Question regarding the Many-Worlds interpretation

[stevendaryl]

MW is NOT my preferred interpretation either - I hold to the Ensemble Interpretation with deocherence.. It just interested me enough to investigate it further. And indeed it has proved quite interesting.

Thanks
Bill

I don't actually see much difference between an ensemble interpretation and a Many Worlds interpretation.

 

[lukesfn]

Well, none of them is accurate. There is no doubling, multiplying or branching going on, really. Those all misleadingly imply discreteness, that there is a number of possibilities and you can just count them. But in general, the wave function is compatible with an infinite number of possibilities. It doesn't make sense to "count" the number of branches.

I would be included to agree with you. However, in the context I was using them, they where definitely meant to be discrete, and I was trying to show that it is wrong to thing of them as such. It isn't helpful to say the word is wrong in that context.

However, on second thought, perhaps split is a good word, because it is easy to conceptualize splitting an apparently continuous entity unevenly since we do it every we slice a piece of pie, or take a bite of food.

edit:

I don't actually see much difference between an ensemble interpretation and a Many Worlds interpretation.

I thought that the ensemble interpretation is mostly a plea of ignorance that doesn't explain help anything... I would love somebody to be able to explain to me otherwise, however, this is off topic.

 

[bhobba]

But obviously, you're in all of them

Not quite.

There is a copy of you in all of them and each copy experiences a different outcome. What copy you are and what you experience is the same thing.

Thanks
Bill

 

[bhobba]

I thought that the ensemble interpretation is mostly a plea of ignorance that doesn't explain help anything... I would love somebody to be able to explain to me otherwise, however, this is off topic.

Actually there is a lot of similarity with the decoherence ensemble interpretation the detail of which you can read about here (as well as decoherence):
http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf

The key difference is the ensemble decoherence interpretation leaves up in the air exactly how a improper mixed state becomes a proper one (I get around that by my version of QM has as a basic postulate observationally equivalent systems are equivalent but that is another story). The MWI has no such issue because all outcomes actually occur and are equally real, its just that each copy of you only experiences one outcome. Slightly different issues - with MW you have this extravagant exponentially increasing worlds and no problem about what is chosen because they all are, and with mine you only have one outcome but the problem of how that outcome is chosen.

Thanks
Bill

 

[stevendaryl]

Slightly different issues - with MW you have this extravagant exponentially increasing worlds

Viewed slightly differently, the number of "possible worlds" remains constant, but associated with each possible world is an amplitude that changes with time. All worlds exist at all times.

 

[bhobba]

Viewed slightly differently, the number of "possible worlds" remains constant, but associated with each possible world is an amplitude that changes with time. All worlds exist at all times.

Sure. Slightly different ways of viewing it certainly exist - its basically what you feel the most comfortable with.

Thanks
Bill

 

[S.Daedalus]

Not quite.

There is a copy of you in all of them and each copy experiences a different outcome. What copy you are and what you experience is the same thing.

This is probably a bit too metaphysical, but if you split into two identical copies, one having experience A, and the other experience B, in order to be able to talk about 'you' being one of these copies (as opposed to the other), you have to introduce some means that picks out your 'you'-ness, i.e. that makes it so that your continuous experience is with copy B, rather than A, say. This is the route Albert and Loewer have considered (but rejected), and under such a constraint, it's indeed possible to make sense of probability in the 'MWI' (though of course it's not really the MWI anymore, but essentially a hidden-variable theory with an observer's mind being the hidden variable), by simply postulating that whatever this 'you'-index is, it 'jumps' stochastically into one of the two copies (as opposed to the other), thus providing a basis for a probabilistic interpretation. But I think that many worlds traditionally does not boil down to such a view; rather, one would typically hold that both copies are indeed equally much you, with no additional distinguishing features (though how this works is a bit of a subtle issue).

Anyway, it seems most people have made up their minds, but I've decided nevertheless to flesh out the reason why Gleason does not help in the MWI a little. I won't really be telling anybody anything new, but I think it helps looking at the story in a different way from how it's usually told.

