A Understanding Barandes' microscopic theory of causality

[gentzen]

The transition probabilities of subsystems given by division events have a nomological status. They are objective, and describe the evolution of subsystems.

But Barandes didn't even clarify this much. And it can make sense not to clarify, because there could be spurious division events.

I suspect Albert just subjectively wants the formalism to look a bit more like Bohmian mechanics.

I suspect that Albert noticed how Barandes answered questions for clarifications, and did what was most consistent with that. (But of course, I basically already wrote that in the post you responded to. However, I just think it is a mistake to assume that Albert would make stupid mistakes for stupid reasons.)

 

[Morbert]

But Barandes didn't even clarify this much.

See section 3.7 here. Transition maps are the entirety of dynamics in this formalism, and system-environment dynamics (equation 44) give rise to transition maps between conditioning times called division events (equations 56-58) in subsystems Albert considers (Napoleon and libraries with books about him). This is clear.

 

[Morbert]

This timestamp might be of interest, where he discusses dynamics of subsystems vs the entire universe.

 

[gentzen]

See section 3.7 here. Transition maps are the entirety of dynamics in this formalism, and system-environment dynamics (equation 44) give rise to transition maps between conditioning times called division events (equations 56-58) in subsystems Albert considers (Napoleon and libraries with books about him). This is clear.

Let me verify:

Thus, the interaction between the subject system S and the environment E up to the time t′ has led to a transition matrix Γ^S (t ← 0) for the subject system that is divisible at t′, which has become a valid conditioning time.
It is therefore natural to refer to the new conditioning time t′ as a division event.

I have to admit that the bold parts (bold by me) suggest that Barandes distinguishes between the t' at which the subsystem is divisible and those which are a conditioning time. This is new in v3, compared to v2 which I once read:

Thus, the interaction between the subject system S and the environment E up to the time t′ has led to a transition matrix Γ^S(t) for the subject system that is momentarily divisible at t′.
It is natural to refer to t′ as a division event.

However, even the new passage suggests that merely being divisible at t' is enough to become a valid conditioning time.

Looks to me as if he intentionally deemphasizes that distinction, since his answers in his videos suggest that he became well aware of complaints like

The difficulty is that in Barandes' formulation, the "hidden variables" can do whatever they want between division events. Only their states at the moments of the division events have causal power (or any other importance).

They can do whatever they want in the sense that the theory doesn't offer a distribution over trajectories, in contrast with e.g. EPE decoherent histories. But is this an objective difficulty or merely subjective distaste for indivisible stochastic maps as dynamical laws?

and his answers generally go into the direction I described in the previous post ("... division events in independent subsystems can be defined purely in terms of the indivisible stochastic dynamics...").

So I still think that David Albert did the right thing with his minimal completion. As long as Barandes insists that there is nothing to clarify, and doesn't want to add "arbitrary" clarifications beyond his indivisible stochastic dynamics, Albert's completion remains a valid one.

 

[Morbert]

("... division events in independent subsystems can be defined purely in terms of the indivisible stochastic dynamics...").

For a subsystem constantly interacting with its environment, the stochastic dynamics will be divisible. They can be defined in terms of undivided stochastic dynamics in the sense that we can ignore the divisibility of the dynamics. But that doesn't mean it isn't there.

And to reiterate: It is true that if we suppose the universe as a whole is an isolated system, then the stochastic map of the entire universe will be indivisible. But Albert cannot use this to then conclude that the dynamics of subsystems like Napoleon's army or libraries recording his deeds are also indivisible. They are clearly divisible.

This isn't something that needs clarification. I wish Albert all the best in attempting a "completion", but it is not necessitated by some obscurity in the formalism.

 

[Sambuco]

PS Keep in mind that your concern is not quite the same as Albert's. Albert is considering high-frequency division events giving rise to a branching structure. You brought up the behavior of the quantum system prior to a sparse division event. They are related, but distinct in important ways. For now I am focusing on Albert's, as I have previously discussed your concern, but we can revisit later.

I agree that, although they are related, they are not the same issue. I'll also focus on what Albert says to make the discussion more organized.

