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

[DarMM]

You've lost me. There's always a physical meaning to the Born Weights. The principle of induction doesn't depend on it though and yes it will mislead you on occasion. I don't know what you're getting at at all.

I will explain this in more detail.

There have been two points here:

1. The point that Wallace leaves you with no idea as to why using the Born weights is more rational, just that in his version of Many-Worlds it is. However he does not provide a structure to the branching multiverse that explains their use. There a several possible handwaving models you could conceive of that would explain their use, but they are all quite different and don't have rigorous mathematical baking as of 2018.
2. The idea that Many-Worlds, ignoring any proofs of the Born rule, like Wallace's and Zurek's will always have some worlds where the Born weights are the ratios of experimental observations. The problem here is that without some proof connecting the weights to physical observations, these worlds are simply (vanishingly rare) flukes, not in any sense common. There will also be worlds where the ratios are $f(\alpha_k)$ rather than $\alpha_k$ and these worlds are no more or less common.
   Hence the Born rule fails to be predictive, regardless of the induction principle.

 

[DarMM]

Well Zurek's proof is very simple.

The idea is take a Schmidt state:

$$\psi = \Sigma_{i} \alpha_{i} |\sigma_{i}\rangle |\eta_{i}\rangle \tag 1$$

with $|\sigma_{i}\rangle$ being states of the microscopic system and $|\eta_{i}\rangle$ being environment states.

The state is said to be envariant under a transformation, $U_{s}$, of the microscopic system, if it can be undone by a transformation, $U_{\eta}$, of the environment .

Lemma 1: For states with $|\alpha_i| = |\alpha_k|$ for some $i,k$, then swapping $|\sigma_{i}\rangle$ and $|\sigma_{k}\rangle$ is an envariant transformation.
That is, for Schmidt states, swaps of microsystem states with equal magnitude coefficients can be undone by the environment.

Next Zurek has four axioms, with a theorem that can be proved from the first three.

A1. To represent an alteration of a system $S$, a unitary transformation must act on its Hilbert Space $\mathcal{H}_{S}$
A2. All information on observables, their values and their probabilities for a system $S$ is fully captured by the state of $S$. This state might be mixed or pure.
A3. The state of a subsystem is fully specified by the state of the total system.

From these three axioms you can conclude:
Theorem 1: Phases do not affect probabilities, i.e. probabilities depend only on $|\alpha_{i}|$.

The final axiom comes in three versions, anyone of them may be added to the list as the final axiom.

B1. If swapping two orthogonal states leaves the state of the system $S$ unchanged, the probabilities of the outcomes associated with those states are the same.
B2. If all unitary transformations within a subsystem $\bar{S}$ of $S$ leave $S$ unchanged, then the probabilities of any state in an orthonormal basis of $\mathcal{H}_{\bar{S}}$ are equal.
B3. The probabilistic meaning of a Schmidt state is that the environment and the state are perfectly correlated, i.e. observing $|\sigma_{i}\rangle$ means $|\eta_{i}\rangle$ will be observed with probability 1.

Any one of these axioms allows him to prove the following:
Theorem 2: Terms with equal amplitudes in Schmidt states like $(1)$ have equal probabilities.
Corollary: In a Schmidt state with all $N$ coefficients equal, all outcomes are equally likely, i.e. $1/N$.

Looking ahead I will say that using B2 permits you to prove this without using A2.

From there he uses the fact that the environment can always be enlarged by adding an extra system to reduce the unequal amplitude case to the equal amplitude case. As an example say we have:

$$\psi = \sqrt{\frac{1}{3}}|\sigma_1\rangle|\eta_1\rangle + \sqrt{\frac{2}{3}}|\sigma_2\rangle|\eta_1\rangle \tag 2$$

We can introduce another system, $\beta$, essentially enlarge the environment so that this becomes:

$$\psi = \sqrt{\frac{1}{3}}|\sigma_1\rangle|\eta_1\rangle|\beta_1\rangle + \sqrt{\frac{2}{3}}|\sigma_2\rangle|\eta_2\rangle|\beta_2\rangle \tag 3$$

Then provided the new environment $\beta$ is large enough so that most of its eigenstates (the states that couple to the microscopic system) are degenerate, we can expand them enough to counterweight the unequal amplitudes:

$$\beta_1 = \gamma_1 \tag 4$$

$$\beta_2 = \sqrt{\frac{1}{2}}\gamma_{2,1} + \sqrt{\frac{1}{2}}\gamma_{2,2} \tag 5$$

and so, substituting $(4),(5)$ into $(3)$:

$$\psi = \sqrt{\frac{1}{3}}|\sigma_1\rangle|\eta_1\rangle|\gamma_1\rangle + \sqrt{\frac{1}{3}}|\sigma_2\rangle|\eta_2\rangle|\gamma_{2,1}\rangle + \sqrt{\frac{1}{3}}|\sigma_2\rangle|\eta_2\rangle|\gamma_{2,2}\rangle$$

Hence this is now an equal amplitude case, and by Theorem 2 all have equal probability $1/3$. Since $\sigma_2$ appears twice, we can say it has $2/3$ chance of being seen.