Let's start classical. Consider a set of classical objects---marbles, say. This set is partitioned into subsets, according to some distinguishing characteristic of the marbles---say, their colour; say furthermore that we have four colours, red, green, blue, and pink. A subset of the total marble set corresponds to a proposition; whether or not a given marble belongs to that subset corresponds to the truth value of that proposition, i.e. 'the marble is red' is true if the marble belongs to the subset of red marbles.

Now, let's assume there's some fixed quantity of marbles---say 100---, and that in each subset, there's a fixed number of marbles, as well---50 red ones, 30 green ones, 19 blue ones and 1 pink marble. This gives us a way to associate a measure with the subsets---the set of red marbles gets measure 0.5, and so on, while the full set obviously gets measure 1. Now, we can make sense of the construction 'the probability that the marble has colour c is p'. If we don't know anything about the marble---draw it at random---, then, for instance, the probability that the marble is green is 30%. (This doesn't consider any subtleties about probability, and isn't meant to; even without the precise definitions, I suppose it's obvious to anybody that we can grasp the extension of the concept of probability.) If we know something---say, that the marble is neither green nor pink---we can adjust our probabilities accordingly. In fact, writing P_R for the proposition that the marble is red, and P_B for the proposition that it is blue, we might represent our knowledge of the situation by the quantity
r=p_1P_R+p_2P_B.
Now let's move over to quantum mechanics. Here, we don't have a set, but a Hilbert space, and we don't have subsets, but (closed) subspaces. And we also don't have propositions in the classical sense, but we can associate to every subspace a special operator, namely the projector on that subspace. So, we can kinda do the same thing we did before, and imagine the Hilbert space partitioned into subspaces, and any state belonging to a certain subspace has a certain property---assume there's four, and call them R, G, B, P. But here's the first problem: the counting measure doesn't really make sense anymore. Luckily, there's a way out: as Gleason showed, there's a unique measure attributable to closed subspaces, and it's given by the squared amplitudes (we don't need to get into any technicalities here, as most people are familiar with them anyway). So, we can play the same game as before---well, almost. We can play the same game if we have some assurance that the state is in one of the subspaces we're considering, that is, determinately has any of the properties R, G, B, P. Then, we can proceed as before, and (again, given knowledge that the state does not have property G or P) write, for instance

\rho=|c_1|^2P_R+|c_2|^2P_B,

which, with P_R=|\psi_R\rangle\langle \psi_R| and P_B=|\psi_B\rangle\langle \psi_B|, corresponds to the proper mixture

\rho=|c_1|^2|\psi_R\rangle\langle \psi_R|+|c_2|^2|\psi_B\rangle\langle \psi_B|.

Up to this point, anything proceeds analogously to the classical case, thanks to Gleason. However, in general, we don't have the assurance that the state will have any of the properties determinately; in general, we're faced with a superposition such as

|\psi\rangle=c_1|\psi_R\rangle + c_2|\psi_B\rangle.

The question now is, how do we get from |\psi\rangle to \rho? And on the standard interpretation, the answer to this is the collapse; \rho is just the state after a measurement has occurred on |\psi\rangle, but before anybody had a look, so to speak. This is the key point: we must get to \rho before Gleason is of any help.

However, on the many worlds interpretation, no such mechanism is available. The state never collapses; we're left with |\psi\rangle at all times. Hence, Gleason doesn't say anything about the probabilities we ought to expect---in contrast to the case of the collapse. Appealing to Gleason's theorem, then, to the best of my ability to tell, is simply a nonstarter in the case of many worlds; it simply doesn't apply.

 

[kith]

It allows to formulate theories that predict amplitudes, and gives a method to do hypothesis testing based on those predictions.

The Hamiltonian of the system can be used in principle to predict the amplitudes. Does every Hamiltonian define one of these theories or what kind of theories do you have in mind?

