you would assume always produce definite outcomes, and regardless of indivisibility or divisibility, your stochastic process always produces definite outcomes. I still don't understand how the distinction between beable and emergeable is anything other than perspectival.

Regarding your second quote, I don't see you refuting the idea that the configuration space can't be describing a counterfactual ontology like the fisherman example.

 

[Morbert]

But it doesn't seem to me that there is any preferred basis of configurations. Surely, Barandes formulation doesn't stop you from creating beables with a configuration space for any quantum observable? Moreover, the diagonal vs. non-diagonal aspect I am not sure is relevant because under Barandes' formulation, beables and emergeables act similarly with regard to the measurement device and you would assume always produce definite outcomes, and regardless of indivisibility or divisibility, your stochastic process always produces definite outcomes. I still don't understand how the distinction between beable and emergeable is anything other than perspectival.

Barandes's formalism involves a standard classical configuration space $\mathcal{C}$. Whether it is, for example, a space of particle or field configurations depends on the theory you are constructing a correspondence to. Observables that are not diagonal wrt these configurations cannot be read off from these configurations, and are hence not beables.

Regarding your second quote, I don't see you refuting the idea that the configuration space can't be describing a counterfactual ontology like the fisherman example.

You are free to construct alternative ontological models, just as you are free to construct them for ordinary Markovian stochastic processes. For the purposes of this thread I am discussing the one presented by Barandes.

 

[iste]

Barandes's formalism involves a standard classical configuration space $\mathcal{C}$. Whether it is, for example, a space of particle or field configurations depends on the theory you are constructing a correspondence to. Observables that are not diagonal wrt these configurations cannot be read off from these configurations, and are hence not beables.You are free to construct alternative ontological models, just as you are free to construct them for ordinary Markovian stochastic processes. For the purposes of this thread I am discussing the one presented by Barandes.

Yes, but my point is that surely position and momentum basis in quantum mechanics would each be translated to respective classical configuration bases where they are beables.

Again, I don't think diagonal nature matters because indivisibility and interference itself is characterized by non-diagonality and yet the stochastic process produces definite realizations of configurations regardless. In DOI: 10.31389/pop.186 Barandes describes a beable as having a non-diagonal density matrix several times with regard to coherence and uncertainty principle sections.

Barandes does not present a specific ontological model other than the use of classical configurations. My point is that there is nothing stopping you using them to represent counterfactual classical configurations describing something that can only said to exist as a consequence of a measurement interaction.

 

[Morbert]

Yes, but my point is that surely position and momentum basis in quantum mechanics would each be translated to respective classical configuration bases where they are beables.

Show me what that configuration space (note, not phase space) would look like.

Again, I don't think diagonal nature matters because indivisibility and interference itself is characterized by non-diagonality and yet the stochastic process produces definite realizations of configurations regardless. In DOI: 10.31389/pop.186 Barandes describes a beable as having a non-diagonal density matrix several times with regard to coherence and uncertainty principle sections.

It's the observable itself that is diagonal or not diagonal. See equation 19.

Barandes does not present a specific ontological model other than the use of classical configurations. My point is that there is nothing stopping you using them to represent counterfactual classical configurations describing something that can only said to exist as a consequence of a measurement interaction.

Barandes's kinematic axiom is clear. Let's stick to it for this thread.

 

[iste]

Show me what that configuration space (note, not phase space) would look like.

It's the observable itself that is diagonal or not diagonal. See

Barandes's kinematic axiom is clear. Let's stick to it for this thread.

i'm thinking that maybe the issue is that the stochastic correspondence clearly overs more than just what you are referring to in terms of configuration basis then. Clearly you can give momentum and anything else you want a representation which is as a beable in the configuration basis is described, using the dictionary; and from the indivisible perspective, that must actually be the explanation for different measurement bases (all representable as indivisible stochastic processes). But once you are able to do this, I think it does really make it questionable whether the stochastic process in the formalism always has to have a stringently realist ontology if plausibly you can use it to describe things which may not have that interpretation in a straightforward way (or its at least ambiguous whether they do). Sure you can postulate about a preference for a configuration basis and so thats where the only real beables are, but I guess thats a difference betwern an assumption about what you  want your formulation to represent and the capabilities of what the formulation can represent; after all, Barandes thinks this formulation can potentially be used to describe systems in the special sciences from neuroscience to psychology to even more abstract things like financial systems perhaps.