Consider for example a simple model of the Stern-Gerlach Experiment with initial particle, pointer and environment states |ψ⟩, |ω⟩ and |ϵ+⟩ respectively.U(t′←0)|ψ⟩|ω⟩|ϵ⟩=12(|z+⟩|ω+⟩|ϵ+⟩+|z−⟩|ω−⟩|ϵ−⟩)The corresponding transition matrix of the particle and pointer device is Γ(t←0)=Γ(t←t′)Γ(t′←0)The correlation between the pointer device and the particle spin with the environment will give rise to a division event at t′. The transition matrix Γ(t←t′) will have the formΓij(t←t′)=p(ziωi,t|zjωj,t′)=δijI.e. The probability for jumping from branch i to branch j at any time t after t′ is 0.

More generally, correspondence between this formalism and QM is exact, and QM does not predict our quasiclassical experiences jumping from branch to branch.

Maybe I'm misunderstanding something, but let me explain my point. Assuming that a division event occurs at $t'$, you say that the following equality holds $p(z_i\omega_i, t | z_j\omega_j,t') = \Gamma_{ij}(t\leftarrow t')$. For me, this equation defines the probability of occurrence of event $i$ at $t$ given the occurrence of event $j$ at $t'$. Assuming this, I agree with you that the correspondence between Barandes's formulation and textbook QM is proven. This is because this definition of probability is equivalent to the postulate of wave function collapse, in the usual language of Hilbert space.

However, Barandes argues that his formulation does not include a "collapse". He said: "Although generically always approximate, division events will become nearly exact when the environment is sufficiently macroscopic, for precisely the same reasons that decoherence becomes nearly exact in such cases. Any resulting discrepancies in the effective stochastic laws will therefore be minuscule in real-world cases. These tiny discrepancies in the effective laws for subsystems are inevitable in all no-collapse formulations or interpretations of quantum theory (...) indivisible quantum theory does not invoke any fundamental violations of unitarity".

I have the feeling that both statements cannot be true at the same time. If the "collapse" induced by the occurrence of a division event at $t'$ is only effective (as in Bohmian mechanics), then it is not strictly true that $p(z_i\omega_i, t | z_j\omega_j,t') = \Gamma_{ij}(t\leftarrow t')$. In other words, if the collapse is only effective, then to calculate the probability that the system will be in some configuration at time $t$, we must consider the transition matrix corresponding to the statistical mixture (the different branches of the wave function) and not just the collapsed state. In Bohmian mechanics, this isn't a problem, since the presence of a particle on a branch makes the others completely irrelevant, as the trajectory depends on the value of the wave function at the point where it is located, that is, solely on the branch corresponding to that configuration. To achieve something like this in Barandes's formulation, it is necessary to assume that statistical conditioning plays exactly the same role as the collapse of textbook QM, that is, that the other branches no longer exist, by definition.

Lucas.

 

[Fra]

David Albert writes:

"It gives us (in particular) probabilities that the configuration of the world will be this or that at any time t given its configuration at the initial time t0 – but (unlike, for example, Bohmian Mechanics) it tells us nothing about what path the world may have taken, through the space of possible configurations, to get there. Here’s another way to put that: Jacob’s theory, as it stands, gives us probabilities that the configuration of the world is this or that at any time t given its configuration at the initial time t0 – but for any two times ta and tb, neither of which is the unique and metaphysically privileged “initial” time t0, it tells us nothing whatever about how the configuration of the world at tb depends on the configuration of the world at ta.

Jacob, in so far as I can tell, does not regard this as a particularly worrisome shortcoming.
...
And the business of telling ourselves stories about what path the world might be taking from one of those times to another – once those probabilities have been specified - is idle, speculative, unconfirmable, fluff
...

Consider (for example) the following way – you might call it the minimal way - of completing Jacob’s theory:

(1) The probability that the configuration of the world at any time t is this or that, given its configuration at t0, is the one we get from my original version of Jacob’s original theory.

(2) Given any two times ta and tb, neither of which is t0, the probability-distribution over possible configurations of the world at tb is independent of the configuration of the world at ta."

The difficulty is that in Barandes' formulation, the "hidden variables" can do whatever they want between division events. Only their states at the moments of the division events have causal power (or any other importance).

But clarifying the precise role of the configurations is both closely related to his proposal itself, and something Barandes could be able to do, and what he really should try to do.