And so Zurek obtains the Born rule for amplitudes that are roots of rationals.

To obtain it for all reals, he uses the fact that $\mathbb{Q}$ is dense in $\mathbb{R}$ and that an "arbitrarily" fine grained* larger environment can be found.

Thus we have the Born rule. Issues to follow.

*In the sense of having as large as necessary expansion in the form of $(5)$

 

[Derek P]

I will explain this in more detail.

There have been two points here:

1. The point that Wallace leaves you with no idea as to why using the Born weights is more rational, just that in his version of Many-Worlds it is. However he does not provide a structure to the branching multiverse that explains their use. There a several possible handwaving models you could conceive of that would explain their use, but they are all quite different and don't have rigorous mathematical baking as of 2018.
2. The idea that Many-Worlds, ignoring any proofs of the Born rule, like Wallace's and Zurek's will always have some worlds where the Born weights are the ratios of experimental observations. The problem here is that without some proof connecting the weights to physical observations, these worlds are simply (vanishingly rare) flukes, not in any sense common. There will also be worlds where the ratios are $f(\alpha_k)$ rather than $\alpha_k$ and these worlds are no more or less common.
   Hence the Born rule fails to be predictive, regardless of the induction principle.

Obviously we need a "proof connecting the weights to physical observations"! If Wallace fails to provide it, why are we even discussing his work?

 

[DarMM]

Obviously we need a "proof connecting the weights to physical observations"! If Wallace fails to provide it, why are we even discussing his work?

First of all, because it is one of the two major purported proofs of the Born rule in Many Worlds. The thread is about "in what sense does MWI fail to predict the Born rule", answering that requires addressing the failures of the major approaches.

Also he does provide one, but it requires features (e.g. branching structure without using the normal decoherence formalism, powerful erasure operations) that have not been themselves proven to be valid. Hence in a Wallacean Multiverse, provided it can actually be shown to arise from pure unitary QM with no assumptions, does have physically relevant Born weights.

 

[Derek P]

Well Zurek's proof is very simple.

The idea is take a Schmidt state:
[...]
Thus we have the Born rule. Issues to follow.

Excellent. Thanks. I eagerly await the "issues" because, from where you say "From there ...", it looks remarkably like what I have been calling "state counting" and which Price's proof (which was immediately pooh-poohed) uses and which @stevendaryl seems to be heading towards and which I had, until this thread, assumed was the only one anyone would ever consider to be mainstream. I guess I was misled by the obvious fact that orthogonal vectors add by Pythagoras but their probability measures add linearly - ergo Born.

I think the fact that Zurek uses a Schmidt decomposition needs an axiom or three - certainly his use of it would explain why at some point he says MWI requires a postulate that systems exist. With such an axiom you can create the necessary composite system out of subsystems. Otherwise I suppose you would have to prove that a sufficiently large system's state space can be factorized in a way recognisable as subsystem state spaces. Or something like that, no doubt expressed more elegantly. As a non-mathematician I imagine it would be another can of worms.

Anyway, the thing that strikes me immediately is that axioms B mention probability without saying what it means. I don't think Zurek would be silly enough to introduce a Deus ex Machina. But before quantifying it as amplitude squared, B needs to be related to some definition of probability. Otherwise we are back to "If there is zippettybopp and the following axioms which quantify zippettybopp apply, then zippettybopp follows the XYZ Rule". Not much use in a model of the real world if nobody knows what zippettybopp means. And I'm not going to tell you.

 

[bhobba]

The point that Wallace leaves you with no idea as to why using the Born weights is more rational, just that in his version of Many-Worlds it is.

That is the big problem with Wallace - it took me about 6 months to fully study it. It is very mathematical and I could not break any of it's theorem's - but its just that - very mathematical. In fact it helped me understand decoherent histories a lot better. I have said its mathematically very beautiful - but math is not physics. The big issue is can the assumptions it makes ie can you use decision theory to deduce the Born Rule. If you accept yes its fine - if not its challengeable. But if you have a deterministic theory that does not give the version of you that is in a particular world then what can you do? If you reject it then the whole thing goes down the gurgler in the sense in MW all you have is a wave-function and that's it.

