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

[Derek P]

The OP can look up its objections and make up their own mind.

The OP has looked it up. Long ago. It struck me as absurd then and it strikes me as absurd now!

 

[Derek P]

Then it doesn't agree with experiment, because the tracing prediction is what we observe.

How can a theorem which ex hypothesi does not exist agree or not disagree with experiment?

 

[bhobba]

The OP has looked it up. Long ago. It struck me as absurd then and it strikes me as absurd now!

Ok - you think its absurd - presumably believing applying probability concepts to a deterministic theory is 'absurd'. You are not the only one. Others, including me, don't think so - but I won't wont argue it - nothing is to be gained from doing so. I just want you to cognate on the deterministic theory of BM and that it assigns probabilities - but the reason is different - or maybe not.

The precise answer to your question is if you accept the tenants of Decision Theory then you can do it - if you don't you can't - not hard really and a long thread is not required.

If you want to comment further might I suggest studying decision theory:
http://web.science.unsw.edu.au/~stevensherwood/120b/Hansson_05.pdf

Thanks
Bill

 

[Derek P]

Okay, but I'm asking what you mean by "world", if it's not a branch.

Same as it has always meant. The way the universe appears to the observer.

 

[stevendaryl]

How can a theorem which ex-hypothesis does not exist agree or not disagree with experiment?

I'm saying that for MWI to be a viable theory of the way the world is, it has to be able to make predictions that agree with our observations.

 

[A. Neumaier]

The Born rule gives the answer.

It postulates it, but only if a and b are orthonormal. Then interpreting the coefficients' absolute squares as probabilities of measuring a or b is usually called Born's rule. But Born actually claimed something different!

It is very little known that in his papers, Born did not relate his interpretation to measurement but to scattering processes. In particular, Born's 1926 formulation,

gives the probability for the electron, arriving from the $z$-direction, to be thrown out into the direction designated by the angles $\alpha, \beta, \gamma$, with the phase change $\delta$

for which he got the 1954 Nobel prize, does not depend on anything being measured, let alone to be assigned a precise numerical measurement value! Instead it has the ring of objective properties of electrons (''being thrown out'') independent of measurement.

 

[stevendaryl]

Same as it has always meant. The way the universe appears to the observer.

You keep making distinctions, but don't explain what you mean by the distinctions. You say that a "branch" is different from a "world", even though I've always thought that those terms were used interchangeably. Then you distinguished "branch counting" from "state counting", but you didn't explain what you meant by the latter.

I'm really not trying to put you on the spot---I just don't know what you're talking about, and I would like to understand.

 

[Derek P]

Ok - you think its absurd - presumably believing applying probability concepts to a deterministic theory is 'absurd'. You are not the only one.

I am not the only one because I am not one at all. It seems to me to be trivially obvious that we can assign probabilities to statements being true regardless of whether the physical system is deterministic.

Others, including me, don't think so - but I won't wont argue it - nothing is to be gained from doing so. I just want you to cognate on the deterministic theory of BM and that it assigns probabilities - but the reason is different - or maybe not.
The precise answer to your question is if you accept the tenants of Decision Theory then you can do it - if you don't you can't - not hard really and a long thread is not required.
If you want to comment further might I suggest studying decision theory:
http://web.science.unsw.edu.au/~stevensherwood/120b/Hansson_05.pdf

Will you give me your word of honour that within it I will find proof that decision theory is the only way one can derive the Born Rule?

 

[bhobba]

It postulates it, but only if a and b are orthonormal. Then interpreting the coefficients' absolute squares as probabilities of measuring a or b is usually called Born's rule. But Born actually claimed something different!

Yes true - but it's meaning has morphed somewhat.

To be precise given the observable O and the state |a> the average of the outcome is trace (Oa). You get a different answer depending on c1 and c2.

But I am being pedantic - you are correct in pointing out I was loose.

And even with Gleason it is a postulate because of that damned non-contextuality thing.

Thanks
Bill

 

[bhobba]

Will you give me your word of honour that within it I will find proof that decision theory is the only way one can derive the Born Rule?

I will only give you my word of honor you will understand it better.

Thanks
Bill

 

[A. Neumaier]

To be precise given the observable O and the state |a> the average of the outcome is trace (Oa).