 

[Morbert]

@iste You keep straying from Barandes's literature.

i) Barandes presents a kinematic axiom which says the system always has a configuration $i,\ldots,N$ in the configuration space $\mathcal{C}$ we use to model the system.
ii) Beables are the random variables $A(t) = \sum_i^Na_i P_i = \mathrm{diag(\ldots,a_i,\ldots)}$ which can be read off from the configuration the system is in.
iii) Emergeables cannot be read off from the configuration the system is in. Instead they mix in dynamical information and determine the probabilities for the configurations an ancillary measurement apparatus can evolve into should it interact with the system.

If you want to posit an alternative model, which places beables and emergeables on equal ontic footing, I wish you the best of luck.

 

[Morbert]

PS this conversation is also straying from the recent paper by Albert. Unless there's something novel and specific in your response I'll leave it here.

 

[iste]

@iste You keep straying from Barandes's literature.

i) Barandes presents a kinematic axiom which says the system always has a configuration $i,\ldots,N$ in the configuration space $\mathcal{C}$ we use to model the system.
ii) Beables are the random variables $A(t) = \sum_i^Na_i P_i = \mathrm{diag(\ldots,a_i,\ldots)}$ which can be read off from the configuration the system is in.
iii) Emergeables cannot be read off from the configuration the system is in. Instead they mix in dynamical information and determine the probabilities for the configurations an ancillary measurement apparatus can evolve into should it interact with the system.

If you want to posit an alternative model, which places beables and emergeables on equal ontic footing, I wish you the best of luck.

You don't need another model because its in the theory. The stochastic-quantum correspondence surely says that momentum basis and any other observable are describable as and translatable to an indivisible stochastic process in the same way as one would for the configuration basis beable. This would then give you the division events for these other "emergeables".

 

[Morbert]

You don't need another model because its in the theory. The stochastic-quantum correspondence surely says that momentum basis and any other observable are describable as and translatable to an indivisible stochastic process in the same way as one would for the configuration basis beable. This would then give you the division events for these other "emergeables".

I wish you the best of luck in showing this. Please be specific with your example. And please cite the relevant literature.

 

[Morbert]

@iste Rereading the convo, I think the confusion might be you think a classical configuration space is like a Hilbert space with many spectral representations where different observables are diagonal.

My classical physics is rusty, but: While you can do coordinate transformations on a configuration space, none will diagonalize emergeables.

 

[PeterDonis]

surely position and momentum basis in quantum mechanics would each be translated to respective classical configuration bases where they are beables.

Unfortunately that's not possible.

In classical physics, position and momentum each have their own independent configuration spaces. The full state of a single particle is described by a 2-tuple of points, one in the position configuration space and one in the momentum configuration space. In other words, in three spatial dimensions, the classical configuration space of a single particle is a 2-tuple of points, each point in a 3-dimensional space.

In quantum physics, position and momentum are operators on Hilbert space (not configuration space). The Hilbert space is the space of square integrable functions on configuration space. But the configuration space even for a single particle is not the classical configuration space described above, because position and momentum don't each have their own configuration spaces in QM. For a single particle in three spatial dimensions, the configuration space is simply the set of points in 3-dimensional space. Whether each of those points designates a position or a momentum (or something in between) depends on what basis you choose for the Hilbert space, the space of square integrable functions on configuration space.

So there's simply no way to "translate" position and momentum from a quantum state into classical configuration spaces. It won't work.

 

[iste]

I wish you the best of luck in showing this. Please be specific with your example. And please cite the relevant literature.

@iste Rereading the convo, I think the confusion might be you think a classical configuration space is like a Hilbert space with many spectral representations where different observables are diagonal.

My classical physics is rusty, but: While you can do coordinate transformations on a configuration space, none will diagonalize emergeables.

No, I did not have this confusion. I did have a confusion in the sense I was thinking about configuration as something arbitrary in regard to an arbitrary generalized stochastic process that could represent anything, not necessarily about physics even. I was not thinking in the sense about specific physical configuration space that can be contrasted to velocity or momentum.