Personally even without the non-contextuality proof I don't think contextuality makes much sense in MW so you have Born via Gleason. But like all interpretations I guess it's a matter of personal taste. I think MW is just too silly to believe - that's not scientific - just my gut feeling. I remember a long discussion with a philosopher who thought I could not think like that - I must have a rational reason. In the end all I could say is this is science - not philosophy. There are certain indefensible beliefs such I think most scientists think we are slowly getting closer and closer to some objective truth - I certainly do. I can't prove it, but I believe it very strongly. Guess that's the difference between science and philosophy.

Thanks
Bill

 

[AlexCaledin]

I think MW is just too silly to believe - that's not scientific - just my gut feeling.

I think MW is just requiring to take it easy! If a theorist says there are "really" many worlds - and many "real" variants of me - then, the important thing is to remember that that "realness" (and anyone's "belief" in it) is theoretical. My actual business, anyway, is to see that, in the world I am observing, I be not too bad - so, to calculate what I need, I have to add the wavefunction collapse to that theory and "return to Copenhagen" - with better understanding as to whence that collapse comes (namely, from the rules of the actual business of life).

 

[Stephen Tashi]

I guess the issue is whether the rational agent would continue to believe the Born rule after you've explain "Many Worlds" to him.

Is the rational agent supposed to know the Born weights? For example, if he is an agent in a "maverick" world, do we assume he knows the correct values of the Born weights? - or would he believe some erroneous (from a global viewpoint) values for them?

 

[Derek P]

I think MW is just requiring to take it easy! If a theorist says there are "really" many worlds - and many "real" variants of me - then, the important thing is to remember that that "realness" (and anyone's "belief" in it) is theoretical. My actual business, anyway, is to see that, in the world I am observing, I be not too bad - so, to calculate what I need, I have to add the wavefunction collapse to that theory and "return to Copenhagen" - with better understanding as to whence that collapse comes (namely, from the rules of the actual business of life).

Judging from the discussion of Wallace here, it may very well be that, to make any decisions at all in life, the best strategy is always to act "as if" Born probabilities apply to the outcome of events - i.e. we have the appearence of Copenhagen collapse even if that is a bit of a legal fiction. The trouble is, deciding what is best may depend on whether you think MWI is true. So if you think you live in an MWI-universe and you think that quantum immortality/suicide makes sense, you will arrange for a spectacular but painless death confident that you will survive and be feted as The Man Who Cheated Death. You are still using Born, it's just that the cost in worlds where you die is rated at precisely zero. Under Copenhagen, you cannot argue this way. unless you have a reckless disregard for death. That may sound like a good reason for believing the CI :)

 

[Stephen Tashi]

I think MW is just too silly to believe - that's not scientific - just my gut feeling.

Without digressing into metaphysics, we can look at the assumptions made about the sensation of self - namely that we think of ourselves (at time t) as being a unique physical phenomena. So, assuming our sensation of self is implemented by physical phenomena, we assume that we are not being implemented by two distinct phenomena. For example, if somehow an exact duplicate of our bodies was created we would assume that we would remain ourself. The duplicate would think it was us, but, from our point of view, be mistaken. It seems to me that in mathematical discussions about a rational agent "the agent" (at time t) denotes a unique physical phenomena and the agent makes decisions thinking of itself as a unique physical phenomena. Is that correct? Or do the technicalities of macroscopic vs microscopic phenomena undermine the uniqueness of an agent as a unique physical phenomena?

 

[Derek P]

Is the rational agent supposed to know the Born weights? For example, if he is an agent in a "maverick" world, do we assume he knows the correct values of the Born weights? - or would he believe some erroneous (from a global viewpoint) values for them?

I have been asking myself the same thing. Reading between the lines - and there are many of them - of what has been posted here, it seems that Wallace's scenario is of living in an MWI-universe but not necessarily realising it. And not necessarily knowing how to compute Born weights. Instead, the best expectation - as calculated by "us" on the rational agent's behalf - would be when the agent's decisions are based on estimates of probabilities which are numerically equal to the Born weights. So if he lives in a maverick world, his empirical estimates of the Born weights which he thinks of as probabilities, will simply be wrong. He will doubtless use them as they are presumably all he has but his strategy in life will be sub-optimum.

