In what sense does MWI fail to predict the Born Rule?

[Michael Price]

I'm being a bit dense but where does the N*N come from?

N is whatever you choose it to be. If someone gives me the Zurek red I'll double check and report back.

 

[A. Neumaier]

I am sure there are some standard theorems, not invented specially for proving the Born Rule, which say whether and when continuous variables can be represented to any desired accuracy by discrete values.

To have a valid claim that you are sure, you must be able to point to the sources. They don't exist. See the discussion here.

 

[Michael Price]

I believe Zurek's derivation will require the following property of the environment (=rest of the universe), namely that you can decompose it into an orthonormal basis dim(N*N) :
$|env\rangle=\frac{1}{N} Σ ^{N^2}_{i=1}|env^N_i\rangle$
which is why I'd like the explicit ref so I can check.

Just to answer my own question, the above is what Carroll and Sebens use in "Many Worlds, the Born Rule, and Self-Locating Uncertainty". And they credit Zurek for the insight, so I am happy that Zurek uses the same property of the environment vector.
So the question is answered. MWI does predict the Born rule.

 

[DarMM]

Just to answer my own question, the above is what Carroll and Sebens use in "Many Worlds, the Born Rule, and Self-Locating Uncertainty". And they credit Zurek for the insight, so I am happy that Zurek uses the same property of the environment vector.
So the question is answered. MWI does predict the Born rule.

Zurek reference is here:
https://arxiv.org/abs/quant-ph/0405161

However Zurek himself acknowledges that the derivation contained in the paper is circular, relying on a well-defined branching structure that has decohered already. Something that can only be shown using the Born rule.

Hence, I do not how you can claim that MWI does predict the Born rule.

 

[Michael Price]

Zurek reference is here:
https://arxiv.org/abs/quant-ph/0405161

However Zurek himself acknowledges that the derivation contained in the paper is circular, relying on a well-defined branching structure that has decohered already. Something that can only be shown using the Born rule.

Hence, I do not how you can claim that MWI does predict the Born rule.

Zurek might be being a bit over-cautious there. Decoherence is a property of many body systems becoming progressively more entangled as time passes. The branching occurs in an irreversible fashion as a result of the decoherence, even without weighting the branches. All the Born rule is doing is supplying a weighting to the already-defined branches. Anyway, Carroll and Sebens don't seem to share Zurek's reservations, stating it works for the full range of classical to quantum.
https://arxiv.org/abs/1405.7577

 

[Derek P]

Zurek might be being a bit over-cautious there. Decoherence is a property of many body systems becoming progressively more entangled as time passes. The branching occurs in an irreversible fashion as a result of the decoherence, even without weighting the branches. All the Born rule is doing is supplying a weighting to the already-defined branches. Anyway, Carroll and Sebens don't seem to share Zurek's reservations, stating it works for the full range of classical to quantum.
https://arxiv.org/abs/1405.7577

It occurred to me a while back that if the environment provides enough branches then ordinary statistics kick in and things like the Central Limit Theorem apply to bundles of branches.. But I can see that you may have to axiomatize the conditions to avoid pathological distributions.

 

[Michael Price]

It occurred to me a while back that if the environment provides enough branches then ordinary statistics kick in and things like the Central Limit Theorem apply to bundles of branches.. But I can see that you may have to axiomatize the conditions to avoid pathological distributions.

I don't think we have to worry about such things. The derivation works due to the mere presence of the environment in the background - but the observer-observed pair don't have to interact, in any way, with the environment to get this result. The environment is not supplying the decoherence or branches - although if you do interact with the environment (as we would in real life) then more branching and decoherence occurs.

 

[Derek P]

I don't think we have to worry about such things. The derivation works due to the mere presence of the environment in the background - but the observer-observed pair don't have to interact, in any way, with the environment to get this result. The environment is not supplying the decoherence or branches - although if you do interact with the environment (as we would in real life) then more branching and decoherence occurs.

Hmm, I think that is where people will disagree. You can extend the state with dummy environmental states but that won't give you any of the dynamics of world separation. (Obviously, since a beam splitter does not give you separation of worlds despite an entire universe minus one photon, in the background.) I think you need to allow the interaction and show that the Schmidt terms are degenerate. But what would I know?