We have the theorem saying that "every generalized stochastic system corresponds to a unitarily evolving quantum system on a Hilbert space". Barandes constructs a Hilbert space representation for positions / configurations in his examples. Given that the theorem is bi-directional, why can't you translate the Hilbert space representation of momentum into a generalized stochastic process where momentum effectively serves as the configuration for a generalized stochastic process? Where is the asymmetry that stops you doing that? In quantum mechanics, the position basis is not inherently mathematically fundamental if I'm not mistaken.

This paper is very obviously not about Barandes' formulation and I don't intend it to prove anything about Barandes' formulation:

https://link.springer.com/article/10.1007/s10701-024-00757-7

But it does give a good example of the kind of thing I am talking about. They have this method of constructing a stochastic process that corresponds to a quantum mechanical system. Usually, in this type of theory it is only applied in regard to position, acting as the preferred representation of the system. But they decide they want no preferred representation so they apply the method to momentum, where momentum acts as the co-ordinate; they say this on left hand side of page 4. Section III.C they describe their momentum representation which is a stochastic process for a momentum co-ordinate / configuration. This is the kind of thing I mean.

With the way Barandes constructs a position Hilbert-space representation for position from a stochastic process, the existence of an equally fundamental Hilbert representation for momentum in QM, and the bi-directionality of the stochastic-correspondence theorem, it seems to me that there should be a generalized stochastic process describing momentum (at least thats what its Hilbert space representation should correspond to) analogous to what they are doing in the above paper where the momentum space is represented by its own stochastic process.

Edit: Another short way to put the argument is maybe that: if the Hilbert space representation Barandes constructs for a generalized stochastic process is about the configuration of that process, then the bi-directionality of the stochastic-quantum correspondence implies that the quantum Hilbert space representation of momentum must also be about a generalized stochastic process whose configuration is the momentum.

 

[Morbert]

@iste In a previous post, I wondered out loud about other such correspondences.

In all the literature I am skimming, configuration space is presented as an intrinsic manifold, and phase space is constructed as a cotangent bundle on this manifold. I cannot find an instance, for example, of constructing a phase space from an intrinsic momentum manifold, so I do not know if classical physics is as protean as quantum physics. I also do not know if there are other correspondences, or a more general correspondence than the one Barandes presents. It is an interesting question.

But nevertheless, Barandes presents a correspondence where standard configuration spaces model the kinematics on the classical side of the correspondence. This will very clearly render some observables as intrinsic beables, and some as emergeables.

 

[iste]

@iste In a previous post, I wondered out loud about other such correspondences.

Aha, interesting. Seeing that post would have saved some time. I feel we have different interest.

But nevertheless, Barandes presents a correspondence where standard configuration spaces model the kinematics on the classical side of the correspondence. This will very clearly render some observables as intrinsic beables, and some as emergeables.

I am not sure I agree. I think Barandes has presented a correspondence that is the highest level of generality and it is not about physics inherently. He then uses this correspondence to construct a quantum theory which is as you say and has some preferred beables in contrast to emergeables.

My perspective would then be that Barandes' theory as you present it in terms of a physical interpretation is not satisfactory imo. Using Barandes' correspondence to produce a formulation agnostic about physical interpretation would be satisfactory when ignoring questions about underlying ontology. But I don't think the correspondence in and of itself adequately motivates the physical interpretation in a sufficiently compelling way even if I probably think there are some things I would agree with.

 

[Morbert]

I think Barandes has presented a correspondence that is the highest level of generality

He doesn't. He presents a correspondence between quantum theory and a unistochastic processes involving ordinary classical configuration spaces. If you want to generalize further, you're moving beyond his correspondence.

 

[iste]

He doesn't. He presents a correspondence between quantum theory and a unistochastic processes involving ordinary classical configuration spaces. If you want to generalize further, you're moving beyond his correspondence.

It is very general. His theorem is to show that "Every generalized stochastic system can be regarded as a subsystem of a unistochastic system.", with the entries of a unistochastic being modulus square of a unitary matrix. This covers any generalized stochastic process. This generality is why Barandes can and has stated that one of his interests is to see how it applies to areas outside of physics to see if quantum representations to those things has interesting implications (and quantum cognition is actually already a field of describing psychology using quantum representation, not because the mind is quantum but because some human behavior can be described in terms of something similar to contextuality or incompatibility). The theorem is more general than physics, but clearly he can use it to give a new description of quantum physics.

