A Understanding Barandes' microscopic theory of causality

[Morbert]

Why will this be the case?

Equation (56) from this paper is an example of a subsystem's transition matrix divisible at $t'$. The transition matrix $\Gamma^\mathcal{S}(t\leftarrow t')$ on the left hand side will have the form (as per equation 1)$$\Gamma^\mathcal{S}_{ij}(t\leftarrow t') = p^\mathcal{S}(i,t|j,t')$$In a quasiclassical context, these conditional probabilities encode our quasiclassical expectations, and are how Barandes's formalism recovers quasiclassical physics at the appropriate limit, analogous to the way Everettian quatum mechanics recovers it with quasiclassical, decoherent branches. E.g. The probability that it will rain presently where you are, given there are no clouds where you are, is 0. The earth subsystem will not spontaneously evolve into a configuration where it will in fact immediately start to rain.

I suspect what Albert is doing is considering the transition matrix of the entire universe modeled as an isolated system, which does not contain any division events, and hence the standalone probability does not constrain any quasiclassical evolution. The mistake here is that standalone probabilities, are epistemic, not dynamical/nomological. It is only the directed conditional probabilities that make up transition matrices that are dynamical.

 

[gentzen]

I don't understand this paragraph

Division events will give rise to more than probability distributions. They will give rise to directed conditional probabilities that make up the dynamics of the theory. And these directed conditional probabilities will prohibit the world from jumping from branch to branch.

Barandes is free to clarify this, if he wants. Perhaps he wants the minimal version, where each division event is global, and has the same status as t=0. Then t=0 would be just one division event among many. And as a further clarification, give time a clear direction, so that each division event is only important for what happens in its future (and not its past), until the next division event happens.

 

[Morbert]

Barandes is free to clarify this, if he wants. Perhaps he wants the minimal version, where each division event is global, and has the same status as t=0. Then t=0 would be just one division event among many. And as a further clarification, give time a clear direction, so that each division event is only important for what happens in its future (and not its past), until the next division event happens.

See this timestamp, where he distinguishes the dynamical conditional (transition) probabilities from the epistemic standalone probabilities.

 

[Morbert]

Since quantum mechanics is non-local, any attempts to "fix" that is bound to be unsatisfactory.

"Bound to be unsatisfactory" needs to be unpacked. There are objective conditions of a good interpretation of quantum mechanics: Empirically adequate, unambiguous, generalizeable to all quantum theories (e.g. QFT). A good few interpretations meet these conditions.

Then there are subjective conditions: Must be realist, must be deterministic, must be local. These subjective conditions are where things become "unsatisfactory".

 

[gentzen]

See this timestamp, where he distinguishes the dynamical conditional (transition) probabilities from the epistemic standalone probabilities.

Are you sure you have the correct timestamp? It doesn't clarify the points where David Albert sees the incompleteness. But I could even find anything where he would distinguish "dynamical conditional" from "epistemic standalone" probabilities.

 

[gentzen]

A good few interpretations meet these conditions.

And some sometimes claim to be local, for example MWI. But this claim risks to make MWI "unambiguous", and hence no longer satisfying all "objective conditions of a good interpretation". Why? Because before that claim, MWI was just a wavefunction in Hilbert space which evolves according to the Schrödinger equation. But such claim that MWI would be local refer to papers where the Heisenberg picture is used instead. It remains unclear whether this is still the same MWI or not.

Then there are subjective conditions: ..., must be local.

And this last one is bound to cause trouble.

 

[PeterDonis]

such claim that MWI would be local refer to papers where the Heisenberg picture is used instead

Can you give any specific references?

 

[gentzen]

Can you give any specific references?

Bell on Bell’s theorem: The changing face of nonlocality
Harvey R Brown, Christopher G Timpson
https://arxiv.org/abs/1501.03521

In our view, the significance of the Bell theorem, both in its deterministic and stochastic forms, can only be fully understood by taking into account the fact that a fully Lorentz covariant version of quantum theory, free of action-at-a-distance, can be articulated in the Everett interpretation.

