Question regarding the Many-Worlds interpretation

[Delta Kilo]

This is quite certainly not accurate. Linearity implies continuity, and if you can arbitrarily close to 0 then you can also reach zero.

Sorry, I guess I wasn't clear enough. I was talking about off-diagonal elements of the reduced density matrix. Say you start with cat in state a|Cdead> + b|Clive> where <Cdead|Clive> = 0 and environment in state |E>, the state of the composite system being a product state. Final state of the system is then a|Cdead>|Edead> + b|Clive>|Elive>, with <Edead|Elive>≈0 FAPP. But while these cross terms <Edead|Elive> are vanishingly small, they can never become exactly 0, otherwise the unitarity is violated.

 

[mfb]

What would that mean for the branch structure in my example?

The weights of the different branches would have the same numerical values as the probabilities in probabilistic interpretations.
As a result, most of the weight would be in branches with approximately 10% X and 90% Y - similar to probabilistic interpretations, where the probability is high to get approximately 10% X and 90% Y.

 

[tom.stoer]

The weights of the different branches would have the same numerical values as the probabilities in probabilistic interpretations.

What does that mean? Are there two branches with these to weights assigned? But after the branching an observer "in" a branch observes not 90% (or 10%) but 100%. So two branches with some weights do not help when counting branches.

What could help are 90 + 10 branches, but that sounds rather strange.

Is it possible for my experiment to count and draw the branches and write down some formulas?

 

[mfb]

Are there two branches with these to weights assigned?

Right

But after the branching an observer "in" a branch observes not 90% (or 10%) but 100%. So two branches with some weights do not help when counting branches.

Counting branches (as 1+1=2) is the wrong approach, nothing will help there. It is like the lottery example: if I just count "I will not win" and "I will win" I don't get a relevant result - I just get a list of all options.

 

[tom.stoer]

Right ... Counting branches (as 1+1=2) is the wrong approach, nothing will help there.

But that is a contradiction.

You agree that there are two branches, but you say that I must not count them. How else but via counting can you say that there are two branches?

There are two branches, therefore two observers, the first observers sees "x", the second one sees "y". How do you avoid the conclusion that 50% of all observers see an event that should have a probability of 10% ?

 

[mfb]

But that is a contradiction.

You agree that there are two branches, but you say that I must not count them. How else but via counting can you say that there are two branches?

I say counting is pointless. You can count them, but the resulting number is not really interesting.

There are two branches, therefore two observers, the first observers sees "x", the second one sees "y". How do you avoid the conclusion that 50% of all observers see an event that should have a probability of 10% ?

How do you get this conclusion? Why do you consider both as equivalent? And how do you use this to calculate a probability of something? Probability of what?

 

[tom.stoer]

How do you get this conclusion? Why do you consider both as equivalent? And how do you use this to calculate a probability of something? Probability of what?

I described my experiment with two possible results "x" and "y". I asked you whether there are two branches. You said "yes". Then you said that one has to associate the probabilities of 90% and 10% to the two branches.

Now MWI says that the detection causes branching in two branches (you agreed to that). That means I have one branch with result "x" and one observer in that branch observing "x". For "y" the same reasoning applies.

I do not use the branches to calculate any probability (this will come later). I simply calculate the 90% and 10% using ordinary QM.

Everything correct so far?

 

[mfb]

Then you said that one has to associate the probabilities of 90% and 10% to the two branches.

Where?

Now MWI says that the detection causes branching in two branches (you agreed to that).

That means I have one branch with result "x" and one observer in that branch observing "x". For "y" the same reasoning applies.

Everything correct so far?

Right.

 

[tom.stoer]

Where?

Here:

The weights of the different branches would have the same numerical values as the probabilities in probabilistic interpretations.
As a result, most of the weight would be in branches with approximately 10% X and 90% Y - similar to probabilistic interpretations, where the probability is high to get approximately 10% X and 90% Y.

Right.

OK. So you agree to having two branches, one branch with result "x" and one observer in that branch observing "x". For "y" the same reasoning applies.

Now repeating the experiment results in a second branching and four branches with four observers in total. On their sheet of paper they have the results "xx", "xy", "xy", "yy".

Still correct?

 

[mfb]

Here:

I said probabilistic interpretations would give those results probability values. MWI does not.