So I still think that David Albert did the right thing with his minimal completion. As long as Barandes insists that there is nothing to clarify, and doesn't want to add "arbitrary" clarifications beyond his indivisible stochastic dynamics, Albert's completion remains a valid one.

It seems to me Albert is disturbed by that fact the indivisible transitions, does not trace out a history in a state space having a timeless dynamical law, where the future depends on any previous state.

This is indeedd the weird part, but this is also the beauty. If Albert is disturbed by this then I think he probably doesn't appreciate or understand the idea behind this perspecive.

The fact that the exact hidden or local configuration history between all the parts interacting with other parts in the universe are not objective beables, and and does not have a place in the system level dynamical description, is I think they key to how system level "non-locality" can be explained, while keeping micro-level causally local "interaction rules". For me this is a key insight, not a problem to be fixed.

My personal choice of understnading here is that the apparent problem IMO lies in that I think the logic associated to the paradigm of system dynamnics, are insuitable to model what is going on at part-part level inteactions. And the insight is the system dynamics ia always "effective", never fundamental, and that what is often understood as dynamical LAW, might be emergent aggregate behaviour, when you consider "parts" or "particles" that can be understood as aggregates of something smaller.

I see it as a feature, that the system level description is indeed asymptotically independent of the micro-level configurations. It helps explain stability of "effective law" precisely beausae it represents a kind of mean field of some underlying chaos - that we do not need the details of.

The unconfirmative fluff lives at a low microlevel, and here there exists no "timeless dynamical law", only except the transition probabilities, because of the indivisibility of interactions. divisibility is emergent at system level only.

This is how I read Barandes
"At the level of dynamics, the microphysical laws consist of conditional or transition probabilities"
-- https://arxiv.org/abs/2402.16935

I honestly think that a change of paradigm is required to really appreciate Barandes perspective. The idea that it needs "completion" in this sense, to me at least, is missing the point.

I also have issues with that Barandes is incomplete, but its in a different way. But I can appreciate his perspective I think thanks to that I see this from an ABM perspective, and here his "subsystems" with indivisiable stochastics are at least "much closer" ontological to the preseumed "agent/parts" than the hilbert space is. Tyring to tame the fluff is not the task I think, I think the fluff is suppose to be there, but it depends on what perspective you have. What is fluff from one perspective, is a definite history from the perspective of the fluff itself. This WERID if you only think in system view. But it is not at all weird if you thing from agent perspective.

/Fredrik

 

[Morbert]

Maybe I'm misunderstanding something, but let me explain my point. Assuming that a division event occurs at $t'$, you say that the following equality holds $p(z_i\omega_i, t | z_j\omega_j,t') = \Gamma_{ij}(t\leftarrow t')$. For me, this equation defines the probability of occurrence of event $i$ at $t$ given the occurrence of event $j$ at $t'$. Assuming this, I agree with you that the correspondence between Barandes's formulation and textbook QM is proven. This is because this definition of probability is equivalent to the postulate of wave function collapse, in the usual language of Hilbert space.

However, Barandes argues that his formulation does not include a "collapse". He said: "Although generically always approximate, division events will become nearly exact when the environment is sufficiently macroscopic, for precisely the same reasons that decoherence becomes nearly exact in such cases. Any resulting discrepancies in the effective stochastic laws will therefore be minuscule in real-world cases. These tiny discrepancies in the effective laws for subsystems are inevitable in all no-collapse formulations or interpretations of quantum theory (...) indivisible quantum theory does not invoke any fundamental violations of unitarity".

I have the feeling that both statements cannot be true at the same time. If the "collapse" induced by the occurrence of a division event at $t'$ is only effective (as in Bohmian mechanics), then it is not strictly true that $p(z_i\omega_i, t | z_j\omega_j,t') = \Gamma_{ij}(t\leftarrow t')$. In other words, if the collapse is only effective, then to calculate the probability that the system will be in some configuration at time $t$, we must consider the transition matrix corresponding to the statistical mixture (the different branches of the wave function) and not just the collapsed state. In Bohmian mechanics, this isn't a problem, since the presence of a particle on a branch makes the others completely irrelevant, as the trajectory depends on the value of the wave function at the point where it is located, that is, solely on the branch corresponding to that configuration. To achieve something like this in Barandes's formulation, it is necessary to assume that statistical conditioning plays exactly the same role as the collapse of textbook QM, that is, that the other branches no longer exist, by definition.