 

[Michael Price]

Hmm, I think that is where people will disagree. You can extend the state with dummy environmental states but that won't give you any of the dynamics of world separation. (Obviously, since a beam splitter does not give you separation of worlds despite an entire universe minus one photon, in the background.) I think you need to allow the interaction and show that the Schmidt terms are degenerate. But what would I know?

The environment states are not dummy states, and the dynamics is not affected by the presence or absence of the Born rule. The dynamics is given by the Schrödinger equation or equivalent EOM. Beam splitting does not split decohered worlds because it is not an irreversible event. The Born Rule doesn't need the splitting to be decohered and permanent.

 

[Derek P]

The environment states are not dummy states, and the dynamics is not affected by the presence or absence of the Born rule. The dynamics is given by the Schrödinger equation or equivalent EOM. Beam splitting does not split decohered worlds because it is not an irreversible event. The Born Rule doesn't need the splitting to be decohered and permanent.

Fairt enough. I was thinking in terms of MWI world splitting.

 

[DarMM]

Zurek might be being a bit over-cautious there. Decoherence is a property of many body systems becoming progressively more entangled as time passes. The branching occurs in an irreversible fashion as a result of the decoherence, even without weighting the branches. All the Born rule is doing is supplying a weighting to the already-defined branches.

And it is the tracing formula, in essence attaining the marginal probabilities for the system, that allows you to show that decoherence occurs. There is currently no derivation of decoherence without the Born rule present, that is decoherence can only be shown to occur if you weight the branches, otherwise it doesn't. This is the point Zurek concedes.

 

[Michael Price]

And it is the tracing formula, in essence attaining the marginal probabilities for the system, that allows you to show that decoherence occurs. There is currently no derivation of decoherence without the Born rule present, that is decoherence can only be shown to occur if you weight the branches, otherwise it doesn't. This is the point Zurek concedes.

No, you have misread Zurek. Zurek is explicit (page 25/6) that his derivation avoids using decoherence precisely because that would be circular. His derivation of the Born rule is fully quantum, being based on entanglement. Only once the Born rule is deduced can the "decoherence toolbox" (his phrase) be employed, if needed.

 

[DarMM]

Earlier Post: https://www.physicsforums.com/threa...dict-the-born-rule.946467/page-9#post-5994645

Throughout his papers Zurek proves his theorem from axioms A1-A3 that I listed earlier combined with a fourth axiom. The fourth axiom may be any of three I listed as B1-B3.

The paper I linked you to has the "best" version of Zurek's derivation, as it adopts axiom B3 as the fourth axiom, the only one with robust experimental confirmation. However this axiom means the proof no longer takes place within a Many-Worlds framework, but in the "Existential Interpretation", Zurek's own interpretation.

See
Zurek, W. (2010). Quantum Jumps, Born’s Rule, and Objective Reality. In: S. Saunders et al, ed., Many Worlds? Everett, Quantum Theory, and Reality, 1st ed. Oxford University Press, pp. 409-432.

 

[DarMM]

Also see the following papers for a discussion of how Zurek is assuming structure only known to be present after decoherence:

Howard Barnum. No-signalling-based version of zurek’s derivation of quantum probabilities:
A note on “environment-assisted invariance, entanglement, and probabilities in
quantum physics”. arXiv quant-ph/0312150, 2003.

C. M. Caves. Notes on Zurek’s derivation of the quantum probability rule.
info.phys.unm.edu/ caves/reports/ZurekBornderivation.ps

My example breakdown of Zurek's proof is based on the latter.

 

[Michael Price]

Earlier Post: https://www.physicsforums.com/threa...dict-the-born-rule.946467/page-9#post-5994645

Throughout his papers Zurek proves his theorem from axioms A1-A3 that I listed earlier combined with a fourth axiom. The fourth axiom may be any of three I listed as B1-B3.

The paper I linked you to has the "best" version of Zurek's derivation, as it adopts axiom B3 as the fourth axiom, the only one with robust experimental confirmation. However this axiom means the proof no longer takes place within a Many-Worlds framework, but in the "Existential Interpretation", Zurek's own interpretation.