 

[Morbert]

But I don't think the correspondence in and of itself adequately motivates the physical interpretation in a sufficiently compelling way even if I probably think there are some things I would agree with.

i) Barandes presents a kinematic axiom which says the system always has a configuration $i,\ldots,N$ in the configuration space $\mathcal{C}$ we use to model the system.

This axiom offers up a natural ontological model: Systems have definite configurations at all times, as they do in Bohmian mechanics or ordinary classical physics.

Since this is interpretational, and quantum theory does not insist on a singular, unique interpretation, we are always free to reject such ontological models. If you are looking for an interpretation that necessarily follows from quantum theory, you will never find it.

 

[Morbert]

It is very general. His theorem is to show that "Every generalized stochastic system can be regarded as a subsystem of a unistochastic system.", with the entries of a unistochastic being modulus square of a unitary matrix. This covers any generalized stochastic process. This generality is why Barandes can and has stated that one of his interests is to see how it applies to areas outside of physics to see if quantum representations to those things has interesting implications (and quantum cognition is actually already a field of describing psychology using quantum representation, not because the mind is quantum but because some human behavior can be described in terms of something similar to contextuality or incompatibility). The theorem is more general than physics, but clearly he can use it to give a new description of quantum physics.

The first line in his abstract:

This paper argues that every quantum system can be understood as a sufficiently general kind of stochastic process unfolding in an old-fashioned configuration space according to ordinary notions of probability.

That is the correspondence he establishes.

 

[iste]

This axiom offers up a natural ontological model: Systems have definite configurations at all times, as they do in Bohmian mechanics or ordinary classical physics.

Since this is interpretational, and quantum theory does not insist on a singular, unique interpretation, we are always free to reject such ontological models. If you are looking for an interpretation that necessarily follows from quantum theory, you will never find it.

Yes, you won't find an interpretation that necessarily follows, but then I would rather support a formulation designed to do what it says it does rather than co-opt a theorem and try to turn it into something that it may not actually be about. If say the generalized stochastic process implied by a quantum system doesn't strictly imply anything more than an (edit: instrumentalist) representation of the measurement process using a stochastic process, then the small leap to representing physical configuration in the world (outside of measurement) feels kind of ugly and arbitrary imo and no less speculative than Bohmian or stochastic mechanical models which nonetheless are more ambitious and try to give a more complete physical account.

That is the correspondence he establishes.

Apologies I meant arXiv:2309.03085v1. And the quote in my post was from the statement of the theorem. The correspondence is between indivisible stochastic systems and unitary evolution which could in principle be describing things that have nothing to do with physics. I feel like for what you are implying to make sense, it would mean that generalized stochastic systems can only be describing physics, which clearly can't be the case imo. Its an abstract mathematical formalism. You can use a stochastic process to describe anything you want.

edited, noted in text

 

[Morbert]

If say the generalized stochastic process implied by a quantum system doesn't strictly imply anything more than an (edit: instrumentalist) representation of the measurement process using a stochastic process

A formalism does not imply any unique interpretation. A unistochastic process with a standard classical configuration space providing the kinematics can be interpreted as an operational model for instrumentalists or it can be interpreted ontologically. The latter requires no measurement contexts.

Apologies I meant arXiv:2309.03085v1.

From section III:

A generalized stochastic system will be defined to mean a tuple of the form (C, T , Γ, p, A) that consists of the following data.
• The symbol C will denote a set called the system’s configuration space, and the elements of C will be called the system’s (allowed) configurations. Configurations and configuration spaces will play an analogous role for generalized stochastic systems that states and state spaces play for generalized dynamical systems.
• The reason for switching the terminology from ‘states’ to ‘configurations’ is conceptual. In applications of the theory of dynamical systems to physical situations, such as in classical Hamiltonian mechanics, the notion of a ‘state’ is often taken to include rates of change or momenta in addition to configurations, because defining a ‘state’ in that way can make it possible to obtain deterministic laws in the form of first-order differential equations. For a generalized stochastic system, by contrast, the system’s probabilistic laws may mean that rates of change and momenta are not well-defined in the absence of specifying a trajectory. In that case, the only available notion of a ‘state’ is limited to the more rudimentary notion of a ‘configuration.