So here we have a claim that MWI is local, at least that is how David Marcus interpreted that paper here. Not sure how they argue themselves, but they certainly cite and comment on
Rubin, Mark A. 2001. Locality in the Everett Interpretation of Heisenberg Picture Quantum Mechanics. Foundations of Physics Letters, 14, 301–322.
Deutsch, David, and Hayden, Patrick. 2000. Information Flow in Entangled Quantum Systems. Proceedings of the Royal Society of London A, 456, 1759–1774.

We also had a thread "How Does the Many-Worlds Interpretation Address Nonlocality?" recently, with examples of claims that MWI is a "local interpretation":

To follow up, David Deutsch argues for a local interpretation here. I note this is an old paper. https://arxiv.org/pdf/quant-ph/9906007

Many "Oxford Everettians" (Deutsch, Wallace, Vaidman et al) hold a local (in the sense of no spooky action) account of MWI.

(I don't think that Vaidman himself is an "Oxford Everettian", but maybe one of his "et al" coauthors was.)

 

[PeterDonis]

here we have a claim that MWI is local

By "local" here is meant "free of action-at-a-distance", which, if you unpack further based on what's said in section 9 of the paper, means that the dynamics is derived from a Hamiltonian that only includes point interactions. But in the very next sentence, the paper says that the fundamental states of the theory are non-separable, due to entanglement--which is exactly where things like correlations that violate the Bell inequalities come from.

In other words, the claim the MWI is local is based on picking a definition of "local" that makes it true. But that might not be the definition of "local" that other people actually care about. Much of the argument in the literature about this, and indeed about QM interpretations in general, is based on different people picking different definitions of a key term and then talking past each other.

 

[RUTA]

From SEP on Many-Worlds (written by Vaidman):

Although the MWI removes the most bothersome aspect of nonlocality, action at a distance, the other aspect of quantum nonlocality, the nonseparability of remote objects manifested in entanglement, is still there. A “world” is a nonlocal concept. This explains why we observe nonlocal correlations in a particular world.

 

[Fra]

Many subquestions here, and Alberts paper was long i skimmed it yeasterday but i think one of his issues is that Barandes seems to not release himself from the hamiltonian flow in the hilbert space, thus the lack of first principle construction of dynamical law.

I think this is correct, but it's because Barandes only provides the correspondence. He does not supply a complete first principle construction of the timedependent transition matrix(gamma). And in Baranders view, it is excatly the transition matrix that encodes the information that is equivalence to the system dynamical law. Ie the time dependence of the transition matrix, replaces all the information encoded in schrödinger equation and hamiltonian.

I think that is how it is, something is missing I agree.
My own opinion on what is missing, and what would in principle resolve this, is as i mentioned in other posts.

While Barandes does not mention observers/agents, he instead considers a sufficienty complex subsystem to take that role. This is a LIMIT.

My conceptul perspective/interpretation of this and how it sees Barandes work is this
- Baranded indivisiable stochastics, encoded by the time dependent transition matrix encodes corresponds to a LIMIT of an agent based model (that we do not have yet!) where in the limit of many observers and t -> infinity, we get the transitiion matrix of barandes that represents an asymptotic behaviour of the interactions.

Ie the missing "first principle" construction of gamma, in order NOT to still rest on the information encoded in the hilbert space hamiltoniani, would be to explain the emergence of this limit, from some yet unknonwn microscopic agent agent interations. It is IMO at THIS level that the true causality is encoded. In principle that is, because this is not yet unlocked. But it is the "completion" I see as missing to make full sense of Barandes view.

This is what mean before when I've said that I view both quanutm mechanics as it stands; and the reformulation via indivisiable stochastics as a limiting case (effective description) or a more complex agent based model, corresponding to an asymptotic steady state. And if we can understand that in detail, we should get that missing first principle "explanattion" of the "effective dynamical law" we are used to see at system dynamics level. This also merges fined with "effective theory" perspectives of modern physics. And before the limit is taken; we might expect an "evolving law", like a PDE/ODE with external explicit time dependence, that would relate to some "absolute time".

