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

[bhobba]

I know the proof of Gleason's theorem, but it has never genuinely helped me comprehend the MWI arguments as it comes from a very different direction, Wallace argues that his proof is a separate line of argumentation to Gleason. I think I need to read Wallace's book in full perhaps.

Look into the non-contextuality theorem in the appendix of Wallace. It may have an error - but I couldn't find it. That means Gleason applies.

IMHO the real issue is, yes there are physical reasons one can ague about regarding the proof but the real bug bear is - how to introduce probabilities into a deterministic theory. I won't say what I think, you probably have guessed it, arguing positions is not something I enjoy that much. A little bit is OK. I think its much better to understand the pro's and con's of different views - to that end form and elucidate your view but arguing just seems to go on and on not really getting anywhere. Just my view of course - mentors as a group will ensure it all remains under control.

Interesting to hear what you think after reading Wallace. I actuay found it very illuminating of decoherent histories as much as MW - interesting. I like its theorem/proof approach due to my math background but I know its not every-ones cup of tea.

Thanks
Bill

 

[stevendaryl]

Okay. So let's pretend there isn't a theorem and tracing is not justified.

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

 

[bhobba]

Tracing is required to obtain decoherence and tracing requires the Born Rule. Hence decoherence does require the Born rule. I'm reading Wallace's book right now, so I'll see what he says.

Tracing does require the Born rule - and other things - but the mixed state after decoherence may or may not be what we call an observation. One out, one I actually like, is simply define it that way - but it just semantics.

Decoherent histories replaces measurement with consistency - but the details require to study a text on it.

I personally like Schlosshauer's book - it looks at a lot of interpretations and decorehrence implications:
https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20

Thanks
Bill

 

[stevendaryl]

No, it's state counting.

What do you mean by "state counting"? In the MWI, there is only one state for the entire universe, so counting states always gives you the answer 1.

Do you mean counting the number of terms in the superposition? If so, then how is that different from branch counting?

 

[bhobba]

Okay. So let's pretend there isn't a theorem

But there is a theorem - you can't waive away Gleason. The only out with Gleason is non-contextuality - non-contextual interpretations usually have there own way of handling the measurement issue.

The measurement issue has problems - but we know things about it the early pioneers did not - there is no getting around it. Progress will not be made by - let's suppose something we know is true is not.

As I have said please please try to understand interpretations pro's and cons. I think your question has been answered. But if you or anyone wants to keep it going - go ahead. It will be shut down if it becomes simply argumentative.

Thanks
Bill

 

[DarMM]

Okay, just to say, I'm going to stay away from this thread until I finish Wallace and Schlosshauer's books. I'm reading Wallace's book using Mandolesi's papers as a guide, both the 2015 and 2018 ones.

 

[Derek P]

What do you mean by "state counting"? In the MWI, there is only one state for the entire universe, so counting states always gives you the answer 1.

Do you mean counting the number of terms in the superposition? If so, then how is that different from branch counting?

There are googols of terms in "the superposition" but only a small number of worlds. It's up to you which one you call "branches" but they are certainly very different.

 

[bhobba]

Okay, just to say, I'm going to stay away from this thread until I finish Wallace and Schlosshauer's books. I'm reading Wallace's book using Mandolesi's papers as a guide, both the 2015 and 2018 ones.

Its a good approach, but you chose a tough one. I don't know if I could do it. I would post my latest thinking as I learn more. You have my very humble admiration.

Thanks
Bill

 

[A. Neumaier]

You can (I believe) derive the Born rule by expanding the state as a sum of components.

This only gives a decomposition but assigns no meaning or frequencies/probabilities to the terms. You have to postulate that each term has a given frequency/probability. This is a postulate far worse than Born's rule since Born's rule has at least an empirical support, but assigning probabilities to worlds of which only one is observed is completely arbitrary.

 

[A. Neumaier]

Okay, just to say, I'm going to stay away from this thread until I finish Wallace and Schlosshauer's books. I'm reading Wallace's book using Mandolesi's papers as a guide, both the 2015 and 2018 ones.

Schlosshauer's book is probably the best modern discussion of the measurement problem.

 

[Derek P]

But there is a theorem - you can't waive away Gleason. The only out with Gleason is non-contextuality - non-contextual interpretations usually have there own way of handling the measurement issue.

The measurement issue has problems - but we know things about it the early pioneers did not - there is no getting around it. Progress will not be made by - let's suppose something we know is true is not.

As I have said please please try to understand interpretations pro's and cons. I think your question has been answered. But if you or anyone wants to keep it going - go ahead. It will be shut down if it becomes simply argumentative.

Thanks
Bill

I don't think it has been answered. If you decide to moderate it please just remove the offending subthreads.
I would agree that Gleason is conclusive but just saying so doesn't account for why the claim is so often made that MWI fails to deliver Born.

 

[Derek P]

This only gives a decomposition but assigns no meaning or frequencies/probabilities to the terms. You have to postulate that each term has a given frequency/probability. This is a postulate far worse than Born's rule since Born's rule has at least an empirical support, but assigning probabilities to worlds of which only one is observed is completely arbitrary.

I have no idea what you mean. I would not assign probabilities to worlds. As far as I know it cannot be done, and I think it is provably impossible.

 

[bhobba]

This only gives a decomposition but assigns no meaning or frequencies/probabilities to the terms. You have to postulate that each term has a given frequency/probability. This is a postulate far worse than Born's rule since Born's rule has at least an empirical support, but assigning probabilities to worlds of which only one is observed is completely arbitrary.

Gleason is the only generally accepted one.

It requires a few assumptions:
1. Assigning a probability to an outcome of an observation is the correct way to go.
2. Non-Contextuality.
3. The principle that any superposition is at least in principle possible - called if I remember the strong principle of superposition.

There may be others - but I just can't recall them.

The decision theoretic approach is much more controversial than Gleason. The OP can look up its objections and make up their own mind. There are others such as Quantum Darwinism that think they can prove it from entanglement - again its controversial.

Thanks
Bill

 

[stevendaryl]

There are googols of terms in "the superposition" but only a small number of worlds. It's up to you which one you call "branches" but they are certainly very different.

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

 

[bhobba]

I have no idea what you mean. I would not assign probabilities to worlds. As far as I know it cannot be done, and I think it is provably impossible.

The principle of superposition says given any two states |a> and |b> then another state is c1*|a> + c2*|b> ie it forms a vector space what is the physical meaning of c1 and c2?

The Born rule gives the answer.

Thanks
Bill

 

[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.