This is probably a bit too metaphysical, but if you split into two identical copies, one having experience A, and the other experience B, in order to be able to talk about 'you' being one of these copies (as opposed to the other), you have to introduce some means that picks out your 'you'-ness, i.e. that makes it so that your continuous experience is with copy B, rather than A, say. This is the route Albert and Loewer have considered (but rejected), and under such a constraint, it's indeed possible to make sense of probability in the 'MWI' (though of course it's not really the MWI anymore, but essentially a hidden-variable theory with an observer's mind being the hidden variable), by simply postulating that whatever this 'you'-index is, it 'jumps' stochastically into one of the two copies (as opposed to the other), thus providing a basis for a probabilistic interpretation. But I think that many worlds traditionally does not boil down to such a view; rather, one would typically hold that both copies are indeed equally much you, with no additional distinguishing features (though how this works is a bit of a subtle issue).

Maybe this is related to the point mfb is trying to make: if you want to talk about probabilities, you can do so by using the 'jumping you'. But if you consider both yous to be qually real it doesn't make sense to talk about probabilities.

The question now is, how do we get from |\psi\rangle to \rho?

Decoherence?

 

[stevendaryl]

|\psi\rangle=c_1|\psi_R\rangle + c_2|\psi_B\rangle.

The question now is, how do we get from |\psi\rangle to \rho? And on the standard interpretation, the answer to this is the collapse; \rho is just the state after a measurement has occurred on |\psi\rangle, but before anybody had a look, so to speak. This is the key point: we must get to \rho before Gleason is of any help.

That was actually the chief mathematical result that Everett derived in his first paper on the Many Worlds Interpretation (which he didn't call that--that name was coined by DeWitt). He showed that density matrices arise naturally from pure wave functions in cases of entanglement.

 

[mfb]

The Hamiltonian of the system can be used in principle to predict the amplitudes. Does every Hamiltonian define one of these theories or what kind of theories do you have in mind?

Different Hamiltonians that lead to different evolutions of wavefunction amplitudes are different theories.

 

[stevendaryl]

That was actually the chief mathematical result that Everett derived in his first paper on the Many Worlds Interpretation (which he didn't call that--that name was coined by DeWitt). He showed that density matrices arise naturally from pure wave functions in cases of entanglement.

Here's a sketch from memory:

Suppose that you have a system in state |\Psi \rangle that is made up of two subsystems. We can write |\Psi \rangle as a superposition of product states of the two subsystems:

|\Psi \rangle = \sum_{i, a} C_{i a} |\varphi_i \rangle | \chi_a \rangle

Now suppose that we have an operator O that depends only on one of the subsystems. In other words:

\langle \varphi_i' | \langle \chi_a' | O | \chi_a \rangle | \varphi_i \rangle = O_{a a'} \delta_{i i'}

In that case, the expectation value of O in state \Psi is given by:

\langle \Psi |O|\Psi \rangle = \sum_{i, a, a'} C^*_{i a'} C_{i a} O_{a a'}

This is the same result you would get using a density matrix \rho with components

\rho_{a a'} = \sum_i C^*_{i a'} C_{i a}

As long as you are only talking about measurements of one of the two subsystems, you can treat the system as if it were in a mixture, rather than a pure state.

 

[S.Daedalus]

Decoherence?

That was actually the chief mathematical result that Everett derived in his first paper on the Many Worlds Interpretation (which he didn't call that--that name was coined by DeWitt). He showed that density matrices arise naturally from pure wave functions in cases of entanglement.

This is why I specified proper mixture: improper mixtures that arise via tracing out a part of the system look mathematically indistinguishable from 'true' mixtures that arise from uncertainty about what the actual state is, but can't be given an ignorance interpretation (this error is also at the root of Art Hobson's recent 'resolution' of the measurement problem which we've discussed here).