You are critiquing your very very loose and speculative conceptualization of a more general correspondence that might or might not exist. One that Barandes explicitly guards against here. Taken on its own terms, the correspondece actually existing in literature offers a natural ontological model if we want to avail of it.

 

[iste]

A formalism does not imply any unique interpretation. A unistochastic process with a standard classical configuration space providing the kinematics can be interpreted as an operational model for instrumentalists or it can be interpreted ontologically. The latter requires no measurement contexts.

Sure, formalism does not imply interpretation. But then, if formalism doesn't imply interpretation, it doesn't make sense to me to use the formalism to guide interpretation. Its kind of circular, pulling itself up by its own bootstraps, which doesn't seem coherent if the formalism was ambiguous in the first place. And Barandes' formulation does depend on the formalism because of the stochastic-quantum correspondence which implies QM corresponds to these stochastic processes. If QM doesn't say much more than what we see empirically, then the related indivsible stochastic processes might arguably be models about what we see empirically, without saying much else. If that is the case, it feels dissonant to me to use that model as a basis of ontology, as opposed to other kinds of models which are explicitly about ontology. Sure, as you say, you can interpret formalism any way you want and squeeze things out of it any way you want. But the real question is whether doing so is well-motivated. Bohmian mechanics is designed to be about some ontology of the world. Barandes' is using a ready-made formulation and squeezing ontology out of it such that the ontology is contingent on a model which is not suited for that, leading to ontologies which are inherently limited and incomplete.

 

[iste]

You are critiquing your very very loose and speculative conceptualization of a more general correspondence that might or might not exist. One that Barandes explicitly guards against here. Taken on its own terms, the correspondece actually existing in literature offers a natural ontological model if we want to avail of it.

No, I have stated the exact correspondence in the paper in quotes: a generalized stochastic system is a subsystem of a unistochastic system whose entries are the modulus square of a unitary matrix. You can use a unitary matrix to describe anything you want. You can use a generalized stochastic system to describe anything you want. I can show you papers using unitary and unistochastic transition matrices to model psychological phenomena, also using the same interference terms Barandes uses.

I don't really understand what you are trying to convey by the quotations you use or what Barandes is guarding against here. What you are calling a "natural ontological model" is just the sample space associated to random variables and can be used to describe anything you want. Its a triviality that random variables are about definite outcomes. There is no constraint on what they can be used to represent. You can even find generalized coordinates in neuroscience. As you say, formalism doesn't imply interpretation. Using the word or a formalism associated with 'configuration' doesn't necessarily imply anything until you have specified what you want it to describe. As said before, these same systems have already been used to describe things in psychology. The correspondence is fully general insofar that there is no entailment that a unistochastic process or a unitary matrix can only be used to describe a physical system. That is demonstrably false.

 

[Morbert]

No, I have stated the exact correspondence in the paper in quotes: a generalized stochastic system is a subsystem of a unistochastic system whose entries are the modulus square of a unitary matrix. You can use a unitary matrix to describe anything you want. You can use a generalized stochastic system to describe anything you want. I can show you papers using unitary and unistochastic transition matrices to model psychological phenomena, also using the same interference terms Barandes uses.

A generalized stochastic system will be defined to mean a tuple of the form (C, T , Γ, p, A).
[...]
A generalized stochastic system (C, T , Γ, p, A) whose stochastic map Γ defines a unistochastic matrix Γ(t) at every time t will be called a unistochastic system.

Note again that Barandes's reformulation involves a configuration space C for both general stochastic systems and unistochastic systems. This doesn't mean you can't try to find other uses for such transition matrices, but this thread is about what Barandes has published.

I don't really understand what you are trying to convey by the quotations you use or what Barandes is guarding against here. What you are calling a "natural ontological model" is just the sample space associated to random variables and can be used to describe anything you want. Its a triviality that random variables are about definite outcomes. There is no constraint on what they can be used to represent. You can even find generalized coordinates in neuroscience. As you say, formalism doesn't imply interpretation. Using the word or a formalism associated with 'configuration' doesn't necessarily imply anything until you have specified what you want it to describe. As said before, these same systems have already been used to describe things in psychology. The correspondence is fully general insofar that there is no entailment that a unistochastic process or a unitary matrix can only be used to describe a physical system. That is demonstrably false.