Note that, this is a genereal interpretion. I have not actual on the table ABM models for this (so it's not a speculation just to be belar, but this my conceptual understnading of Barandes work in the context of how to complete it.

Edit: this is also I think what Barnades in one of his other videos hints that the nothion of "causality" is highly suspicious or questionable in the system dynamical view if the law is timeless; then the future is alreadty determined from the past. So the notion of causal mechanism is hidden. So I agree with Barandes that the non-locality notion Bell uses is not very useful thinking tool, so I am with Barandes on this one.

/Fredrik

 

[gentzen]

Many subquestions here, and Alberts paper was long i skimmed it yeasterday but i think one of his issues is that Barandes seems to not release himself from the hamiltonian flow in the hilbert space, thus the lack of first principle construction of dynamical law

No, even so this is a point which came up in the discussions here (on PhysicsForums), the two main objections raised by David Albert (incompleteness and locality) are not directly related to this. The incompleteness he highlights is that the „causal power“ of the current configuration is not clarified sufficiently. To illustrate this, he discusses a minimal completion, where not even the configurations at division events have „causal power“.

 

[Fra]

No, even so this is a point which came up in the discussions here (on PhysicsForums), the two main objections raised by David Albert (incompleteness and locality) are not directly related to this. The incompleteness he highlights is that the „causal power“ of the current configuration is not clarified sufficiently. To illustrate this, he discusses a minimal completion, where not even the configurations at division events have „causal power“.

But in the traditional view (system dynamics) where else is causal power encoded, if not in initial conditions and dynamical law?

As I see it the topics are related. IMOH the true causal power, would be best understood in an expanded ABM model if which not onlt barandes formulation but also QM itself is a limitind case.

But lets me digest Alberts words again to see if there is som subquestions here. But to me "causal power" can sit on two places depending on perspective.

1)In system dynamics perspectice, we have dynamical law, either inform of transition matrix or differential equations.

2) In agent based model (meaning not "observers" necessarily if that is annouing, but seeing the system as interacting PARTS) the causal power likes in the decentralized agent update rules (which could very very be reduce to evolving stochastics themselves -> meaning it's fundamentally non-markovian interactions)

And then of course the two views are not in contradiction. But (1) is a limiting case of (2), and at least in MY understnading the true causal power lies at part-part inteactions, not system level laws. (The general perspective are in the links of https://www.physicsforums.com/threa...-quantum-interpretations.1082027/post-7283566)

I'll read the paper again later... it was quite long and i skimmed it.

/Fredrik

 

[gentzen]

But in the traditional view (system dynamics) where else is causal power encoded, if not in initial conditions and dynamical law?

What you are missing is that Barandes’ proposal is not just a different perspective, but incomplete in some very concrete ways.

One can argue about whether staying silent about the dynamical laws is a problem, but at least it is clear that those laws are unrelated to his proposal.

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.

 

[Morbert]

Are you sure you have the correct timestamp? It doesn't clarify the points where David Albert sees the incompleteness. But I could even find anything where he would distinguish "dynamical conditional" from "epistemic standalone" probabilities.

At the timestamp, Barandes remarks on the distinction between standalone probabilities and transition probabilities, the latter of which make up the objective dynamics of subsystems Albert concerns himself with, like the earth and Napoleon. The "jumping around" from branch to branch that Albert asserts is, as far as I can tell, inferred from some standalone probability of the entire universe, devoid of any consideration of the transition probabilities of subsystems which would prevent such jumping around.

An analogous objection to Everettian QM would look like this: In this paper, David Wallace sketches how quasiclassical behavior is found in the Everettian formalism.

So: if we pick a particular choice of system-environment split, we find a “strong” form of quasi-classical behaviour: we find that the system is isomorphic to a collection of dynamically independent simulacra of a classical system. We did not find this isomorphism by some formal algorithm; we found it by making a fairly unprincipled choice of system-environment split and then noticing that that split led to interesting behaviour. The interesting behaviour is no less real for all that.

Albert might object by saying if you don't consider such system-environment splits, and only consider the universe as a whole, you lose the quasiclassical branch structure.