But this is a completely different statement than what you claimed before, and does not give a meaning to the coefficients, unless you interpret ''average'' in a frequentist and hence probabilistic way.

Moreover, it is not what Dirac, or Weinberg, or Wikipedia formulate as postulates.

 

[bhobba]

But this is a completely different statement than what you claimed before, and does not give a meaning to the coefficients, unless you interpret ''average'' in a frequentist and hence probabilistic way. Moreover, it is not what Dirac, or Weinberg, or Wikipedia formulate as postulates.

I gave it my like but don't quite follow. Can you clarify. What I gave is the modern version of the Born rule I think formulated by Von-Neumann - it's also in Ballentine. It is actually provable using Gleason with the assumptions I gave. If you change c1 and c2 you get a different average.

I haven't read Weinberg's text yet - I should but I have been told he believes the Born Rule is separate from the observation rule - ie the eigenvalues of the observation are possible outcomes. But I do not know his exact reasoning - mine is non-contextuality.

Thanks
Bill

 

[Derek P]

You keep making distinctions, but don't explain what you mean by the distinctions. You say that a "branch" is different from a "world", even though I've always thought that those terms were used interchangeably. Then you distinguished "branch counting" from "state counting", but you didn't explain what you meant by the latter.

I'm really not trying to put you on the spot---I just don't know what you're talking about, and I would like to understand.

I know you're not. MW predicts different observer experiences. These are the worlds. The end result of an observation is to create an entanglement with countless terms since it includes the environment. These are the states

 

[stevendaryl]

I know you're not. MW predicts different observer experiences. These are the worlds. The end result of an observation is to create an entanglement with countless terms since it includes the environment. These are the states

Okay, so if I could be a little mathematical:

Let our observer be Alice. Let me assume, as I did in another thread, that Alice's experience is a coarse-grained fact about the universe. So I'm assuming that there is a projection operator $\Pi_j$ such that if $|\psi\rangle$ is a possible state of the universe in which Alice definitely has experience state $j$, then $\Pi_j |\psi\rangle = |\psi\rangle$ if it is a possible state of the universe in which Alice definitely does not have that experience state, then $\Pi_j |\psi\rangle = 0$. (This is a little bit problematic, because in a universe that is too different from the current one, there might not be a clear observer that corresponds to Alice. So maybe we just let $\Pi_j |\psi\rangle = 0$ for all worlds that are sufficiently macroscopically different from ours.)

So let's assume that initially Alice has a definite experience state, $j$. That means that $\Pi_j |\psi_0\rangle = |\psi_0\rangle$. Later, there is a splitting for Alice. Letting $|\psi(t)\rangle$ be the state of the universe after some time $t$ has passed, we will in general have:

$|\psi(t)\rangle = \sum_k \alpha_{jk} |\psi_k\rangle$

Where for each $k$, $\Pi_k |\psi_k\rangle = |\psi_k\rangle$

This split into Alice experience states is not exactly the same as splitting based on decoherence. Definitely, if $i \neq k$ then interference between $|\psi_i\rangle$ and $|\psi_k\rangle$ will be unobservable because of decoherence, but decoherence might split things further.

Anyway, would you say (assuming you understand what I'm doing and don't object to it) that "counting states" means counting the number of nonzero terms $\alpha_{jk}$?

 

[A. Neumaier]

What I gave is the modern version of the Born rule I think formulated by Von-Neumann[ - it's also in Ballentine.

Von Neumann's 1932 book (in German) contains in Chapter III a Section 1 titled (in the 1955 English edition by Princeton University Press) ''The statistical assertions of quantum mechanics'' in which he starts by postulating the spatial probability density interpretation of the multiparticle wave function. He attributes it to Born, Dirac, and Jordan. He then derives the formula for expectations (p.203 in the Princeton edition).

Wikipedia is surely modern and also defines Born's rule as a probability statement.

Ballentine does not mention Born in the context of his postulates. He postulates his statistical interpretation in Postulates 1 (measurable values are eigenvalues) and 2 (defining the expectation). Thus what you call the modern version of Born's rule is actually Ballentine's postulate. He doesn't call it Born's rule.

 

[bhobba]

what you call the modern version of Born's rule is actually Ballentine's postulate.