The ontological model is not a sample space. It is a conceptual model of what exists.

A generalized stochastic system will be defined to mean a tuple of the form (C, T , Γ, p, A) that consists of the following data.
• The symbol C will denote a set called the system’s configuration space, and the elements of C will be called the system’s (allowed) configurations. Configurations and configuration spaces will play an analogous role for generalized stochastic systems that states and state spaces play for generalized dynamical systems.
• The reason for switching the terminology from ‘states’ to ‘configurations’ is conceptual. In applications of the theory of dynamical systems to physical situations, such as in classical Hamiltonian mechanics, the notion of a ‘state’ is often taken to include rates of change or momenta in addition to configurations, because defining a ‘state’ in that way can make it possible to obtain deterministic laws in the form of first-order differential equations. For a generalized stochastic system, by contrast, the system’s probabilistic laws may mean that rates of change and momenta are not well-defined in the absence of specifying a trajectory. In that case, the only available notion of a ‘state’ is limited to the more rudimentary notion of a ‘configuration.

Please explicitly address what is said here. In particular the bits in bold.

 

[iste]

Note again that Barandes's reformulation involves a configuration space C for both general stochastic systems and unistochastic systems. This doesn't mean you can't try to find other uses for such transition matrices, but this thread is about what Barandes has published.The ontological model is not a sample space. It is a conceptual model of what exists.Please explicitly address what is said here. In particular the bits in bold.

It seems pretty clear to me that in the context of a stochastic process, the "configuration" is just what the sample space represents, but the notion of a generalized stochastic system isn't required to represent any specific kind of physical configuration space. No stochastic process is required to do this. A stochastic process is an abstract mathematical formalism that doesn't need to be about a physical system. Barandes is modelling a physical system using a stochastic process but the stochastic-quantum correspondence is a formal result about math which transcends what that math represents.

In the paper: arXiv:2507.21192v1

He says in section 2:

"In defining a stochastic process to serve as such a model, one can take the sample space to be the system’s configuration space C"

Its literally just semantic. Call the sample space whatever you want, it doesn't change that you are talking about an abstract stochastic process.

Yes, you can use it to represent a physical system easily, thats fine. I am not saying you can't use it for that reason. I am saying it is poorly motivated for that reason that orthodox quantum theory is ambiguous about ontology. If orthodox quantum theory is ambivalent about ontology and you are pulling these indivisible stochastic processes out of a correspondence with these quantum formalisms that are ambiguous about ontology, this is poor motivation for me. Yes, I could just take the configurations as a physical ontology; but given the stochastic-correspondence theorem, doing so would personally require me to do mental gymnastics and ignoring certain features of the correspondence in an ad hoc way. I can't take it as an interpretation seriously if I have to do that.

 

[Morbert]

Yes, you can use it to represent a physical system easily, thats fine. I am not saying you can't use it for that reason. I am saying it is poorly motivated for that reason that orthodox quantum theory is ambiguous about ontology. If orthodox quantum theory is ambivalent about ontology and you are pulling these indivisible stochastic processes out of a correspondence with these quantum formalisms that are ambiguous about ontology, this is poor motivation for me. Yes, I could just take the configurations as a physical ontology; but given the stochastic-correspondence theorem, doing so would personally require me to do mental gymnastics and ignoring certain features of the correspondence in an ad hoc way. I can't take it as an interpretation seriously if I have to do that.

You're looking at it backwards. Barandes starts with unistochastic processes defined by stochastic maps across classical configurations, and recovers a quantum theory from it, including all observables, establishing a correspondence.

You are free to instead swap out the configuration space with an arbitrary sample space and try to recover a quantum theory from it, including all observables. But i) It is not clear that this is possible, as classical phase space is derived from configuration space, an intrinsic manifold, not from momentum space. And ii) It will likely not offer up an intuitive ontology like configuration space does.

On a personal level, you are allowed to be unmoved by the correspondence between general stochastic processes on configuration space and quantum theory, but that's not a substantive critique.

 

[iste]

You're looking at it backwards. Barandes starts with unistochastic processes defined by stochastic maps across classical configurations, and recovers a quantum theory from it, including all observables, establishing a correspondence.