Lucas.

The near exactness of divisibility, analogous to the near exactness of decoherence, means that for a toy system I described previously, with a single division event, we have $\Gamma(t\leftarrow 0)\approx \Gamma(t\leftarrow t')\Gamma(t'\leftarrow 0)$ and for a system with regular division and a characteristic time scale $\delta t$ we have $\Gamma(n \delta t\leftarrow 0) \approx (\Gamma)^n$. These are equations (56) and (57) in Barandes's paper, modified to express the technical near exactness. At this timestamp, Barandes and Carroll discuss the ramifications of this inexactness of division events

Note that this doesn't save Albert's objection. Nearly exactly divisible laws of subsystems still establish very strong correlations between (to use Albert's examples) the existence of the earth in 1804, Napoleon's campaigns, and history books today. I still suspect Albert's inferences follow from his insistence in only considering the stochastic map of the universe as a whole and as an isolated system (what he calls "the world").

 

[Sambuco]

The near exactness of divisibility, analogous to the near exactness of decoherence, means that for a toy system I described previously, with a single division event, we have Γ(t←0)≈Γ(t←t′)Γ(t′←0) and for a system with regular division and a characteristic time scale δt we have Γ(nδt←0)≈(Γ)n. These are equations (56) and (57) in Barandes's paper, modified to express the technical near exactness.

I'm not objecting to any of this, that's not my point. I'm not referring to a case with imperfect/unsharp division event, but rather to how conditional probabilities are determined when a division event occurs. My point is that, to show that there is no "jump" between branches, one must find an expression for the probability $p(q_i,t|q_j,t')$ that shows that such a jump is extremely unlikely.

Let's take the case of the Stern-Gerlach experiment you mentioned earlier. Asumming that at a time $t'$, the particle, detector, and environment interact enough for decoherence to occur, the transition matrix between times $0$ and $t$ becomes $$\Gamma(t\leftarrow 0) = \Gamma(t\leftarrow t')\Gamma(t'\leftarrow 0)$$ Now my question is, how do we calculate the probabilities of events for $t > t'$? You wrote $$\Gamma_{ij}(t\leftarrow t') = p(z_i\omega_i, t | z_j\omega_j,t') = \delta_{ij}$$ which defines the probability in an unambiguous way.

My point is that this definition of the conditional probability is exactly the collapse postulate, written in the language of the indivisible stochastic formalism. I have no problem with this, in fact, it eliminates any possibility of "jump" between different branches (only one branch remains, the others no longer exist). Therefore, there is full correspondence with the predictions of quantum mechanics. However, Barandes says that his formulation is a "no-collapse formulation" and that there are no "fundamental violations of unitarity". Well, both things can't be true at the same time!

As an aside, I think Barandes's formulation has a lot of merit, given that his way of introducing the concept of "collapse" is very natural and not at all mysterious, it's simply statistical conditioning!. Furthermore, he elegantly resolves certain aspects of the measurement problem, such as why a superposition of alternative outcomes is equivalent to a statistical mixture. Of course, I'm talking about the formalism including the definition of probabilities described above, i.e. with "collapse".

Lucas.

 

[PeterDonis]

his way of introducing the concept of "collapse" is very natural and not at all mysterious, it's simply statistical conditioning!

But it can't be just that, if your other criticism is valid. If it's just statistical conditioning, then there is no actual collapse. That's what Barandes seems to be claiming when he says that there is no fundamental violation of unitarity. But, as you've pointed out, that's inconsistent with there not being any jumps between branches (or even with them being extremely improbable).

Perhaps a simpler way to put your objection would be this: if there's no fundamental violation of unitarity, then the fundamental dynamics can't be stochastic. Those two things are inconsistent, because unitary dynamics is fully deterministic; there's no stochasticity anywhere.

 

[Sambuco]

But it can't be just that, if your other criticism is valid. If it's just statistical conditioning, then there is no actual collapse. That's what Barandes seems to be claiming when he says that there is no fundamental violation of unitarity. But, as you've pointed out, that's inconsistent with there not being any jumps between branches (or even with them being extremely improbable).