(In fact, I've sometimes thought that the confusion behind probability in the MWI looks a lot like the confusion between proper and improper mixtures; there's an argument due to Albert and Barrett that's structurally very similar to d'Espagnats argument regarding improper mixtures, producing a contradiction between considering each term of the superposition as a world in itself and the predictions of quantum mechanics taking into account he complete quantum state.)

 

[bhobba]

but can't be given an ignorance interpretation (this error is also at the root of Art Hobson's recent 'resolution' of the measurement problem which we've discussed here).

Sorry - but since it is observationally equivalent to a proper mixture a perfectly valid interpretation is to simply assume it is - nothing can prove you wrong. See the paper I posted previously about decoherence where this is examined in detail. You may not like the ignorance interpretation - but valid it is.

Thanks
Bill

 

[tom.stoer]

Seems that I am the only one where "the penny did not drop".

I still have the same problems - or even more.

In experiments we can make fundamental individual observations and we can derive statistical frequencies. In many interpretations of QM we can both derive matrix elements and interpret them as probabilities to be compared with the statistical frequencies.

Now I expected that MWI - being a different interpretation of the same underlying formalism - allows for the same calculations and experimental tests. But what I read is very confusing (for me):
1) MWI is talking about branches and relies on decoherence to identify them, but is not able to count them or to derive a corresponding measure
2) My simple question regarding the "probability being in a certain branch" which I can identify via a result string seems to become meaningless
3) I still have the feeling that my concerns regarding the "missing link" between the experimentally inaccesable top-down perspective of the full Hilbert space with all its branches and the accessable bottom-up approach restricted to the branch I am observing right now have not been understood
4) We have the above mentioned statistical frequencies, but I learn that MWI does not provide the corresponding probabilities - that there are no probabilities at all
5) It is often claimed that the Born rule can derived, but what does it mean if there are no probabilities?

I guess the answers are there, written down in numerous posts, but I am not able to identify them.

 

[S.Daedalus]

Sorry - but since it is observationally equivalent to a proper mixture a perfectly valid interpretation is to simply assume it is - nothing can prove you wrong. See the paper I posted previously about decoherence where this is examined in detail. You may not like the ignorance interpretation - but valid it is.

Locally, that's true, but once Alice and Bob get together and compare their measurement results, or a measurement is carried out on both parts of the system, you get results that falsify the idea that the parts of the system are in some definite state, and we just don't know which---correlations that we can't account for with such a model in the first case, and interference results in the second. These are perfectly valid observations, so I don't see how it's true that the two states are 'observationally equivalent'.

Seems that I am the only one where "the penny did not drop".

If by the penny dropping you mean understanding how (Born) probabilities arise in many worlds, then no, count me as one as confused as you are.

 

[Jazzdude]

Sorry - but since it is observationally equivalent to a proper mixture a perfectly valid interpretation is to simply assume it is - nothing can prove you wrong.

You keep asserting this, but it's not true. To make this statement true, you have to employ the measurement postulate, which is what we are trying to motivate or prove. It comes in in the construction of ensemble descriptions as density matrices, which is only a sensible construction with the measurement postulate in mind. If you don't have the measurement postulate then the only complete way to describe an ensemble is a list of states along with the probability of finding each.

Cheers,

Jazz

 

[Jazzdude]

Seems that I am the only one where "the penny did not drop".

I see it differently. I think you just don't get confused by shifting the argument between different levels all the time and drawing different conclusions. I have yet to see a way to make MWI work that doesn't rely on obfuscation of the real issues.

Cheers,

Jazz

 

[tom.stoer]

I see it differently. I think you just don't get confused by shifting the argument between different levels all the time and drawing different conclusions. I have yet to see a way to make MWI work that doesn't rely on obfuscation of the real issues.

Cheers,

Jazz

So you say MWI can't answer these valid questions?

 

[Jazzdude]

This is why I specified proper mixture: improper mixtures that arise via tracing out a part of the system look mathematically indistinguishable from 'true' mixtures that arise from uncertainty about what the actual state is, but can't be given an ignorance interpretation (this error is also at the root of Art Hobson's recent 'resolution' of the measurement problem which we've discussed here).