 

[Sambuco]

In a quasiclassical context, these conditional probabilities encode our quasiclassical expectations, and are how Barandes's formalism recovers quasiclassical physics at the appropriate limit, analogous to the way Everettian quatum mechanics recovers it with quasiclassical, decoherent branches.

I think the comparison with Everett is pertinent. In Everett's interpretation, if the wavefunction splits into two decoherent branches with probability 1/2, the theory predicts the superposition of two non-interacting "worlds," and the classical case is recovered (albeit with multiple worlds, of course). On the other hand, in Bohmian mechanics, the particle is located on one of the two branches, and the wavefunction (either the "global" one or the one resulting from the effective collapse) predicts the correct probabilities, so the classical limit is also recovered. In my opinion, this is not the case in Barandes's formulation, and this is precisely the problem Albert points out (and which I also pointed out in post #63 for the two-slit experiment).

Suppose we have an initial one-particle wavefunction at time $t_0$ and consider configurations of the system at two times $t_1$ and $t_2$ $(t_0 < t_1 < t_2)$. If a division event ocurrs at $t_1$, then the probability that the system is in a certain configuration at time $t_2$ can be written as $$P(q_2|q_0) = \sum_{q_1} P(q_2|q_1)P(q_1|q_0)$$ where $q_i$ is the configuration at time $t_i$. The problem I see is that, if the transition matrix (or the wavefunction) plays a nomological role, then, the configuration of the system at $t_2$ does not depend on the configuration at $t_1$ (hidden variables are too hidden!), so the particle can be at $t_1$ within one of the branches and jump at $t_2$ to the other. As we know, this is ruled out by experiments. Nothing in Barandes's formulation prevents this from happening. In line with what Albert says, one can add to the laws a dependence between the configurations at two different times, as occurs in Bohmian mechanics. Another possibility is to simply say that all the possibilities are equally "real", but this is just many-worlds. In my opinion, the problem becomes even clearer if we consider another possible solution: we can postulate that, when a division event occurs, only the branch where the particle is located should be consider when calculating future probabilities, but this is the collapse postulate of textbook QM! Of course, when the collapse postulate is added, the measurement problem is back to life.

Lucas.

 

[Fra]

I think already at the introduction, I can sense the approach and the immediate hunch is that I do not share Alberts analysis...

Albert writes:

"The fundamental ontology of Jacob’s version of quantum mechanics – at least in the non-relativistic first-quantized case - consists entirely and exclusively of some presumably finite collection of particles. Just as in the case of Classical Mechanics, each of these particles invariably has some perfectly determinate position in space – and the history of those positions, the history (that is) of the configuration of that collection of particles, constitutes the complete physical history of the world."

This is indeed makes it sound like Bohmian mechanics, but I do not agree with this characterization. Although in a way, it is the case, the language makes if confusing and IMO misleading.

First of all Barandes considers a system composed of subsystems. To label these particles, IMO induces an mental image that is at least misleading. Because often the notion of particles, make no sense unless given some dynamical law (in the ordinary sense). And it is also not really brownian motion, as the particles are not doing markovian random walks.

This is why I might stretch myself to relabel "the collection of subsystem" as a "collection of composite-agents". While an agent is simply material like a particle, the key difference is that the concept of "agent" makes sense without reference to global dynimcal law - the conditional transition probabilities are rather than corresponds to the decentralized agent rules. This is a key difference. And talking about "particles" is just hiding this difference, causing confusion IMHO at lesat.

Also to say that they all have determinate position and history in space, I find somewhat misleading as well, because while the history of each subsystem is determinate, this context of this history (which I personally attribute to the "agent" corresponding to that syssystem) is different than the context in which a fellow nearby "subsystems" history is determinate. This is why one has a "collection" of determinate histories, but the collection is a unition of "incompatible context". And it's when these contexts (subsystems) interact that things happen.

In this view, the notion of "causality" is also NOT defined at global system level, its defined on subsystem level, in that two distinct subsystems have independent transition probabilities. This is exactly also what on would naturally have in decentralised agent interactions.