Its bidirectional so it doesn't matter. If your indivisible system exactly describes regular quantum theory then it is describing something with an ambiguous ontology.

You are free to instead swap out the configuration space with an arbitrary sample space

Barandes' is just using "configuration" as his word to describe the sample space because when he translates it to a quantum theory, he wants that to be what describes the physical configuration of the system. But as far as the stochastic system is concerned it doesn't matter, its just a word for the sample space.

I will repeat the quote earlier:

"In defining a stochastic process to serve as such a model, one can take the sample space to be the system’s configuration space C"

Clearly states they are the same thing as far as the stochastic process is concerned. He is only calling it a configuration because he intends it to model a physical system, but it has absolutely nothing to do with the validity of the stochastic-quantum correspondence which is about moving from an indivisible system to a unistochastic process to unitary matrices on a purely formal level without regard to interpretation.

But i) It is not clear that this is possible, as classical phase space is derived from configuration space, an intrinsic manifold, not from momentum space.

This is the gymnastics I am talking about that I wouldn't want to do. In order to build a physical theory out of this you have to ignore the fact that the stochastic-quantum correspondence is clearly valid beyond the realms you are willing to use it in.

This is why I am saying its not well motivated. As far as I am concerned it would be a category error to use a formalism that, in the context of quantum (physics), might only be interpretable as describing empirical measurement results instrumentally (because that is what the orthodox quantum formalism arguably does), to then be used as the basis of a hidden variable ontology for the exact same theory. That makes no sense imo.

Edited: (in brackets)

 

[Morbert]

Its bidirectional so it doesn't matter. If your indivisible system exactly describes regular quantum theory then it is describing something with an ambiguous ontology.

As explained already, the configuration space C is one of the fixed ingredients of the model, and provides the model with its kinematics, meaning its elementary physical or ‘ontological’ content.

Bidirectionality does not change this.

Barandes' is just using "configuration" as his word to describe the sample space because when he translates it to a quantum theory, he wants that to be what describes the physical configuration of the system. But as far as the stochastic system is concerned it doesn't matter, its just a word for the sample space.

I will repeat the quote earlier:

"In defining a stochastic process to serve as such a model, one can take the sample space to be the system’s configuration space C"

Clearly states they are the same thing as far as the stochastic process is concerned. He is only calling it a configuration because he intends it to model a physical system, but it has absolutely nothing to do with the validity of the stochastic-quantum correspondence which is about moving from an indivisible system to a unistochastic process to unitary matrices on a purely formal level without regard to interpretation.

This is the gymnastics I am talking about that I wouldn't want to do. In order to build a physical theory out of this you have to ignore the fact that the stochastic-quantum correspondence is clearly valid beyond the realms you are willing to use it in.

This is why I am saying its not well motivated. As far as I am concerned it would be a category error to use a formalism that, in the context of quantum (physics), might only be interpretable as describing empirical measurement results instrumentally (because that is what the orthodox quantum formalism arguably does), to then be used as the basis of a hidden variable ontology for the exact same theory. That makes no sense imo.

A generalized stochastic system will be defined to mean a tuple of the form (C, T , Γ, p, A) that consists of the following data.
• The symbol C will denote a set called the system’s configuration space, and the elements of C will be called the system’s (allowed) configurations. Configurations and configuration spaces will play an analogous role for generalized stochastic systems that states and state spaces play for generalized dynamical systems.
• The reason for switching the terminology from ‘states’ to ‘configurations’ is conceptual. In applications of the theory of dynamical systems to physical situations, such as in classical Hamiltonian mechanics, the notion of a ‘state’ is often taken to include rates of change or momenta in addition to configurations, because defining a ‘state’ in that way can make it possible to obtain deterministic laws in the form of first-order differential equations. For a generalized stochastic system, by contrast, the system’s probabilistic laws may mean that rates of change and momenta are not well-defined in the absence of specifying a trajectory. In that case, the only available notion of a ‘state’ is limited to the more rudimentary notion of a ‘configuration'.

No ignoring anything is necessary. The motivation for configuration spaces providing ontological content is here, which you still have not addressed.

 

[iste]

Bidirectionality does not change this.No ignoring anything is necessary. The motivation for configuration spaces providing ontological content is here, which you still have not addressed.