Got it. All true and verified it in Ballentine. He mentions Born on page 7 but only he came up with the statistical aspect.

Thanks
Bill

 

[A. Neumaier]

Can you clarify.

A very thorough discussion of Born's rule is in Section 4 of my recent draft ''Coherent foundations for quantum mechanics''. I just uploaded an improved version.

 

[A. Neumaier]

Got it. All true and verified it in Ballentine. He mentions Born on page 7 but only he came up with the statistical aspect.

The last statement is of course not true. The statistical aspect was already known in 1932 to von Neumann, and is probably even older.

Ballentine only placed the expectation before probability. This was done earlier by Peter Whittle in the third (1992) edition of his book ''Probability via expectation'' - very recommended reading!

 

[Stephen Tashi]

- presumably believing applying probability concepts to a deterministic theory is 'absurd'.

The precise answer to your question is if you accept the tenants of Decision Theory then you can do it - if you don't you can't

If you want to comment further might I suggest studying decision theory:
http://web.science.unsw.edu.au/~stevensherwood/120b/Hansson_05.pdf

How can a precise discussion arise from using that reference? It is a wide-ranging treatment of many different versions of decision theories.

The paper by Wallace that was recommended ( https://arxiv.org/abs/0906.2718 ) says:

More generally, decision theory mandates that an agent should assign a utility to each payoff, and a probability to each outcome, and that faced with any decision, the agent should choose that option which maximises expected utility with respect to those assignments

That's a conventional version of decision theory and it requires that the "agent" introduce probabilities.

One can always introduce probability into deterministic situations if one takes the view of someone (an agent) who finds himself in the midst of such a situation with only incomplete information. Introducing probability involves philosophically controversial assumptions -e.g. "The Principle Of Indifference" , the Bayesian outlook etc. This is a familiar approach so it's natural to read papers that derive probabilities from MWI as if they are implementing it - e.g. I am observer and the world is about to split, what's the probability that the observer who experiences outcome X will be me?

A harder to comprehend alternative is to take the viewpoint of an individual observer who worries about the welfare of his descendants and wonders what fraction of them will experience outcome X.

Questions about applying decision theory to MWI:

If the observer grants he is in a deterministic setting, I don't see what decisions he has available - except to "make up his mind" about his vision of the future. (Logically this presents a problem since his opinions would be determined by whatever laws rule the deterministic world. However, I suppose we can imagine he has free will.) From the viewpoint of decision theory, does the observer in MWI incur any risk or reward from forming an opinion?

I also don't see to how to apply the Bayesian outlook or The Principle Of Indifference without counting something - how many "worlds" ( or whatever) are in outcome X versus how many are in outcome Y. Some participants of the thread strenuously object to the idea that anything in the MWI is being counted. I don't understand what alternative to counting is proposed.

 

[stevendaryl]

I also don't see to how to apply the Bayesian outlook or The Principle Of Indifference without counting something - how many "worlds" ( or whatever) are in outcome X versus how many are in outcome Y. Some participants of the thread strenuously object to the idea that anything in the MWI is being counted. I don't understand what alternative to counting is proposed.

If using counting arguments or the principle of indifference, you can establish probabilities for primitive events, then the laws of probability would allow you to deduce probabilities for more complicated compound events. So you don't need to do counting on possible worlds; it's enough to know probabilities for particle decays or whatever.

 

[Stephen Tashi]

If using counting arguments or the principle of indifference, you can establish probabilities for primitive events, then the laws of probability would allow you to deduce probabilities for more complicated compound events. So you don't need to do counting on possible worlds; it's enough to know probabilities for particle decays or whatever.

In regard to the MWI, I don't see how that answers my question. (I've lost track of which participants take which viewpoints, so I can't place you answer in a specific context. ) If we assume it is meaningful to talk about the probability of certain simple events in the context of MWI then I agree, it will be meaningful to talk about the probability of more complicated events. However, I thought the main point of discussion is whether it is possible to assign probabilities to any phenomena in the context of the MWI.

Edit: Should I interpret your reply to mean an agent will have the memory of past experiments? These experiments produced specific counts for the various outcomes of some situation, so the agent may infer probabilities from these counts.