 

[Derek P]

Without digressing into metaphysics, we can look at the assumptions made about the sensation of self - namely that we think of ourselves (at time t) as being a unique physical phenomena. So, assuming our sensation of self is implemented by physical phenomena, we assume that we are not being implemented by two distinct phenomena. For example, if somehow an exact duplicate of our bodies was created we would assume that we would remain ourself. The duplicate would think it was us, but, from our point of view, be mistaken. It seems to me that in mathematical discussions about a rational agent "the agent" (at time t) denotes a unique physical phenomena and the agent makes decisions thinking of itself as a unique physical phenomena. Is that correct? Or do the technicalities of macroscopic vs microscopic phenomena undermine the uniqueness of an agent as a unique physical phenomena?

A probabilistic argument won't get very far if it postulates the existence of an entity without defining a measure on the entity. Which is why it is best to decouple who or what an observer is from the phenomena that she observes. I.e. for MWI to stop at the sense organs, thus allowing "what she sees" to be defined in terms of Everettian relative states. Anything else is metaphysics. Whilst it is entirely reasonable to say that an improper mixture, such as Schroedinger's Cat, is a superposition of einselected states, it is complete gibberish to talk about the observer's mind being a superposition of experiences. To mine of the same anyway.

 

[DarMM]

That is the big problem with Wallace - it took me about 6 months to fully study it. It is very mathematical and I could not break any of it's theorem's - but its just that - very mathematical. In fact it helped me understand decoherent histories a lot better. I have said its mathematically very beautiful - but math is not physics. The big issue is can the assumptions it makes ie can you use decision theory to deduce the Born Rule. If you accept yes its fine - if not its challengeable. But if you have a deterministic theory that does not give the version of you that in a particular world then what can you do? If you reject it then the whole thing goes down the gurgler in the sense in MW all you have is a wave-function and that's it.

My biggest problem with Wallace comes early on in the book and is number one on my list above. Currently there is only the standard way of deriving decoherence, via use of the tracing rule. Wallace and many other proponents of MWI can't do this, they basically need to reverse the usual proof order, derive a branching structure first and then the Born rule.

The problem is, they haven't done this, outside of having vague arguments that it could be done by coarse-graining Hilbert space. So Wallace's proof is fine provided you accept the assumptions that a branching structure will derived via other methods at some point. I certainly don't accept the arguments in his book, they're not a calculation proving the emergence of robust quasi-classical branches. And at times the arguments are a little self-contradictory as Mandolesi discusses (others point this out as well).

So my main contention would indeed be with the use of decision theory, but more so in the assumption of a stable background for it to take place in.

 

[Derek P]

My biggest problem with Wallace comes early on in the book and is number one on my list above. Currently there is only the standard way of deriving decoherence, via use of the tracing rule. Wallace and many other proponents of MWI can't do this, they basically need to reverse the usual proof order, derive a branching structure first and then the Born rule.

Regardless of Wallace, where does Zurek's proof assume the tracing rule? Or will you be showing it sneaking in somewhere when you post the "issues"?

 

[ftr]

but math is not physics

Even more importantly can it postdict or predict anything new. Or give some real testable understanding of things like what are particles, what is exactly charge ..etc. Any theory that is of a fundamental nature should contain many of these things as aspects of a coherent concept/theory.

 

[A. Neumaier]

From a paper of Kent about the MWI:

After fifty years, there is no well-defined, generally agreed set of assumptions and postulates that together constitute “the Everett interpretation of quantum theory”. Far from it: Everett[1, 2], DeWitt[7], Graham[8], Hartle[6], Geroch[10], Deutsch[11], Deutsch[12], Saunders[13], Barbour[14] (partly inspired by Bell[15], though Bell’s aim was not to inspire), Albert-Loewer[16], Coleman[17], Lockwood[18], Wallace[19], Wallace[20], Vaidman[21], Papineau[22], Greaves[23], Greaves-Myrvold[24], Gell-Mann and Hartle[25], Zurek[26] and Tegmark[27], among many others, have offered distinctive and often fundamentally conflicting views on what precisely one needs to assume in order to get the Everett programme off the ground, and what precisely an Everettian (or, some say, post-Everettian) version of quantum theory entails.
​

 

[A. Neumaier]

Zurek has four axioms

But you stated only three. Please correct the post.

 

[DarMM]

But you stated only three. Please correct the post.

I think all four are listed. Axioms A1-A3 and then anyone of B1-B3.

EDIT: I see I had said there were four versions of the fourth axiom, there are only three versions of it, so I've corrected that.

 

[A. Neumaier]

I think all four are listed. Axioms A1-A3 and then anyone of B1-B3.