Now repeating the experiment results in a second branching and four branches with four observers in total. On their sheet of paper they have the results "xx", "xy", "xy", "yy".

Still correct?

Sure.

 

[tom.stoer]

Sure.

OK. The story continues. We end up with 2^N observers. Each observer has a list of results, i.e. strings like "xyxxyyxyxxyyxyx...". The first observer has a string "xxx...x" (N * "x"). The Nth observer has a string "yyy...y" (N * "y"). Each observer can calculate the probability of his string (his observed sequence of events) by calculating

$p = 0.9^{n(x)} \cdot 0.1^{n(y)} = 0.9^{n(x)} \cdot 0.1^{N-n(x)}$

Still correct?

 

[mfb]

We end up with 2^N observers. Each observer has a list of results, i.e. strings like "xyxxyyxyxxyyxyx...". The first observer has a string "xxx...x" (N * "x"). The Nth observer has a string "yyy...y" (N * "y").

Sure

Each observer can calculate the probability of his string (his observed sequence of events) by calculating

$p = 0.9^{n(x)} \cdot 0.1^{n(y)}$

Still correct?

There is no probability of his string in MWI.
Or 1, if you like, as all strings occur in some branch.

 

[tom.stoer]

There is no probability of his string in MWI.

Of course there is!

Before I start the experiment I can calculate the probability for the single result "x"; this is 90%. And I can calculate the probability for "xx"; this is 81%. And up to now this has nothing to do with MWI but is simply QM 1st course 1st lesson.

After I completed the experiment I can get a list of all calculations I did before starting the experiment, I look up the string "xyxxyxxxx..." I have written down and then I look up the probability. Or I do the above mentioned calculation afterwards.

 

[tom.stoer]

OK - doing that an observer can check (after the experiments) whether his branch is one with high or low probability according to the QM rules (= whether his branch is a typical one or not). Again this is standard QM.

 

[mfb]

Before I start the experiment I can calculate the probability for the single result "x"; this is 90%.

You can calculate that for probabilistic interpretations, but not for MWI.

And I can calculate the probability for "xx"; this is 81%. And up to now this has nothing to do with MWI but is simply QM 1st course 1st lesson.

That is a common misconception, as the 1st course of QM always uses the (probabilistic) Copenhagen interpretation without even telling that there are other interpretations.

 

[tom.stoer]

This is ridiculous.

I can calculate a statstical frequency f(x) = n(x) / N ≈ 0.9 for these experiments.
I can calculate p(x) = 0.9 approximating f(x) for all past and future experiments.

I works perfectly. Everybody agrees that p(x) is the probability to find "x", which can be derived from some formalism. Now some people ask me what I am doing exactly. I explain all the details of the polarization and how the apparatus works. Fine. I explain that I calculate |<x|ψ>|² = 0.9. Fine? No, one guy from the mathematical faculty asks me why a scalar product should be something like a probability. I have to admit that I have no idea, but I can prove that it works (I have to prove that the scalar product has all properties he expects for a probability). OK, some hard work, but eventually he agrees that it's a probability. Then comes mfb and tells me that it's not a probability; he cannot explain what else p(x) could be and he cannot explain why it behaves as a probability but is something different (what?)

So my question to you is: why is p(x) - which has all properties we expect for a probability - both mathematically and FAPP - not a probability? what is p(x) - if it's not a probability?

(remark: when talking about probabilities I do not mean that there is no other interpretation, I do not talk about a collapse, I do not assume anything like the eigenvalue-eigenvector link, I never talk about a "probability of a system S being in a state |x>"; all what I am saying is that there is a p(x) which acts as a probability of finding "x" simply b/c it allows us to calculate the statistical frequency f(x))

 

[Delta Kilo]

OK - doing that an observer can check (after the experiments) whether his branch is one with high or low probability according to the QM rules (= whether his branch is a typical one or not). Again this is standard QM.

Yes, I think it is fair enough. I'm not sure what are you trying to achieve though, please continue.

 

[Delta Kilo]

There are two branches, therefore two observers

This is incorrect. There are infinitely many(*) observers. Or rather there is a single observer in superposition of infinitely many states. These states exist in infinitely-dimensional space which describes all possible states observer can be in (whether he saw the outcome of X or Y, whether his name is John or Jim, what he had for breakfast, whether the weather was sunny or cloudy etc etc).

