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

[bhobba]

That sounds equivalent to Bryce Dewitt's argument from his 1970 article, Quantum Mechanics and Reality, although he expressed it slightly differently. But, yes, the measure (Hilbert space norm) of worlds where the Born Rule is violated vanishes - hence the Born Rule is obeyed. I've always found this argument quite persuasive and always been somewhat bemused at the complexity of other "derivations".

How can the Born rule be violated in any world since its non-contextual? Wallace proves it, but it doesn't really make any sense for a theory with just the wave function to be contextual.

Thanks
Bill

 

[stevendaryl]

But wave functions convey physics only up to an x-independent phase. Your distance doesn't and hence is physically meaningless.

I think you're confused. The Hilbert space of square-integrable functions includes phase information, and as I said, is only defined modulo sets of measure 0. Then of functions in the Hilbert space, we further say that physical states are rays. But you can't work with Hilbert space by dealing with rays as the basis elements.

 

[stevendaryl]

How can the Born rule be violated in any world since its non-contextual?

I think we might be talking about different things. If I prepare an electron to be spin-up in the z-direction, and then measure its spin in the x-direction, the Born rule says that it's 50% likely to be spin-up or spin-down. If I perform a billion such measurements, then there is nonzero probability that the number of spin-up results will be 0 or a billion or whatever. There is no guarantee, for a finite number of trials, that the relative frequencies will be equal to (or close to) what is calculated using the Born rule.

I guess it's a matter of terminology as to whether you say that that's a violation of the Born rule, or is a violation of the expectation that relative frequencies are approximately equal to theoretical probabilities.

 

[A. Neumaier]

I think you're confused.

No. Any measure that can mean something physical necessary must satisfy that the distance between $\psi(x)$$and $\lambda\psi(x)$ is zero when $\lambda|=1$, since these correspond to identical state. Your measure doesn't satisfy this trivial requirement.,

 

[stevendaryl]

No. Any measure that can mean something physical necessary must satisfy that the distance between $\psi(x)$$and $\lambda\psi(x)$ is zero when $\lambda|=1$, since these correspond to identical state. Your measure doesn't satisfy this trivial requirement.,

I don't know what you're talking about. I'm not proposing a measure on possible worlds, I'm stating a fact about the Hilbert space of square-integrable functions.

 

[bhobba]

And yet the worlds which obey the Born Rule are also infinitely old. Their norm tends to 1, whilst the norm of the worlds where the Born Rule is violated tends to 0. That has to be telling us something.

Well in MW it follows the Schrodinger equation - which usually deals with norms of 1, - so exactly how does it tend to 1? I can see after decoherence a world may have a norm less that 1 so we need to normalize it to apply the Schrodinger equation - but after infinite time without that re-normalization it would tend to zero.

As I have mentioned Gell-Mann thinks his Decoherent-Histories is just MW with different semantics - he calls it a post Everton interpretation or something like that. In Decoherent Histories QM is a stochastic theory of histories but only one history exists - so you don't have this dilution issue of MW - nor this exponentially increasing number of worlds. This guy thinks they are isomorphic:
https://www.hedweb.com/manworld.htm#many

I would not go that far, but there is a strong connection such that the differences SEEM to disappear or be trivial on careful examination.

As I said I learned a lot about decoherent histories from reading Wallace - but have not found a really good book on Decohreret Histories - Griffiths Consistent Histories is a bit too basic for me. Good for those starting out - say after reading Susskind.

Thanks
Bill

 

[bhobba]

I think we might be talking about different things.

Yes in that sense its just the law of large numbers - but since we are taking probabilities it really comes with the territory.

Thanks
Bill

 

[A. Neumaier]

I don't know what you're talking about. I'm not proposing a measure on possible worlds, I'm stating a fact about the Hilbert space of square-integrable functions.

But this fact is irrelevant for physics, which only cares for states (rays in Hilbert space), so your subsequent conclusions about the Born rule are irrelevant for physics, too.