Thanks! I was thinking about the case where transition matrices are redefined as new information is acquired. But, yes, as you say, perhaps "statistical conditioning" isn't the most appropriate term to describe it.

 

[Morbert]

I'm not objecting to any of this, that's not my point. I'm not referring to a case with imperfect/unsharp division event, but rather to how conditional probabilities are determined when a division event occurs. My point is that, to show that there is no "jump" between branches, one must find an expression for the probability $p(q_i,t|q_j,t')$ that shows that such a jump is extremely unlikely.

Let's take the case of the Stern-Gerlach experiment you mentioned earlier. Asumming that at a time $t'$, the particle, detector, and environment interact enough for decoherence to occur, the transition matrix between times $0$ and $t$ becomes $$\Gamma(t\leftarrow 0) = \Gamma(t\leftarrow t')\Gamma(t'\leftarrow 0)$$ Now my question is, how do we calculate the probabilities of events for $t > t'$? You wrote $$\Gamma_{ij}(t\leftarrow t') = p(z_i\omega_i, t | z_j\omega_j,t') = \delta_{ij}$$ which defines the probability in an unambiguous way.

My point is that this definition of the conditional probability is exactly the collapse postulate, written in the language of the indivisible stochastic formalism. I have no problem with this, in fact, it eliminates any possibility of "jump" between different branches (only one branch remains, the others no longer exist). Therefore, there is full correspondence with the predictions of quantum mechanics. However, Barandes says that his formulation is a "no-collapse formulation" and that there are no "fundamental violations of unitarity". Well, both things can't be true at the same time!

As an aside, I think Barandes's formulation has a lot of merit, given that his way of introducing the concept of "collapse" is very natural and not at all mysterious, it's simply statistical conditioning!. Furthermore, he elegantly resolves certain aspects of the measurement problem, such as why a superposition of alternative outcomes is equivalent to a statistical mixture. Of course, I'm talking about the formalism including the definition of probabilities described above, i.e. with "collapse".

Lucas.

The near exactness of divisibility, analogous to the near exactness of decoherence, means that for a toy system I described previously, with a single division event, we have $\Gamma(t\leftarrow 0)\approx \Gamma(t\leftarrow t')\Gamma(t'\leftarrow 0)$

Barandes's formalism is an effective collapse formalism, which means the transition matrix $\Gamma(t\leftarrow 0) = \Gamma(t\leftarrow t')\Gamma(t'\leftarrow 0)$ where $\Gamma_{ij}(t\leftarrow t') = p(z_i\omega_i, t | z_j\omega_j,t') = \delta_{ij}$ nearly exactly represents the dynamics of the Stern Gerlach apparatus. This is why data generated by a competently executed Stern-Gerlach experiment are reliable. There would only be a contradiction if collapse was exact instead of nearly exact/effective, such that a superobserver deity coupling the universe to their much larger apparatus would not be able to erase the data, or shunt the the Stern-Gerlach lab onto a different branch.

 

[Sambuco]

Barandes's formalism is an effective collapse formalism, which means the transition matrix Γ(t←0)=Γ(t←t′)Γ(t′←0) where Γij(t←t′)=p(ziωi,t|zjωj,t′)=δij nearly exactly represents the dynamics of the Stern Gerlach apparatus.

Maybe I'm misunderstanding something, but after a division event isn't the definition $p(z_i\omega_i, t | z_j\omega_j,t') = \Gamma_{ij}(t\leftarrow t')$ equivalent to the collapse postulate? If that is the case, Barandes' formalism is equivalent to quantum mechanics based on Hilbert spaces and wave functions, but at the same time, it cannot be said to be a non-collapse formalism, like Bohmian mechanics or many-worlds.

Lucas.

 

[Fra]

We think differently so I am not sure if I see the objections but they seem to be conceptual and not about the math? but as Barandes says

"Unlike stochastic-collapse theories [14, 15], this paper does not invoke any fundamental violations of unitarity,
nor does it require introducing any new constants of nature to specify dynamical-collapse rates."
-- p25, arXiv:2302.10778v2

As I see it conceptaully, no unitarity violoations refers to the hilber-space side of the correspondence, which as per the correspondence, corresponds to "no deviatons from unistochastics" on the other side of correspondence.