See
Zurek, W. (2010). Quantum Jumps, Born’s Rule, and Objective Reality. In: S. Saunders et al, ed., Many Worlds? Everett, Quantum Theory, and Reality, 1st ed. Oxford University Press, pp. 409-432.

Well, Carroll and Sebens reference the Zurek paper you gave and I read, so it would it seem the definitive one. The derivation requires only one piece of calculation beyond elementary Hilbert space algebra, which I have already given. Namely:
$|env\rangle=\frac{1}{\sqrt{N}}∑^N_{i=1}|env^N_i\rangle$
N is is chosen to produce orthonormal states in the system-environment decomposition.
The environment can simply be the rest of the universe. No properties, including decoherence, are required of it, except that it live in a Hilbert space of exceedingly high dimension, perhaps even infinite.
The Caves notes wouldn't open on my tablet.

 

[DarMM]

Well it requires environmental noncontextuality and a few other assumptions. Caves's paper is in postscript and has a good breakdown of all the things Zurek's paper is assuming.

Again, Zurek has three versions of his proof. Two for Many-Worlds, One not. The one I linked is not a proof for Many-Worlds, but for the "Existential" Interpretation as discussed by Zurek in his chapter in the Oxford text I referenced.

 

[Michael Price]

Well it requires environmental noncontextuality and a few other assumptions. Caves's paper is in postscript and has a good breakdown of all the things Zurek's paper is assuming.

Again, Zurek has three versions of his proof. Two for Many-Worlds, One not. The one I linked is not a proof for Many-Worlds, but for the "Existential" Interpretation as discussed by Zurek in his chapter in the Oxford text I referenced.

Finally opened Caves' notes. He seems unhappy with the whole envariance business, and there's not much I can do about that. As for the MWI vs Existential interpretation - another red herring. The Born rule will work for any no-collapse approach, even the DBB pilot wave crowd.

 

[DarMM]

Finally opened Caves' notes. He seems unhappy with the whole envariance business, and there's not much I can do about that. As for the MWI vs Existential interpretation - another red herring. The Born rule will work for any no-collapse approach, even the DBB pilot wave crowd.

I'm not really sure what you're saying. In DeBroglie-Bohm, the Born rule follows from a form of thermalisation. This isn't really related to any form of demonstration of the Born Rule within MWI, I don't know how you can just declare that the Born rule will work for all no collapse approaches, since they each need very different kinds of demonstrations.

 

[Michael Price]

I'm not really sure what you're saying. In DeBroglie-Bohm, the Born rule follows from a form of thermalisation. This isn't really related to any form of demonstration of the Born Rule within MWI, I don't know how you can just declare that the Born rule will work for all no collapse approaches, since they each need very different kinds of demonstrations.

I'm referring to the idea that all pilot-wave theories are really many-worlds theories in denial - to quote D. Deutsch. The wavefunction never collapses in PW, and hence contains all the information present in Everett's MWI. So this Born Rule derivation should apply to them as well. Anyway, it is just a throw way comment, and I don't really want to take it any further - PW people just seem to get very angry when you try to explain the idea to them.

 

[DarMM]

I'm referring to the idea that all pilot-wave theories are really many-worlds theories in denial - to quote D. Deutsch. The wavefunction never collapses in PW, and hence contains all the information present in Everett's MWI. So this Born Rule derivation should apply to them as well. Anyway, it is just a throw way comment, and I don't really want to take it any further - PW people just seem to get very angry when you try to explain the idea to them.

Well regardless of how Pilot Wave advocates feel, I don't really see how this could be valid. In DeBroglie-Bohm we have a quantum potential $\Psi(x)$ with which the particle interacts. For most initial probability distributions, they will evolve to the distribution $\psi$ as a steady state, which obeys the Schrödinger equation. However at early times they will not.

So although the information is present in DeBroglie-Bohm, it means something very different. They're both psi-ontic interpretations, but that doesn't mean one is "really" the other.

 

[Michael Price]

But, as Everett pointed out, the particle in the pilot wave is superfluous in the sense that it can't be observed in any way. It should therefore be removed, leaving us with the pure MW theory. (Making its observation an axiom in the theory means the theory is inconsistent.)

 

[PeterDonis]

Thread closed for moderation; at this point it seems to be just people continuing to post their disagreement without any progress. The moderators will evaluate whether it's worth keeping the thread open.