Ah yes.

the environment can always be enlarged by adding an extra system to reduce the unequal amplitude case to the equal amplitude case.

This is only true in a formal sense that an environment exists with this property. But to apply to the real world, it wold have been necessary to show that enlarging the actual environment to a bigger actual environment is possible, such that his property holds. This is unlikely to hold.

 

[Derek P]

From a paper of Kent about the MWI:

After fifty years, there is no well-defined, generally agreed set of assumptions and postulates that together constitute “the Everett interpretation of quantum theory”. Far from it: Everett[1, 2], DeWitt[7], Graham[8], Hartle[6], Geroch[10], Deutsch[11], Deutsch[12], Saunders[13], Barbour[14] (partly inspired by Bell[15], though Bell’s aim was not to inspire), Albert-Loewer[16], Coleman[17], Lockwood[18], Wallace[19], Wallace[20], Vaidman[21], Papineau[22], Greaves[23], Greaves-Myrvold[24], Gell-Mann and Hartle[25], Zurek[26] and Tegmark[27], among many others, have offered distinctive and often fundamentally conflicting views on what precisely one needs to assume in order to get the Everett programme off the ground, and what precisely an Everettian (or, some say, post-Everettian) version of quantum theory entails.
​

Please restrict your answers to criticisms of derivations of the Born Rule that are generally accepted by proponents of MWI. Please provide a verbal description of the issue where possible so that people like myself who are certainly graduate-plus* but rusty as hell have a chance of seeing what the point is.
*plus nearly half a century in my case
So then, in what sense does MWI fail to predict the Born Rule?

Someone else's list of other author's different treatments doesn't really answer the question.

 

[A. Neumaier]

Someone else's list of other author's different treatments doesn't really answer the question.

But it shows that the question is far too imprecise to be clearly answered. Talking about MWI without saying whose MWI is meant is very ambiguous.

 

[Derek P]

This is unlikely to hold.

Why?

 

[A. Neumaier]

Why?

Because we can construct a particular environment artificially in mathematical terms, whereas in reality we must pick one from the environments encoded in the universal wave function.

 

[Derek P]

Because we can construct a particular environment artificially in mathematical terms, whereas in reality we must pick one from the environments encoded in the universal wave function.

I assume you mean a sub-system of the total environment?

 

[A. Neumaier]

I assume you mean a sub-system of the total environment?

Of course since one cannot augment the total environment.

 

[Derek P]

Of course since one cannot augment the total environment.

And you feel that we cannot pick/specify a subsystem with enough degrees of freedom to obtain degeneracy? Sorry to sound like a stuck record, but why not?

 

[DarMM]

Well, I was reading a few papers around Zurek's proof to give more context and hadn't finished them yet, but roughly the problems are:

1. Zurek assumes the existence of a pre-existing branching structure of the type usually proven via decoherence. This is the same problem Wallace has. Note that Zurek himself now says this is true and considers the proof not to hold for this reason. This is a major motivation for his Quantum Darwinism project.
2. Of the B1-B3 axioms needed to complete the proof, B1 and B2 have little physical evidence in their favour and it is hard to imagine them holding for arbitrary states.
   B3 is a very general statement about entangled states having the interpretation of correlations between the systems. If you take as your four axioms A1-A3 and B3, then the entire proof is just concerned with the statistics of Schmidt states and doesn't take place in any particular interpretation.
   Basically B1 and B2 have MWI meanings, but are probably not valid on real physical systems. B3 turns the proof into a proof of the full Born rule from a weak Born rule in any interpretation and isn't about MWI per se anymore.
3. The problems A. Neumaier mentioned about the environment extension.

I'd like to give more detail however, including a full discussion of a simple case of the proof, where it can be seen that it isn't really envariance that holds the proof together, but a certain noncontextuality assumption.

 

[PeterDonis]

Thread closed for moderation, in order for the Mentors to consider whether allegations about whether the thread is acceptable or not are off topic.

 

[PeterDonis]

Thread reopened. Some off topic posts have been deleted.

 

[A. Neumaier]

why not?

An alleged proof is supposed to explain why, and not to öeave filling out the gap to the reader.

 

[bhobba]

But it shows that the question is far too imprecise to be clearly answered. Talking about MWI without saying whose MWI is meant is very ambiguous.