 

[DarMM]

To the extent that the limiting case actually makes sense, you can say that there are no worlds that disagree with the Born rule, because sets of measure zero are ignored in the definition of Hilbert space.

I don't remember the details of how this limit actually makes sense, so take it with a grain of salt...

I haven't found much formal treatment of DeWitt's limit in the literature. The usual criticism is that for actual finite measurements the coefficients in front of maverick worlds are still non-zero and without any meaning attached to the coefficients via Born's rule, there is no reason to discount them, they still dominate a pure count of branch terms. DeWitt himself says that one has to introduce some meaning to the branch terms, like a "width" of worlds etc to get his result, but it's never really explained what these sorts of ideas mean physically, i.e. what is the "width of a world"?

 

[DarMM]

You can call them different interpretations, but they all agree that the physical (ontic) situation maps 1-to-1 onto unitary QM. That's what the whole Everett program is about. The extra structure given to it is just descriptive. There are different uses of the Born rule - one about subjective experience of agents and one about the objective coefficients of worlds, so that's a fair point to disagree about.

Why do you think WMI cannot just assume the Born rule, interpreted as a measure of existence?

I hope this answers your question. Originally when you raised this I was discussing Wallace and Zurek assuming the results of decoherence, when they in fact need to derive them. Other versions of Many-Worlds might not need to do this, as they assume the Born Rule or a weaker version of it, etc. Zurek and Wallace do not, they have a different use of the Born rule as you say. This is why they attempt to derive it.
I think you took my statement as concerning all MWI versions rather than those of Wallace and Zurek.

For ones like Zurek's, if their derivation makes use of a structure (post-decoherence branching) that thus far has only been shown to arise via use of Born's rule they are in a bind, as Zurek himself has said, hence his Quantum Darwinism project.

 

[akvadrako]

I hope this answers your question.

It's clear enough now; thanks for the clarifications.

 

[bhobba]

The usual criticism is that for actual finite measurements the coefficients in front of maverick worlds

I have never even heard of them. But did a search and sure Everett talks about them. I got a paper on it:
https://arxiv.org/pdf/1511.08881.pdf

I simply do not get the issue - it says:
Everett defined maverick branches of the state vector as those on which the usual Born probability rule fails to hold – these branches exhibit highly improbable behaviors, including possibly the breakdown of decoherence or even the absence of an emergent semi-classical reality.

I am still in the dark. A world happens after decoherence - how can it loose decoherence - beats me. I think it's to do with his view on the proof of the Born Rule - but Wallace is a more modern approach that doesn't seem to have this issue - if an issue it is.

Zurek and Wallace do not, they have a different use of the Born rule as you say. This is why they attempt to derive it.

That's probably the answer - I have only really studied Wallace.

Thanks
Bill

 

[DarMM]

I am still in the dark. A world happens after decoherence - how can it loose decoherence - beats me. I think it's to do with his view on the proof of the Born Rule - but Wallace is a more modern approach that doesn't seem to have this issue - if an issue it is.

I think what might be meant is stability, that some of the "worlds" will interfere with each other, so there is no long lived classical physics. Although I'd say something like those branches simply aren't worlds, as you mentioned. Everett permitted any arbitrary partitioning of the Hilbert Space, so he allows basis where decoherence doesn't occur.

Maverick Worlds in modern MWI are the stable quasi-classical branches (i.e. decohered branches) along whose history experimental frequencies don't hold to the Born rule, non-decoherent basis aren't considered worlds. Hsu is sticking to Everett's original use of "world", which is in essence, any 1D subspace.

That's probably the answer - I have only really studied Wallace.

Wallace assumes the existence of (highly) stable quasi-classical branches to begin with and hence there are no "coherent worlds". His MWI does have Maverick worlds in the modern sense, but it is always more rational to act is if your world will stay or become non-Maverick.

Zurek similarly already has quasi-classical branches as well and there are maverick worlds but, as his method reduces to a form of branch counting, there is simply less of them.

 

[DarMM]

If anybody wants the short version of either proof, the idea as such.