That's a reasonable approach. Then the controversies about the Born Rule center on issues like - what are we trying to show about the Born rule ? - that "most" agents will find it works? (That seems to involve counting agents.) Are we trying to show most agents will find the "global" Born Rule works? Or are we only trying to show each agent will find a Born Rule works using the equations he develops from the counts he knows about?

 

[PeterDonis]

I would not assign probabilities to worlds. As far as I know it cannot be done, and I think it is provably impossible.

Doesn't this mean that the MWI is not compatible with the Born Rule?

 

[Derek P]

Doesn't this mean that the MWI is not compatible with the Born Rule?

No. It means statements like "the probability of the dead-cat world is 30%" are essentially meaningless since all worlds co-exist. What you must do is create a statement that relates probability to the one world you are in. "The probability that I am in a dead-cat world is 30%" is meaningful.

 

[stevendaryl]

No. It means statements like "the probability of the dead-cat world is 30%" are essentially meaningless since all worlds co-exist. What you must do is create a statement that relates probability to the one world you are in. "The probability that I am in a dead-cat world is 30%" is meaningful.

But if the fateful event of killing or not killing the cat is in the future, then people certainly say things like "The probability that the cat will die is 30%". Are you saying that that is a meaningless claim? It seems like it would be meaningless, according to you, since the probability is 100% that there will be some dead-cat world in the future (and a probability of 100% that there will be some live-cat world).

 

[Derek P]

But if the fateful event of killing or not killing the cat is in the future, then people certainly say things like "The probability that the cat will die is 30%". Are you saying that that is a meaningless claim? It seems like it would be meaningless, according to you, since the probability is 100% that there will be some dead-cat world in the future (and a probability of 100% that there will be some live-cat world).

Taken literally and assuming MWI yes, it is meaningless. (IMO)

 

[DarMM]

Okay I have finished Wallace's book. All I will say is I am very thankful that Mandolesi's papers exist, otherwise it would have taken me a long time! Wallace is a very good writer, but for discussing physics the mix of formal proofs with informal discussions is quite head wrecking in some places.

Just to be clear he doesn't exactly use Gleason's versions of non-contextuality or continuity, but closely related ideas (the decision theoretic analogues in a sense). His basic set up is:

(i) A Hilbert Space of microscopic states
(ii) A set of subspaces, $\mathcal{M}$, which is the set of macrostates. Each $M \in \mathcal{M}$ is a subspace of states that are macroscopically indistinguishable
(iii) A set of events, $\mathcal{E}$, with $E \in \mathcal{E}$ being a subspace of the Hilbert Space. In short $\mathcal{M}$ are "worlds" and $\mathcal{E}$ are branch structures, where multiple worlds are in superposition.
(iv) A set of rewards. These are coarsenings (although Wallace is never quite clear on this) of $\mathcal{M}$ as you might get the same reward in different worlds.
(v) A set of acts $\mathcal{U}_{E}$, for each event, that represent bets or experiments.

He then looks at $>^{\psi}$, a preference ordering on $\mathcal{U}_{M}$. That is a preference ordering on acts available in a macroworld $M$, given the microstate is $\psi$. The idea is to prove that $>^{\psi}$ uses the Born Weights.

He has two sets of axioms. One set that ensures the set of experiments or bets $\mathcal{U}_{E}$ is rich enough without artificial restrictions (for example that it doesn't exclude composition of acts). The other set are conditions on the preference ordering $>^{\psi}$ that supposedly encode an ordering being rational.

He then proves both continuity and non-contextuality of $>^{\psi}$ (non-contextuality is Corolloray 1 on p.19 of Mandolesi's first paper) and then proceeds via a Gleason style argument.

Mandolesi, in his first paper, proves non-contexuality and a sequence of lemma's related to continuity. To be honest, I would read Mandolesi's proof rather than Wallace's, as he has redundant axioms and several points where the proof is not entirely clear. Also Mandolesi uses a simpler ordering of lemmas for the proof.

The end result is that $>^{\psi}$ is given by the Born Rule*.

Mandolesi has two classes of objections. Objections to the axioms and objections to the result. I'll discuss the latter first. Basically just because ordering of preferences for experiments/bets might use the Born Rule, does this imply the Born Rule as normally understood, i.e. would it mean within a given history records are expected to have Born Rule frequencies. One might have a way of acting rationally without this uniquely fixing physical behaviour. This will be the topic of his next paper.