This is a very good thought. If you want to get ensembles out you have to put ensembles in. For example you can assume that the environment is in an unknown (but single) state that you model by an ensemble and then see how interaction with the environment splits up a well defined single observed state into a real ensemble. But then you have to face another problem: The contributing states cannot be recovered from a density matrix representation. Since we're talking about definite single (but unknown) states, we want to track their individual histories. That means you have to rather use a more complete representation of a quantum ensemble. That would be a list of states with associated probabilities.

If you perform a calculation like sketched above for a single qubit with an unknown environment you can express the qubit ensemble as a density function on the Bloch sphere that evolves in time as described by a generalized diffusion equation. Unfortunately the Born rule cannot be recovered from the dynamic, because linearity doesn't allow it.

Cheers,

Jazz

 

[Jazzdude]

So you say MWI can't answer these valid questions?

Yes, that's what I'm convinced of.

 

[tom.stoer]

Nevertheless I would like to give mfb et al. a chance!

 

[bhobba]

Locally, that's true, but once Alice and Bob get together and compare their measurement results, or a measurement is carried out on both parts of the system, you get results that falsify the idea that the parts of the system are in some definite state, and we just don't know which---correlations that we can't account for with such a model in the first case, and interference results in the second. These are perfectly valid observations, so I don't see how it's true that the two states are 'observationally equivalent'.

That was dealt with in the thread you mentioned. The answer is in the reference I gave there - the book on Consistent Quantum Theory by Griffiths - it has to do with the necessity of requiring a consistent framework.

Thanks
Bill

 

[Jazzdude]

That assumption, via Glaeson, means you are abandoning basis independence. Why do you want to choose one basis over another? These are man made things introduced for calculational convenience - why do you think nature should depend on such a choice?

Measurement does single out a basis, that's the whole point of it and the heart of the preferred basis problem. The Hilbert space structure is motivated by unitary evolution, not measurement. And using the construction of the space for the theory that describes an observed phenomenon as the reason for this phenomenon is highly circular!

Cheers,

Jazz

 

[bhobba]

You keep asserting this, but it's not true.

It is true - simple as that. There is no way to observationally tell the difference. If you know of one do tell.

Thanks
Bill

 

[Jazzdude]

It is true - simple as that. There is no way to observationally tell the difference. If you know of one do tell.

Thanks
Bill

You're misunderstanding the point. If we perform a measurement then the measurement postulate comes in and you cannot distinguish the two. But that's not what we're arguing about. We're talking about a situation where the measurement postulate is NOT there and we try to derive or motivate it, by means of an interpretation or theory. In this case you cannot assume the equivalence, because it's precisely what are are intending to show!

Cheers,

Jazz

 

[bhobba]

Measurement does single out a basis, that's the whole point of it and the heart of the preferred basis problem.

That is at odds with standard textbooks like Decoherence and the Quantum-to-Classical Transition by Schlosshauer. The factoring problem naysayers must provide a proof it is purely an artifact of decomposition - so far they haven't.

The truth of the matter is detailed in the the paper I linked to before, and I gave in the thread previousoly mentioned:
https://www.physicsforums.com/showthread.php?t=707290

As I posted in that thread:
'Basically he is a holding to the decoherence ensemble interpretation as do I. Rather than me go through its pro's and con's here is a good paper on it:
http://philsci-archive.pitt.edu/5439...iv_version.pdf
'Postulating that although the system-apparatus is in an improper mixed state, we can interpret it as a proper mixed state superficially solves the problem of outcomes, but does not explain why this happens, how or when. This kind of interpretation is sometimes called the ensemble, or ignorance interpretation. Although the state is supposed to describe an individual quantum system, one claims that since we can only infer probabilities from multiple measurements, the reduced density operator SA is supposed to describe an ensemble of quantum systems, of which each member is in a definite state.'