Zurek is basically saying, if quantum states are already associated with probabilities in some way and the world post-measurement is known to evolve into a decoherent form to near perfect accuracy, then environmental noncontextuality (environment has no affect on probabilities) allows us to demonstrate that terms with equal magnitude coefficients are equiprobable. This provides us all we need to prove Born's rule in general via branch counting, as the general case can always be reduced to the equal magnitude case by using the environment to split branches with larger coefficients into multiple branches with equal coefficients. That is in $\sqrt{\frac{1}{3}}|\uparrow\rangle + \sqrt{\frac{2}{3}}|\downarrow\rangle$ the environment can be used to show that there are two down worlds.

Wallace says, if we live in a multiverse with highly decohered parallel branches to begin with and agents in those branches have a certain control over the environment (erasure axiom), then under a certain definition of rationality*, it is always more rational to act as if the Born Rule were true.

*Rationality here has been criticised, but essentially the agent doesn't care about superpositions or branching in and of themselves, i.e. two worlds where you won the lottery are as valuable as one world where you did. Also the agent values more experiments whose average payout among his post-measurement selves is highest. Agents who value things like "best highest payout world" or "best worst outcome world" are excluded from the definition of rational.

 

[stevendaryl]

Wallace assumes the existence of (highly) stable quasi-classical branches to begin with and hence there are no "coherent worlds". His MWI does have Maverick worlds in the modern sense, but it is always more rational to act is if your world will stay or become non-Maverick.

I don't see the latter. If you imagine a world in which relative frequencies for events turn out to not be those predicted by QM, then in that world, nobody would have the slightest reason to develop QM in the first place. So rational agents certainly would not have reason to think the world would soon become non-Maverick.

 

[DarMM]

Yes, I should point out, the proof implicitly assumes they know QM in full, i.e. they are aware of their Maverick status.

 

[Jehannum]

In the Transactional Interpretation the Born Rule arises quite naturally.

 

[Derek P]

In the Transactional Interpretation the Born Rule arises quite naturally.

If you can call messages from the future "natural". But anyway, the ingredients for calculating a density matrix may all be there but the mechanism for selecting one interaction is, correct me if I'm wrong, a random choice "agreed" by the two participants. So it's a bit irrelevant to MWI, which is deterministic and does not have any such choice. Is that fair?

 

[Jehannum]

If you can call messages from the future "natural". But anyway, the ingredients for calculating a density matrix may all be there but the mechanism for selecting one interaction is, correct me if I'm wrong, a random choice "agreed" by the two participants. So it's a bit irrelevant to MWI, which is deterministic and does not have any such choice. Is that fair?

Yes, it's fair. I was actually replying to this much earlier post, which I should have quoted:

In what sense does any approach to QM derive the Born Rule? - as opposed to taking it as an assumption.

Perhaps that's a good topic for another thread.

And yes, I suppose the advanced waves do take a little getting used to.

 

[Derek P]

Zurek is basically saying, if quantum states are already associated with probabilities in some way and the world post-measurement is known to evolve into a decoherent form to near perfect accuracy, then environmental noncontextuality (environment has no affect on probabilities) allows us to demonstrate that terms with equal magnitude coefficients are equiprobable. This provides us all we need to prove Born's rule in general via branch counting, as the general case can always be reduced to the equal magnitude case by using the environment to split branches with larger coefficients into multiple branches with equal coefficients.

Okay. So we need a set of axioms to ensure environmental noncontextuality. I need to go over what you've posted but what's the short version of how Zurek does this? I am thinking that it might depend on some assumptions about how particles interact - locally, reversibly etc - to prove the existence of an ensemble from which the actual environment provides an unpredictable sample?

 

[Derek P]

And yes, I suppose the advanced waves do take a little getting used to.

Backwards causality always does!

 

[Derek P]

Yes, I should point out, the proof implicitly assumes they know QM in full, i.e. they are aware of their Maverick status.