Now, as a result of decoherence, terms corresponding to inconsistent states (where observer's left eye sees X and right eye sees Y) will have vanishingly small amplitude and the remaining terms can be divided into two non-overlapping sets: one set has observer and the environment consistent with the result X, another with result Y. We can call these two sets of terms "branches". Each branch then describes infinitely many observers, all of them having seen the same result (either X or Y) but differing in all other aspects.

(*) Or at least a very large and hard-to-quantify number, depending on where you choose to draw the line. Observers are by definition macroscopic, which means having lots and lots of coupled degrees of freedom.When you add the environment, the number becomes truly mind-boggling.

 

[Nugatory]

This is ridiculous.

Indeed it is, and that is the natural end point of just about any argument about interpretations (it would be mischievious of me to point out that this statement has the same single-branch-probability problem as discussed above - just as outcomes with a 9:1 ratio of x:y are most probable, so too are outcomes in which arguments about interpretations lead to ridiculousness).

As far as I can tell, the only way to avoid ridiculousness is to adopt the "Shut up and calculate" interpretation; this position can be veiled in the more respectable minimal statistical interpretation if you want more delicacy than just telling people to shut up. It's still ridiculous (Schrodinger's cat, long-range entanglement, ...), but at least you can tell the ridiculers to shut up.

 

[Quantumental]

Shut up and calculate isn't satisfactory though. Reality has to have a fundamental realness. I Guess the most sensible thing would be to postulate that there has to be a deeper theory that we can't currently Access.

So the real question is: is this postulate more or less troublesome than MWI's born rule issues?

 

[Nugatory]

Shut up and calculate isn't satisfactory though.

Of course it's not. But "shut up and calculate" can be loosely paraphrased as "Shut up! Who cares whether you're satisfied? That's your problem not mine, as long as the calculations match observation. What part of 'shut up' don't you understand? Quit this whining about wanting something 'satisfactory'"

 

[Quantumental]

That's like saying "**** science"
Science is about finding the truth, not stopping when **** gets complicated.

 

[tom.stoer]

Indeed it is, and that is the natural end point of just about any argument about interpretations ...

I think you did nor read carefully what I wrote. I never talked about any interpretation. All what I did was to observe that there are experimental results with some statistical frequency, and that there is a formalism which allows us to calculate the related probability. This is not interpreting the formalism. And in the end even MWI is talking about the Born rule, so denying th existence of probabilities in QM is ridiculous.

 

[jartsa]

Why is it so that in many universes there are beings that have ideas about "probability"?

It's a result of evolution.

How does evolution work in MWI?

Beings that have the ability to control the branching of their environment in that way that they don't lose the ability to control the environment, retain their abilities. Ability to think about probabilities is one of the good abilities to have and keep.

 

[tom.stoer]

Yes, I think it is fair enough. I'm not sure what are you trying to achieve though, please continue.

It's rather simple.

As said I continue with N identical experiments, with the statistical frequency one observer can derive from her result string "xyxxyxxxx...", and a formula which allows her to predict or post-dict this result. Again this is not about any interpretation but about a correct application of some (yet uninterpreted) formalism; "correct" to be understood in the sense that we can verify / falsify this application by comparing p(x) with f(x) and finding nearly perfect agreement.

Now back to interpretation: we agree that for very simple experiments we know the branch structure. Of course I agree that as soon as the observer as a macroscopic system enters the stage we have to talk about decoherence and a far more complicated branch structure. But we can use a trick and prepare the N experiments such that all N branchings are caused far away from the observer such that she does not affect this "primary' branching" (she could in principle decide not to interact with the microscopic subsystem at all). I think this "primary branching" would not called branching at all but is simply a coherent superposition of all possible results of the N experiments. Right?

What we have achieved so far is that we agree on the "primary branch structure" including its counting. But applying simple branch counting forces us to conclude that most observers reside in branches with rather low QM probability (see my example: only 25% of all observers will reside in a branch with 81% probability).

The conclusion is rather simple: either MWI with its branches is wrong, or my simple branch counting applied to the full branch structure is wrong (or meaningless) and we have to correct it in some way. I would like to agree to the latter position, but then your idea that one should count all branches and that the majority of branches have very small measure means that MWI must provide a means to define full branch counting and a derivation of the QM probabilities (in agreement with the observed statistical frequencies).