The bottom line is, the naysayers are correct - without additional assumptions decoherence does not solve the measurement problem. That's true. But there is another part to it - with additional assumptions it does. That's the key point and a number of interpretations like decoherent histories, MWI, and the Ensemble ignorance interpretation do have additional assumptions and that's what makes them viable.

The guy that wrote the paper in the thread above was wrong - he needed additional assumptions that he didn't make explicit - however with those additional assumptions its valid.

Thanks
Bill

 

[Jazzdude]

That is at odds with standard textbooks like Decoherence and the Quantum-to-Classical Transition by Schlosshauer. The factoring problem naysayers must provide a proof it is purely an artifact of decomposition - so far they haven't.

That's not what I have quoted or referred to. My reply was specifically about your claim that it makes no sense to have a special basis in the process of observation.

And the factoring problem is already one step to far. In most cases there are no sensible factors at all. If you are able to specify a tensor factor space for an observed electron let me know. Practically all the interesting (for description of observation) systems do not have the structure of a tensor factor space of the universal Hilbert space.

Cheers,

Jazz

 

[bhobba]

You're misunderstanding the point. If we perform a measurement then the measurement postulate comes in and you cannot distinguish the two. But that's not what we're arguing about. We're talking about a situation where the measurement postulate is NOT there and we try to derive or motivate it, by means of an interpretation or theory. In this case you cannot assume the equivalence, because it's precisely what are are intending to show!

You are missing my point, and the point of the decoherence adherents. There is no circularity in explicitly stating it gives the appeance of wave function coolapse and because of that actual collapse is a non issue. By this is meant since you can't tell the difference it's not something to worry about.

The Wikipedisa article on it explains it quite well:
http://en.wikipedia.org/wiki/Quantum_mind%E2%80%93body_problem
'Decoherence does not generate literal wave function collapse. Rather, it only provides an explanation for the appearance of wavefunction collapse, as the quantum nature of the system "leaks" into the environment. That is, components of the wavefunction are decoupled from a coherent system, and acquire phases from their immediate surroundings. A total superposition of the universal wavefunction still exists (and remains coherent at the global level), but its fundamentality remains an interpretational issue. "Post-Everett" decoherence also answers the measurement problem, holding that literal wavefunction collapse simply doesn't exist. Rather, decoherence provides an explanation for the transition of the system to a mixture of states that seem to correspond to those states observers perceive. Moreover, our observation tells us that this mixture looks like a proper quantum ensemble in a measurement situation, as we observe that measurements lead to the "realization" of precisely one state in the "ensemble".'

To put it another way, except for people like the person that wrote the article claiming the issue had been solved (and he was wrong), decoherence adherents freely admit it only gives the appearance of collapse, for them that's is good enough - but for others like you it isn't.

Thanks
Bill

 

[Jazzdude]

You are missing my point, and the point of the decoherence adherents. There is no circularity in exlicitly stating it gives the appeance of wave function coolapse and because of that actual collapse is a non issue. By this is meant since you can't tell the difference it's not something to worry about.

I've not been missing your point. I'm intimately familiar with the arguments used in decoherence, and I still disagree. The way you lay it out the argument depends highly on the construction of the density matrix to encode quantum ensembles. And this construction is only motivated if you assume that upon observations quantum probabilities mix with classical (ensemble) probabilities. It doesn't matter then how you construct improper ensembles or if they're sensible constructs because the real ensembles are already problematic.

If you can motivate the construction of a density matrix encoded ensemble without referring to outcomes probabilities and/or the measurement postulate (which includes references to observations of the same) then please share your wisdom.

The Wikipedisa article on it explains it quite well: ...

It's no surprise that you find something like that on Wikipedia. There are still many decoherence misinterpretations found in literature. And yes, there are supporters of this view, but that doesn't make it any more correct.

Cheers,

Jazz

 

[mfb]

Nevertheless I would like to give mfb et al. a chance!

I think it is all written in the thread now. It would be pointless to repeat it.