Bidirectionality does not change this.

Its just the sample space of the stochastic process. It doesn't matter what the 'ontological content' of the sample space, the stochastic
-quantum correspondence will be able to provide a quantum description of its behavior as long as the stochastic process is generically indivisible. Of Barandes wants to create a physical theory where this sample space is physical configurations (which is obviously why he uses the word), then this is a postulate extraneous to the content of the stochastic-correspondence theorem which is entirely a statement about mathematical objects, namely indivisibility, unistochasticity, unitary matrices.

 

[Morbert]

Its just the sample space of the stochastic process. It doesn't matter what the 'ontological content' of the sample space, the stochastic
-quantum correspondence will be able to provide a quantum description of its behavior as long as the stochastic process is generically indivisible. Of Barandes wants to create a physical theory where this sample space is physical configurations (which is obviously why he uses the word), then this is a postulate extraneous to the content of the stochastic-correspondence theorem which is entirely a statement about mathematical objects, namely indivisibility, unistochasticity, unitary matrices.

Where we are so far:
Barandes establishes a correspondence between quantum theory a general stochastic process with an ordinary classical configuration space, and hence, according to him "every quantum system can be understood as a sufficiently general kind of stochastic process unfolding in an old-fashioned configuration space according to ordinary notions of probability."

You object by saying we can consider stochastic processes with some other sample space and establish an exact correspondence with quantum theory, allowing us to understand quantum theory as a theory of these other stochastic processes.

I point out that while you might be able to do that for some alternative sample spaces if by some Pontryagin duality they are also a "configuration" space (see my earlier post speculating about momentum space), it amounts to an alternative interpretation as the ontological content will be different, even there is some commonality in the formalism.

You say this is external to the content of the theorem. Yes, it is an interpretational matter, and there are interpretational motivations for building a specific ontological model, just as there are in other realist interpretations. No formalism will insist upon a singular interpretation. So this does not amount to a criticism of the correspondence in the papers, nor does it recast the formalism as instrumentalist.

 

[iste]

Barandes establishes a correspondence between quantum theory a general stochastic process with an ordinary classical configuration space, and hence, according to him "every quantum system can be understood as a sufficiently general kind of stochastic process unfolding in an old-fashioned configuration space according to ordinary notions of probability."

You object by saying we can consider stochastic processes with some other sample space and establish an exact correspondence with quantum theory, allowing us to understand quantum theory as a theory of these other stochastic processes.

No, my objection is that with regard to the generalized stochastic process, "configuration space" is just a label for a generic sample space. He is using that label not because it is inherent to the stochastic-correspondence but because of the intention to produce a physical theory.

it amounts to an alternative interpretation as the ontological content will be different, even there is some commonality in the formalism.

Well just think that things like this make it an undesirable basis for a physical theory.

You say this is external to the content of the theorem. Yes, it is an interpretational matter, and there are interpretational motivations building an ontological model, just as there are in other realist interpretations. No formalism will insist upon a singular interpretation. So this does not amount to a criticism of the correspondence in the papers, nor does it recast the formalism as instrumentalist.

It just doesn't sit well with me to take something that I believe is effectively designed, structured to do one thing and then repurpose it to do something which I believe conflicts with that initial purpose. My instrumentalism view of it conflicts with a repurposing in terms of ontology.

 

[Morbert]

No, my objection is that with regard to the generalized stochastic process, "configuration space" is just a label for a generic sample space. He is using that label not because it is inherent to the stochastic-correspondence but because of the intention to produce a physical theory.

Configuration space is not just a label for a generic sample space. A configuration space is an established concept in classical physics, and so when it is used as a sample space, you get a stochastic process with clear foundational physical significance.

It just doesn't sit well with me to take something that I believe is effectively designed, structured to do one thing and then repurpose it to do something which I believe conflicts with that initial purpose. My instrumentalism view of it conflicts with a repurposing in terms of ontology.

An instrumentalist view will obviously clash with a realist view. This is true of every formalism. The point is that the stochastic systems specified by an ordinary classical configuration space offer a realist view that is consistent and intuitive even if the dynamics are counterintuitive.

[edit] - Also, this general stochastic formalism is perfectly suited to stochastic processes in configuration spaces.