Yes I now realize MW has an (maybe more than one) issue - not with its math - it's tight - many other issues are the same as decoherent histories and answered the same way. But you just can't get around this idea - its deterministic - how do you introduce probabilities. I think the bet/wager idea is OK meaning decision theory is ok - but can I prove it -it just seems reasonable to me - but reasonable and true are two different things. BTW the reason I find it too weird to believe in is this exponentially increasing number of worlds - there is no end and it keeps increasing faster and faster - that just doesn't seem right.

Thanks
Bill

 

[DarMM]

That's funny, as a result of this thread and what I've read, I've come to round to a different point of view. I don't mind it too much conceptually, but I find it mathematically inconsistent/circular, due to the assumption of branches and macro structures without any proof that these arise. Although maybe you are including this in conceptual issues.

I think there are two MWIs.

1. Pure unitary QM MWI, i.e. a "literally" interpreted QM with only Hilbert Spaces and unitary evolution
2. MWIs with various extra features already assumed to be present. The latter is really a family, being different elements of Kent's list quoted by A. Neumaier. Wallace has the extra features of a fuzzy branch structure and clear macrostate partitioning of the Hilbert Space. Zurek has a precise branching structure and powerful assumptions about quantum subsystems.

The theory is often discussed and promoted as 1, but the proofs all take place in some form of 2.

EDIT: I hope to get to a longer post about Zurek's proof sometime this week.

 

[bhobba]

The latter is really a family, being different elements of Kent's list quoted by A. Neumaier. Wallace has the extra features of a fuzzy branch structure and clear macrostate partitioning of the Hilbert Space.

I find decoherent histories has that same issue and seems resolved there - however a discussion of that (mentors hat on) requires a new thread. Here is a book on consistent histories but I find it a bit basic:
http://quantum.phys.cmu.edu/CQT/index.html

But getting more advanced stuff on it I don't find easy - what I know of it at an advanced level I got mostly from Wallace.

Zurecks stuff is the same - interesting, and I don't think it has been discussed as much as MW here - a bit - but not much - also really needs its own thread.

Thanks
Bill

 

[DarMM]

I find decoherent histories has that same issue and seems resolved there - however a discussion of that (mentors hat on) requires a new thread.

I won't go into decoherent histories as you suggest, only to contrast. I think the difference is that the use of decoherence is valid there as they can use the Born rule, MWI cannot unless it proves it first. So you have to show a fuzzy branching structure and macrostate partition arises without Born's rule.

 

[akvadrako]

I won't go into decoherent histories as you suggest, only to contrast. I think the difference is that the use of decoherence is valid there as they can use the Born rule, MWI cannot unless it proves it first. So you have to show a fuzzy branching structure and macrostate partition arises without Born's rule.

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

 

[bhobba]

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

It can just assume it, or equivalently non-contextuality. The trouble is it ruins the 'beauty' of the interpretation that attracts many to it - you just have the Schrodinger's equation and that's it - you derive the Born rule. But, as a number of posts have shown here, before anything significant can be derived such as say the emergence of a classical world you need the Born Rule to do it - eg see my post on decoherence and its math - the Born rule is inherent in it. You can, like Decoherent Histories define a history without the Born Rule - but using the concept you need to be able to assign a probability to a history - that requires the Born rule. Note - I am using trace (OP) where O is the observable and P the state as what I call the Born Rule.

Thanks
Bill

 

[akvadrako]

The trouble is it ruins the 'beauty' of the interpretation that attracts many to it - you just have the Schrodinger's equation and that's it - you derive the Born rule.

Maybe it ruins the beauty for you, but I don't think it's a popular opinion for those working on it. It also seems clear that it's impossible without making some additional assumptions. So, it basically comes down to what are you trying to show? Not that the Born rule is incompatible with WMI and not that it has to be derived for the theory to make sense. But that you would like it to be derived with no additional assumptions and it can't be.

 

[StevieTNZ]

I've come across this article -- http://iopscience.iop.org/article/10.1088/1367-2630/aabe12/meta (https://phys.org/news/2018-05-quant...e=menu&utm_medium=link&utm_campaign=item-menu) which those who have participated in this thread might find interesting. (Mentors: please create a new thread if you think the above should belong in a separate thread)

 

[Mentz114]

I've come across this article -- http://iopscience.iop.org/article/10.1088/1367-2630/aabe12/meta (https://phys.org/news/2018-05-quant...e=menu&utm_medium=link&utm_campaign=item-menu) which those who have participated in this thread might find interesting. (Mentors: please create a new thread if you think the above should belong in a separate thread)

This looks like entirely new stuff. Is the journal peer reviewed ?

The PDF is here

 

[StevieTNZ]

Is the journal peer reviewed?