But on the stochastic side there must obviously be minor jumps within each subsystem, but as as long as they are constrained by unistochastics, it does not imply unitary violations on the hilbert space side. The dynamics in hilbert space of the whole system of all parts is smooth and conceptaully corresponds to an average of the underlying "chaos". So the subsystem hidden beable jumpts are invisibile at the hilbert level.

The question for me is rather, but this does beyond beyond (extend or complete how one ones to put it) Barandes picutes is: Why is fundamental unistochastics obvious starting point? IMHO, it's not. But it can possible be arguey that unistochastics is emergent from a deeper level where unistochastics and then thus at fine scale the unitarity is violated. So I read Barandes corresponendence to sit at this intermediate level, then it is ifine for me. So "fundamental" may also be scale dependent or effective.

/Fredrik

 

[Morbert]

Maybe I'm misunderstanding something, but after a division event isn't the definition $p(z_i\omega_i, t | z_j\omega_j,t') = \Gamma_{ij}(t\leftarrow t')$ equivalent to the collapse postulate? If that is the case, Barandes' formalism is equivalent to quantum mechanics based on Hilbert spaces and wave functions, but at the same time, it cannot be said to be a non-collapse formalism, like Bohmian mechanics or many-worlds.

Lucas.

It would only be equivalent to the collapse postulate if it was claimed that the transition matrix $p(z_i\omega_i, t | z_j\omega_j,t') = \Gamma_{ij}(t\leftarrow t')$ represented the exact dynamics. Instead, in contexts like ours where the system has interacted with a macroscopic environment we have marginalized over, it represents near exact dynamics, consistent with the formalism being a non-collapse formalism.

 

[Sambuco]

It would only be equivalent to the collapse postulate if it was claimed that the transition matrix $p(z_i\omega_i, t | z_j\omega_j,t') = \Gamma_{ij}(t\leftarrow t')$ represented the exact dynamics. Instead, in contexts like ours where the system has interacted with a macroscopic environment we have marginalized over, it represents near exact dynamics, consistent with the formalism being a non-collapse formalism.

I'm not sure which of the following two possibilities you're referring to when you say that $p(z_i\omega_i, t | z_j\omega_j,t') = \Gamma_{ij}(t\leftarrow t')$ doesn't represent the exact dynamics:

1. The transition matrix doesn't accurately reflect the dynamics because the division event doesn't correspond to an ideal projective measurement.

2. The transition matrix doesn't accurately reflect the dynamics because there is still a possibility (albeit remote) that the other branches of the wave function could generate interference, as in Wigner's friend's experiments.

Lucas.

 

[Morbert]

@Sambuco your case 2 is the relevant one. But

2. The transition matrix doesn't accurately reflect the dynamics because there is still a possibility (albeit remote) that the other branches of the wave function could generate interference, as in Wigner's friend's experiments.

The transition matrix nearly exactly reflects the dynamics of the particle + apparatus subsystem. I would not sign off on any description of it as not accurately reflecting those dynamics.

 

[Sambuco]

@Sambuco your case 2 is the relevant one. But The transition matrix nearly exactly reflects the dynamics of the particle + apparatus subsystem. I would not sign off on any description of it as not accurately reflecting those dynamics.

If I understand correctly, you're saying that, in some way, all the branches are still present, even though the probability of interference is negligible. Now, how does the model account for the system's configuration corresponding to only one of those branches? How does it differ from a model like the one Albert points out, where the configuration "jumps" between branches?

If it were that easy, why does textbook QM add the collapse postulate, instead of arguing the same as Barandes?

Lucas.

 

[PeterDonis]

If I understand correctly, you're saying that, in some way, all the branches are still present, even though the probability of interference is negligible. Now, how does the model account for the system's configuration corresponding to only one of those branches? How does it differ from a model like the one Albert points out, where the configuration "jumps" between branches?

Let me try to rephrase this question for @Morbert.

The "branches" being talked about here are branches of the wave function. The probability of interference between them is negligible because of decoherence. So in any interpretation where the wave function is the physical state of the system (for example, the MWI), it's straightforward to explain why "jumping" between branches doesn't happen.