Mandolesi's second paper contains the objections to the axioms**, which I will discuss in detail tomorrow. My basic assessment is that some of Mandolesi's objections are possible just a result of Wallace being unclear, or in need of tightening his statements. However others point to deep inconsistencies in the axioms or require the branching structure to have properties that are difficult to justify. In essence some of his axioms presume that branching has a small amount of the Born rule built in.

* There has been a disagreement about what the Born rule is here, that I haven't had time to read, apologies. I mean the squares of the branch coefficients in this case. I'll read that discussion tomorrow.

** Some of which are already found in Kent and others, but not as clearly or comprehensively.

 

[DarMM]

Derek P, in line with the thread title, what theorem is it that shows to you Many-Worlds does predict the Born Rule? Zurek's? Wallace's? Something else?

 

[DarMM]

Actually, tracing and decoherence do not involve measurement, hence tracing is strictly speaking an additional assumption independent of the Born rule in its conventional formulation. The latter is a statement about measurement results, but things not yet measured have no results, hence the conventional form of Born's rule is inapplicable.

Perhaps I am misunderstanding things. I thought tracing was justified as it is the only way to prefer Born statistics, for example as discussed on p.107 of
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information 10th Anniversary Edition (Cambridge University Press, Cambridge, England, 2010

Since tracing is required to demonstrate decoherence, decoherence requires the Born Rule. This is acknowledge by Zurek for example in his attempts to derive the Born rule via envariance, where he says:
'We shall, however, refrain from using the “trace” and “reduced density matrix.” Their physical significance is based on Born’s rule'

Probabilities from entanglement, Born's rule ${p}_{k}=|{\psi}_{k}|^{2}$ from envariance, Zurek, Wojciech Hubert, Phys. Rev. A, 71, 5.

I suspect I am missing something.

 

[bhobba]

The last statement is of course not true. The statistical aspect was already known in 1932 to von Neumann, and is probably even older. Ballentine only placed the expectation before probability. This was done earlier by Peter Whittle in the third (1992) edition of his book ''Probability via expectation'' - very recommended reading!

The texts all say Born came up with it in 1926 - and famously omitted the absolute square in an earlier paper eg:
http://www.math.ru.nl/~landsman/Born.pdf

However, like you, I think Von-Neumann and others knew of it earlier - it must have been in the air at the time so to speak - he only published his famous book in 1932 - he knew its contents a lot earlier. For example I think both he and Hilbert (who if I remember he was assistant to) tried to 'get' mathematically the Dirac Delta function - but never could - so used Hilbert Spaces - which I think he named after the great man.

However as far as history goes I am with Feynman - the history really belongs to historians, physicists often get more of a 'fable' handed down from physicist to physicist.

I want to add Gleason proves the trace formula I mentioned earlier - not what was stated by Born - although you can look on it as sort of a generalization. Wallace is the worst - get this - here is his version:
There is a utility function on the set of rewards, unique up to affine transformations, such that one act is preferred to another iff its expected utility, calculated with respect to this utility function and to the quantum-mechanical weights of each reward, is higher.

As I said - Wallace is very formal mathematically. I much prefer the non-contextuality theorem in the appendix of his book and Gleason - its result is much more 'usual'.

Thanks
Bill

 

[bhobba]

Since tracing is required to demonstrate decoherence, decoherence requires the Born Rule.

Of course it does. You will even see why in Susskinds book its that basic:
https://www.amazon.com/dp/0465062903/?tag=pfamazon01-20

The very simplest version is if you have an entangled system say of two subsystems and you view either subsystem it acts as a mixed state. The proof requires the Born Rule, not only to prove it but even state it and interpret it - here I mean the trace version which as has been discussed is not actually the Born Rule proposed by Max Born.

Tracing simply allows you to include an environment and 'trace out' its effects - sort of - don't hold me to any rigor in such a statement. Its just to show you can't escape it despite what some might say.

Thanks
Bill

 

[bhobba]

I suspect I am missing something.