To NOT use the word agent(which Barandes does not) is probably a good way to avoid people confuising it will all the other stuff like conscioussness etc(!), so lets just call it "subsystem". But also to avoid calling it "particle" is also I think good as it makes people associate to bohmian mechanics.

The only bohmian mechanics interpretation I've aware of that at least conceptaully might be compatible with this, is Demystifiers version of "solipsist HV", where the HV are associated to the content of the "bohmian particles" themselves. That is essentially, conceptually very close IMO. But the math may be different. So - had I been forced to convert to a Bohmian, that is the hole I would dig in.

This is how I view it and it's why I overally disagree with Alberts paper.

/Fredrik

 

[Morbert]

I think the comparison with Everett is pertinent. In Everett's interpretation, if the wavefunction splits into two decoherent branches with probability 1/2, the theory predicts the superposition of two non-interacting "worlds," and the classical case is recovered (albeit with multiple worlds, of course). On the other hand, in Bohmian mechanics, the particle is located on one of the two branches, and the wavefunction (either the "global" one or the one resulting from the effective collapse) predicts the correct probabilities, so the classical limit is also recovered. In my opinion, this is not the case in Barandes's formulation, and this is precisely the problem Albert points out (and which I also pointed out in post #63 for the two-slit experiment).

It is the case in Barandes's formalism.

Consider for example a simple model of the Stern-Gerlach Experiment with initial particle, pointer and environment states $\ket{\psi}$, $\ket{\omega}$ and $\ket{\epsilon}$ respectively.$$U(t'\leftarrow 0)\ket{\psi}\ket{\omega}\ket{\epsilon} = \frac{1}{\sqrt{2}}(\ket{z_+}\ket{\omega_+}\ket{\epsilon_+} + \ket{z_-}\ket{\omega_-}\ket{\epsilon_-}
)$$The corresponding transition matrix of the particle and pointer device is $$\Gamma(t\leftarrow 0) = \Gamma(t\leftarrow t')\Gamma(t'\leftarrow 0)$$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 $\Gamma(t\leftarrow t')$ will have the form$$\Gamma_{ij}(t\leftarrow t') = p(z_i\omega_i, t | z_j\omega_j,t') = \delta_{ij}$$I.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.

[edit] - 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.

[edit 2] - Added explicit reference to environment.

 

[gentzen]

At the timestamp, Barandes remarks on the distinction between standalone probabilities and transition probabilities, the latter of which make up the objective dynamics of subsystems Albert concerns himself with, like the earth and Napoleon.

Thanks, now I see what you mean. And based on Barandes' answer (he doesn't want to make any "arbitrary" clarifications beyond what is enforced by his definition of indivisible stochastic dynamics), I also see why David Albert had to use his minimal completion, which doesn't give any special status to division events.

Both independence of subsystems and division events in independent subsystems can be defined purely in terms of the indivisible stochastic dynamics. And in the end, this must be Barandes' intention, independent of whether he says so explicitly or not. Hence David Abert has to reason that since division events are just a property of an indivisible stochastic dynamics (especially they are not given externally), they should have no special ontological status, unless Barandes clarifies that he wishes to given them such a special ontological status.

 

[Morbert]

Thanks, now I see what you mean. And based on Barandes' answer (he doesn't want to make any "arbitrary" clarifications beyond what is enforced by his definition of indivisible stochastic dynamics), I also see why David Albert had to use his minimal completion, which doesn't give any special status to division events.

It's quite simple: Transition probabilities, not standalone probabilities, make up the nomological content of the reformulation. No completion is needed. I suspect Albert just subjectively wants the formalism to look a bit more like Bohmian mechanics.

Both independence of subsystems and division events in independent subsystems can be defined purely in terms of the indivisible stochastic dynamics. And in the end, this must be Barandes' intention, independent of whether he says so explicitly or not. Hence David Abert has to reason that since division events are just a property of an indivisible stochastic dynamics (especially they are not given externally), they should have no special ontological status, unless Barandes clarifies that he wishes to given them such a special ontological status.

The transition probabilities of subsystems given by division events have a nomological status. They are objective, and describe the evolution of subsystems. We should not contrive to ignore them and then declare some absurdity based on this contrivance.

 

[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.