But in Barandes's formulation, the wave function is not the physical state of the system. It's a mathematical convenience. So it can't physically constrain the state of the system from "jumping" between branches. So what does?

 

[Morbert]

If I understand correctly, you're saying that, in some way, all the branches are still present, even though the probability of interference is negligible. Now, how does the model account for the system's configuration corresponding to only one of those branches? How does it differ from a model like the one Albert points out, where the configuration "jumps" between branches?

If it were that easy, why does textbook QM add the collapse postulate, instead of arguing the same as Barandes?

Lucas.

The collapse postulate is motivated by the fact that, for all practical purposes, the statistics of a quantum system + pointer degrees of freedom are described by a simple mixture. It is as if collapse has occurred. The distinction between effective collapse and true collapse becomes relevant if we consider e.g. hypothetical superobservers.

 

[Morbert]

Let me try to rephrase this question for @Morbert.

The "branches" being talked about here are branches of the wave function. The probability of interference between them is negligible because of decoherence. So in any interpretation where the wave function is the physical state of the system (for example, the MWI), it's straightforward to explain why "jumping" between branches doesn't happen.

But in Barandes's formulation, the wave function is not the physical state of the system. It's a mathematical convenience. So it can't physically constrain the state of the system from "jumping" between branches. So what does?

In Barandes's formalism, interference terms in a quantum state encode the indivisibility of a stochastic process. From his correspondence paper:

One sees from this analysis that interference is a direct consequence of the stochastic dynamics not generally being divisible. More precisely, interference is nothing more than a generic discrepancy between the actual indivisible stochastic dynamics and a heuristic-approximate divisible stochastic dynamics. Interference encodes the fact that the underlying stochastic dynamics is indivisible, despite the way that unitary time-evolution operators look superficially divisible.

A scenario where interference terms are suppressed (e.g. irreversible measurement) will correspond to a highly divisible stochastic process, and the stochastic maps across division events are what prevent jumping. I.e. They will yield a jumping probability of effectively 0.

Without these division events and their transition maps, jumping is not prevented. So e.g. considering the universe as an isolated system, textbook QM might let us write down a unitarily evolving pure state, without any system-environment partitioning to suppress interference terms. In Barandes's formalism, this corresponds to a strictly indivisible transition map of the entire universe. I.e. $\Gamma(t\leftarrow 0)$ of the universe will have no division events to constrain jumping. As mentioned in my earlier post, I suspect this indivisible transition map of the universe is where Albert is getting his inferences.

 

[PeterDonis]

considering the universe as an isolated system, textbook QM might let us write down a unitarily evolving pure state, without any system-environment partitioning to suppress interference terms. In Barandes's formalism, this corresponds to a strictly indivisible transition map of the entire universe. I.e. $\Gamma(t\leftarrow 0)$ of the universe will have no division events to constrain jumping.

But you can't have it both ways. That indivisible transition map of the entire universe will have to include all the subsystems. Which makes it seem like there are two different transition maps, one indivisible and one divisible, and Barandes is simply declaring by fiat that only one is relevant to the cases we actually have experimental data on.

 

[Morbert]

But you can't have it both ways. That indivisible transition map of the entire universe will have to include all the subsystems. Which makes it seem like there are two different transition maps, one indivisible and one divisible, and Barandes is simply declaring by fiat that only one is relevant to the cases we actually have experimental data on.

Both transition maps are nomologically valid for their purposes, and while the universe contains subsystems, this does not mean the transition map of the universe contains the transition map of subsystems. Only the latter models the quasiclassical, divisible stochastic dynamics of subsystems like Earth. This does not mean the transition map of the universe is wrong. It just means it is silent on the matter, just as the transition map of a subsystem is silent on environmental degrees of freedom that were marginalized over to derive it. It is Albert who seems to be assuming that only the transition map of the universe is nomologically valid.

 

[PeterDonis]

while the universe contains subsystems, this does not mean the transition map of the universe contains the transition map of subsystems

Why not? That doesn't seem right.

 

[Sambuco]

A scenario where interference terms are suppressed (e.g. irreversible measurement) will correspond to a highly divisible stochastic process, and the stochastic maps across division events are what prevent jumping. I.e. They will yield a jumping probability of effectively 0.