Doubt has been cast on that by many who say its circular. I personally am with them. But why bother - we have Gleason so I don't get the fuss. Its from Quantum Darwinism and like MW I don't think contextuality makes much sense in that interpretation - but I have not seen a formal proof like Wallace.

Thanks
Bill

 

[bhobba]

Okay I have finished Wallace's book. All I will say is I am very thankful that Mandolesi's papers exist, otherwise it would have taken me a long time! Wallace is a very good writer, but for discussing physics the mix of formal proofs with informal discussions is quite head wrecking in some places.

I am literally in awe.

It took me about 6 months of a long hard slog to get through it. It is very mathematically formal. I studied math - not physics so it should have been natural for me - but wasn't.

For me the key (as far as Born goes) is the non-contextuality theorem in the appendix. You obviously had a look at it. I went through it carefully and could not break it. Did you spot an issue? Then again there may be none - it may simply be what you say at the stqart:

Just to be clear he doesn't exactly use Gleason's versions of non-contextuality or continuity, but closely related ideas (the decision theoretic analogues in a sense).

This would seem to imply the whole thing boils down to - can you use decision theory in MWI?

Thanks
Bill

 

[bhobba]

How can a precise discussion arise from using that reference?

I think it provides enough BACKGROUND to tackle the paper I referenced by Wallace. Deric seems to doubt it's even applicable - what I want to know is why he says that - exactly what part of decision theory can't be used and why. That obviously requires an acquaintance with decision theory.

Currently as far as I can see he is not saying anything specific - just things like:

No. It means statements like "the probability of the dead-cat world is 30%" are essentially meaningless since all worlds co-exist. What you must do is create a statement that relates probability to the one world you are in. "The probability that I am in a dead-cat world is 30%" is meaningful.

Such statements are meaningful in terms of decision theory. In MWI the world a hypothetical observer would find themselves in is considered like a bet or wager and that of course is exactly what decision theory is designed to sort out - what is the probability of the outcome of that wager. That's the way its presented but as to an actual observer etc - I am not so sure - see my later comments.

Ok - to the other stuff:

1. Does the Bayesian interpretation of probability require an observer? It is the degree of plausibility according to the cox axioms:
http://ksvanhorn.com/bayes/Papers/rcox.pdf

Here its looked on as a generalization of logic. Does the rules of logic/plausibility require an observer? I simply pose these as issues - the answer likely would take us deep into philosophy which isn't really what we are concerned with here. Indeed is it anymore concrete that the notion of probability in the Kolmogerov axioms. That is one reason I am not a fan of the Bayesian view of probability - really what does it resolve? Sometimes its more useful to look at it that way in applied problems - he decision theory is used - but it's not better or worse than other views - just what helps best in solving something. Here decision theory was chosen by Wallace and others as best for MW. I am starting to get the feeling the real issue is - can you do that? If so another implied axiom of MW is you can use decision theory to calculate probabilities and again a long philosophical argument about it that really foes nowhere.

2. Can you expand on exactly what you mean by counting in this context. Of course numbers are required from the very definition of probability, but counting I do not quite get.

Thanks
Bill

 

[A. Neumaier]

The texts all say Born came up with it in 1926

But the paper they quote does not contain more than what I had quoted in post #111 - i.e., only the interpretation of the scattering amplitudes, which is a by far weaker (and much more defendable) statement. The generalized form seems to have first been stated by Dirac in his Lectures on Quantum Mechanics, but I don't have a copy of the first edition, hence cannot check.

 

[A. Neumaier]

This is acknowledge by Zurek for example in his attempts to derive the Born rule via envariance, where he says:
'We shall, however, refrain from using the “trace” and “reduced density matrix.” Their physical significance is based on Born’s rule'

I suspect I am missing something.

Well, the question is what precisely is taken to be Born's rule. If one takes the definition in Wikipedia (which seems to be the most prevalent form) then it is a statement about measurement.

But nothing can follow from a statement about measurement about processes not involving measurement by the rules of logic alone. This is a basic inconsistency in all derivations of the ensemble mean, used in decoherence. Born's rule in the form stated by Wikipedia only allows conclusions about the expectation of sample means from actual measurements! On the other hand, the ensemble mean is a purely theoretical construct (part of shut-up-and-calculate) abstracted from the latter by analogy.