This is what I've found:
http://ioppublishing.org/news/peer-review-information-now-available-for-online-articles/
https://publishingsupport.iopscience.iop.org/questions/general-review-procedure-on-iop-journals/

 

[DarMM]

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

Let me be clear here. I am discussing the form of Many Worlds advocated by Wallace, Deutsch, Zurek (who no longer advocates it) and others, which is basically just the Hilbert Space of states and unitary evolution with the correct Hamiltonian. The claim being that this contains in some sense the Born rule as an effective subjective rule for observers. I find the current attempts at showing this mathematically circular.

I'm not really discussing versions of MWI where the Born rule is assumed, hence the focus on the proofs and I think from the thread title they wouldn't be the topic (they can't fail to predict the Born rule and the proofs don't concern them). I do have some thoughts on them, but perhaps another thread.

EDIT: I will say that a common "promotion" of Many-Worlds is that it is just Unitary QM without extra assumptions, so I do think this is a fairly common version of MWI. As Kent's list above shows though there are many versions of MWI and I think it can be confusing as these are always presented as one interpretation.

 

[akvadrako]

Let me be clear here. I am discussing the form of Many Worlds advocated by Wallace, Deutsch, Zurek (who no longer advocates it) and others, which is basically just the Hilbert Space of states and unitary evolution with the correct Hamiltonian. The claim being that this contains in some sense the Born rule as an effective subjective rule for observers. I find the current attempts at showing this mathematically circular.

I'm not really discussing versions of MWI where the Born rule is assumed, hence the focus on the proofs and I think from the thread title they wouldn't be the topic (they can't fail to predict the Born rule and the proofs don't concern them). I do have some thoughts on them, but perhaps another thread.

EDIT: I will say that a common "promotion" of Many-Worlds is that it is just Unitary QM without extra assumptions, so I do think this is a fairly common version of MWI. As Kent's list above shows though there are many versions of MWI and I think it can be confusing as these are always presented as one interpretation.

Thanks for the clarification, but I don't really think we are talking about different theories here. Work done in WMI uses the Born rule for calculations as always. Instead they are attempting to find the most acceptable axiomatic basis and a conceptual way to introduce probabilities into a deterministic theory. I would say those physicists thought they could derive the Born rule with a certain set of axioms; in at least some cases their derivations had flaws. But they didn't "have" to use those axioms and it turns out most of the derivations apply equally well to collapse theories.

Also, since all the derivations we have mentioned include more assumptions than Hilbert spaces and unitary evolution, it doesn't make much sense to say they are promoting it without extra assumptions.

 

[stevendaryl]

I will say that a common "promotion" of Many-Worlds is that it is just Unitary QM without extra assumptions, so I do think this is a fairly common version of MWI. As Kent's list above shows though there are many versions of MWI and I think it can be confusing as these are always presented as one interpretation.

I think that the Born rule doesn't by itself contradict the assumption that evolution is purely unitary. What you don't have, with purely unitary evolution, is the "collapse" after an observation is made.

 

[DarMM]

Thanks for the clarification, but I don't really think we are talking about different theories here.

I think different axioms are different theories, if you have different fundamental base assumptions, I would view that as a separate version of MWI. For example Zurek's version of MWI has no concepts like "Weights of Existence". Regardless of this point, I think a theory which derives the Born rule as an effective result is quite different from one where it is a base feature, regardless of if they share another feature, i.e. multiple classical worlds.

Also, since all the derivations we have mentioned include more assumptions than Hilbert spaces and unitary evolution, it doesn't make much sense to say they are promoting it without extra assumptions.

Wallace describes that as one of the virtues of Many-Worlds in his book. My point isn't how much sense it makes, it's that it is a claim made. He also doesn't view the assumptions in his proof as assumptions, more details of pure unitary MWI to be filled in, i.e. there will eventually be a pure unitary proof of decoherence.

I think it makes sense to say it as it is literally claimed in books and papers by MWI proponents.

I think it is a mistake to notice the extra assumptions and think this means they must not be claiming it is only the Hilbert Space + Unitary evolution part of QM. Unsubstantiated claims can easily be made, or people can view their assumptions as "obvious" or "trivial" details (or yet to be proved Lemmas), when in fact they change the physical picture.

This is also a widespread misunderstanding see this stack exchange question and its accepted answer:
https://physics.stackexchange.com/q...on-mwi-cannot-derive-the-born-rule-would-that

It is a commonly held and promoted view of MWI.

 

[DarMM]

I think that the Born rule doesn't by itself contradict the assumption that evolution is purely unitary. What you don't have, with purely unitary evolution, is the "collapse" after an observation is made.