I think you're confusing two different things. For interference to occur, the different branches of the wave function must recombine into a single one, to put it briefly. That's not my argument. What I'm saying is that, even with the branches quite far apart, the configuration of the system can "jump" from one to the other, unless postulated otherwise. The collapse postulate does exactly that in textbook QM. How are these "jumps" avoided in Barandes's formulation?

Lucas.

 

[Morbert]

Why not? That doesn't seem right.

We can derive the subsystem transition map from the larger unistochastic system a la section 3.7 in Barandes's paper. We can say it "includes" the subsystem transition map in this sense. But when we do this, the map will be divisible. I.e. There is no indivisible transition map of the subsystem embedded in the indivisible transition map of the universe. However we extract the transition map of the subsystem, it will be divisible.

 

[Morbert]

I think you're confusing two different things. For interference to occur, the different branches of the wave function must recombine into a single one, to put it briefly. That's not my argument. What I'm saying is that, even with the branches quite far apart, the configuration of the system can "jump" from one to the other, unless postulated otherwise. The collapse postulate does exactly that in textbook QM. How are these "jumps" avoided in Barandes's formulation?

Lucas.

A scenario where interference terms are suppressed (e.g. irreversible measurement) will correspond to a highly divisible stochastic process, and the stochastic maps across division events are what prevent jumping. I.e. They will yield a jumping probability of effectively 0.

The transition maps across division events will prevent jumping.

 

[Sambuco]

The collapse postulate is motivated by the fact that, for all practical purposes, the statistics of a quantum system + pointer degrees of freedom are described by a simple mixture.

I think the main motivation for the collapse postulate is that repeated measurements should yield the same outcomes.

The transition maps across division events will prevent jumping.

That's true if, knowing that the system configuration is $q_j$ at $t'$, we postulate the transition map $p_i(t) = \Gamma_{ij}(t \leftarrow t')$, but you said that this is not exactly true. Maybe there's an additional assumption about the continuity of trajectories or something like that that I'm missing? Otherwise, I don't know how to enforce the condition that the configuration at $t'$ restricts future configurations to stay within the same branch (of course, I'm not considering superobservers and all that stuff).

Lucas.

 

[Morbert]

That's true if, knowing that the system configuration is $q_j$ at $t'$, we postulate the transition map $p_i(t) = \Gamma_{ij}(t \leftarrow t')$, but you said that this is not exactly true. Maybe there's an additional assumption about the continuity of trajectories or something like that that I'm missing? Otherwise, I don't know how to enforce the condition that the configuration at $t'$ restricts future configurations to stay within the same branch (of course, I'm not considering superobservers and all that stuff).

Lucas.

No continuity is assumed in this formalism. The near exactness of the divisible dynamics is enough to prevent jumping.

 

[Sambuco]

I will attempt to describe a position that seems reasonable to me. First, transition matrices define a nomological entity that is only relevant to the extent that we have (contingent) information about the outcome of an event. That is, given a division event at $t'$, the standalone probability $p_i(t)$ can be written as $p_i(t) = \sum_j \Gamma_{ij}(t \leftarrow t') p_j(t')$, where $p_j(t')$ are, in Barandes's words, "the contingent ingredient". Then, if we have precise information about the system configuration at $t'$, all of the probabilities $p_j(t')$ become zero, with the exception of one, which corresponds to the observed value. Applying this information to our expression for the standalone probability $p_i(t)$, we obtain $p_i(t) = \Gamma_{ij}(t \leftarrow t')$ where $j$ refers here to the system configuration $q_j(t')$ for which we have information. That's the "collapse" in this stochastic formulation.

Now, how is it possible for all the other branches still exist if the aforementioned standalone probability only includes one of them? In other words, how is that possible for different branches to recohere (as in Wigner's friend thought experiments) if they are not included in this standalone probability? Well, the answer lies in the fact that what is "objective" is the transition matrices, not the standalone probabilities, since the latter also depend on contingent information about the ocurrence of other events. In cases such as Wigner's friend scenarios, the recoherence of different branches is intrinsically related with the loss of information about the system configuration at $t'$. Without this information, we have to use again the $\Gamma(t \leftarrow 0)$ transition matrix because, in addition, the dynamics are no longer divisible at $t'$.

@Morbert, I think you'll agree. Am I right?

Lucas.