Does it? How can the best strategy depend on the agent's knowledge? I thought the best strategy was to use the numbers that we know are given by the Born Rule whether it seems to fit with the agent's experience or not and whether the agent knows how to calculate them or not. They might not even know what the numbers are if, when the Mavericity has abated, they pick a strategy "at random". The best ones after would still be the ones that reflected the Born Rule even if it was a mystery to the agent as to why picking a strategy out of a hat has proved so successful. Or am I missing the point completely?

 

[Michael Price]

If anybody wants the short version of either proof, the idea as such.

Zurek is basically saying, if quantum states are already associated with probabilities in some way and the world post-measurement is known to evolve into a decoherent form to near perfect accuracy, then environmental noncontextuality (environment has no affect on probabilities) allows us to demonstrate that terms with equal magnitude coefficients are equiprobable. This provides us all we need to prove Born's rule in general via branch counting, as the general case can always be reduced to the equal magnitude case by using the environment to split branches with larger coefficients into multiple branches with equal coefficients. That is in $\sqrt{\frac{1}{3}}|\uparrow\rangle + \sqrt{\frac{2}{3}}|\downarrow\rangle$ the environment can be used to show that there are two down worlds.

...

Can you provide a link or ref to Zurek's paper where he does this? It sounds like the right approach to me, and merits further detailed examination.

 

[ftr]

I have indicated about this issue before, but this time I will ask a direct question.

In standard QM Born rule follows on axiom of the wavefunction( squaring), so shouldn't the derivation be for Schrodinger equation first.

Secondly, in physics we have equations then we interpret its part as corresponding to some elements in reality( which themselves are interpretations of measurement), so the interpretation of QM seems dubious. It is like people in Faraday's days spending all their time interpreting the ontology of the lines of force, while the correct way would have been to find the "more correct" equations/relations that correspond to reality, shouldn't it.

 

[Derek P]

I have indicated about this issue before, but this time I will ask a direct question.

In standard QM Born rule follows on axiom of the wavefunction( squaring), so shouldn't the derivation be for Schrodinger equation first.

Secondly, in physics we have equations then we interpret its part as corresponding to some elements in reality( which themselves are interpretations of measurement), so the interpretation of QM seems dubious. It is like people in Faraday's days spending all their time interpreting the ontology of the lines of force, while the correct way would have been to find the "more correct" equations/relations that correspond to reality, shouldn't it.

Well, I put this in an A level thread with a request (largely respected) to keep the maths in check. So the vector state and Hilbert space formalism can be taken for granted. Indeed, they have not been queried. And in MWI they are a "given". Zurek's proof of the Born Rule is simple once he establishes that the Schmidt decomposition is degenerate. The difficulty is in establishing axioms to prove that the environment does have the necessary characteristics. It seems clear to me that any kind of proof is going to involve the way individual interactions within the environment occur. Which is an eye-opener.

 

[Michael Price]

Well, I put this in an A level thread with a request (largely respected) to keep the maths in check. So the vector state and Hilbert space formalism can be taken for granted. Indeed, they have not been queried. And in MWI they are a "given". Zurek's proof of the Born Rule is simple once he establishes that the Schmidt decomposition is degenerate. The difficulty is in establishing axioms to prove that the environment does have the necessary characteristics. It seems clear to me that any kind of proof is going to involve the way individual interactions within the environment occur. Which is an eye-opener.

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.

 

[A. Neumaier]

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.

The environment contains particles, hence is necessarily represented by an infinite dimensional Hilbert space.

 

[Derek P]

The environment contains particles, hence is necessarily represented by an infinite dimensional Hilbert space.

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. It is a very plausible conjecture given the linearity of QM, at least to my non-mathematical mind. So a finite dimensional Hilbert space should be just fine unless the state space representation introduces unnecessary restrictions.

 

[Derek P]

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.

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

 

[Michael Price]

The environment contains particles, hence is necessarily represented by an infinite dimensional Hilbert space.

The Fock space is infinite dimensional, although the Bekenstein bound would cap that dimensionality, according to the entropy. But these are red herrings, as far as In understand it.