I don't necessarily think it contradicts unitary evolution either, there's just no proof it's a consequence of unitary evolution.

 

[akvadrako]

I think different axioms are different theories, if you have different fundamental base assumptions, I would view that as a separate version of MWI.

So if I had a theory which assumed $a = b$ and $b = c$ and proved $a = c$, then I switched an axiom with the proof, would you call that a different theory?

I think it is a mistake to notice the extra assumptions and think this means they must not be claiming it is only the Hilbert Space + Unitary evolution part of QM. Unsubstantiated claims can easily be made, or people can view their assumptions as "obvious" or "trivial" details (or yet to be proved Lemmas), when in fact they change the physical picture.

If those assumptions change the physical picture, then they are mistaken; I think Wallace and Zurek would agree with that. The physical picture, at least the objective viewpoint of the whole system, should get all it's dynamics from unitary evolution. It doesn't even depend on decoherence. The additional thing which is needed, is some way to measure the system (not necessarily a particular way). Maybe it's so obvious it's overlooked, same as when numbers are assumed to be real instead of p-adic.

It's a common misunderstanding that WMI somehow depends upon a definition of world (or decoherence or branching structure). What defines WMI best is probably:

Without drawing on any external metaphysics or mathematics other than the standard rules of logic, EWG are able, from these postulates, to prove the following metatheorem: The mathematical formalism of the quantum theory is capable of yielding its own interpretation.

The further assumptions in the derivations are only used to show how a subjective agent within the WMI would experience it and if that would correspond with the Born rule. They are, as you said, (approximate) details for our convenience.

This is also a widespread misunderstanding.

I do agree that's another widespread misunderstanding, which is what prompted me to respond to your claim that WMI has to derive the Born rule.

 

[DarMM]

So if I had a theory which assumed $a = b$ and $b = c$ and proved $a = c$, then I switched an axiom with the proof, would you call that a different theory?

I don't really think this is analogous to having the Born rule as an effective approximate rule for agents versus having it as a primary aspect of your theory.

I do agree that's another widespread misunderstanding, which is what prompted me to respond to your claim that WMI has to derive the Born rule.

I didn't claim it needed to.

I'm discussing versions of MWI where no assumptions are made beyond unitary evolution and a derivation of the Born rule is attempted. If the Born rule isn't assumed, it needs to be derived.

Other versions of MWI do assume it and of course they don't need a derivation of it.

Again when discussing derivations of the Born rule I can only discuss those versions of MWI where it is derived and not assumed. This is not a "claim" that all versions of MWI need to derive the rule.

 

[akvadrako]

I don't really think this is analogous to having the Born rule as an effective approximate rule for agents versus having it as a primary aspect of your theory.

I think it's analogous - what matters is that the physical situation is the same and that is always the unitary evolution. It certainly can't depend on decoherence.

Again when discussing derivations of the Born rule I can only discuss those versions of MWI where it is derived and not assumed. This is not a "claim" that all versions of MWI need to derive the rule.

I'm saying these are not different versions of MWI, but different attempts to derive the Born rule for subjective observers, which doesn't define the theory - it works within it.

 

[DarMM]

I think it's analogous - what matters is that the physical situation is the same and that is always the unitary evolution. It certainly can't depend on decoherence.

Really? In Wallace's version the Born rule is only an effective result which can fail when certain features are absent, e.g. the control over the environment assumed by erasure for example.
In one form of Zurek's MWI, it would fail for example when a system includes no subsystems whose invariance under unitary transformations implied equal probabilities for their outcomes.

To me, these are both different from each other (different regimes where Born rule fails) and different from a theory where it isn't an effective rule, but assumed from the start.

 

[akvadrako]

Really? In Wallace's version the Born rule is only an effective result which can fail when certain features are absent, e.g. the control over the environment assumed by erasure for example.
In one form of Zurek's MWI, it would fail for example when a system includes no subsystems whose invariance under unitary transformations implied euqal probabilities for their outcomes.

To me, these are both different from each other (different regimes where Born rule fails) and different from a theory where it isn't an effective rule, but assumed from the start.

So these theories are about the experience of an subjective observer within the universal wavefunction. This seems to be necessarily an approximate concept and they make different assumptions to show it kinda makes sense. But WMI is not QBism - it isn't about the subjective experiences of observers.

It wouldn't make sense to say the evolution of the objective state depends upon control of the environment or choice of subsystems. And the physical situation is given fully by that state (assuming some fixed properties).