How is this achieved?

(what I read so far is that nobody has a clear idea how to count branches and how to derive the QM probabilities; some claim that counting is useless, others claim that its not useless but one has to count in the correct way; some claim that the are no probabilities at all - even if there are statistical frequencies; all this seems to be in contradiction with Everett's original idea of a full realistic and straightforward interpretation of the QM formalism b/c all what I read so far says that I am getting it wrong, but nobody is able to tell be how to get it right)

 

[Delta Kilo]

Now back to interpretation: we agree that for very simple experiments we know the branch structure.

Well, ok. Not sure what it was I have just agreed to :)

But we can use a trick and prepare the N experiments such that all N branchings are caused far away from the observer such that she does not affect this "primary' branching" (she could in principle decide not to interact with the microscopic subsystem at all). I think this "primary branching" would not called branching at all but is simply a coherent superposition of all possible results of the N experiments. Right?

It is not sufficient to prevent the observer from interacting directly with the system under observation. In fact it never happens like that, in real life observer interacts with the system through a whole bunch of intermediates. To do what you want, you would have to cut all interactions through the common environment. The only way to achieve that would be to put the experiments into their individual Shroedinger Cat boxes, isolating them completely from the observer and from each other. I'm not sure it is physically or even theoretically possible.
1. CMB or gravitation might be sufficient to cause decoherence.
2. If your state is entangled with the content of the box, it will remain entangled after you close the lid. You can try to disentangle yourself by interacting with your environment but this will not change the amount of correlations between the inside and the outside of the box, it will only spread those correlations all over your environment.
3. The process of decoherence requires access to environment which has nearly infinite number of degrees of freedom where the cross-correlation terms can spread out and dissipate. When you put your experiment into a box, the content of the box has large but still finite number of degrees of freedom. I think if you separate the content of the box into microscopic system under test, measuring apparatus and environment, with environment being much larger that the other two, then looking just at the measuring apparatus, the branches will be split (wavefunction collapsed if you prefer), but when looking at the content of the box as a whole it will still be in superposition with all cross-terms intact.

What we have achieved so far is that we agree on the "primary branch structure" including its counting.

No, sorry, I don't think we did. No branching happens until you open the lid and look inside. Until then we just have boxes with stuff in some unknown superposition state in them.

...most observers reside in branches...

Here you seem to implicitly assume "One branch - one observer". There is no such rule.

either MWI with its branches is wrong, or my simple branch counting applied to the full branch structure is wrong (or meaningless) and we have to correct it in some way.

I think the branch count is meaningless but whether it can be corrected, I don't know. That would require showing that the the branches are in some way have equal measure (from symmetry point of view) or subdivide it into large number of small branches and then somehow invoke central limit theorem.

MWI must provide ... a derivation of the QM probabilities.

Indeed, wouldn't it be nice? But MWI is only an interpretation. Any such derivation must be done using ordinary QM formalism. If such a derivation can be done without reference to "Measurement Apparatus" or "Ensemble of Identically Prepared Systems" or "Pilot Wave", it will benefit MWI indirectly by providing some room to swing Occam's razor about, otherwise it won't change the status-quo.

 

[tom.stoer]

Up to now you only told me what I did wrong.

Just for curiosity it would be nice to explain how it should be done correctly ;-)

Suppose I have a Schrödinger-like cat + box; outside the box we have vacuum, and everything is pitch black; the box opens automatically after half-life-time; at the same time a light inside the box is switched on. So there is no interaction between a far distant observer and the box after preparing the experiment and before the first photons arrive at the observer.

Of course there is decoherence due to the entanglement of the cat with the air molecules inside the box.

Could you please write down
- what the observers will see
- how the total state and the reduced density matrix (partial trace) look like
- how the branch structure looks like and how it is related to the total state and/or the density matrix
(formulas a preferred)

EDIT: Could you please write down the corresponding formulas and information
- observation
- state, density matrix
- branch structure, its relation to state and density matrix
for N such experiments in parallel

 

[Jazzdude]

Sorry, I guess I wasn't clear enough. I was talking about off-diagonal elements of the reduced density matrix. Say you start with cat in state a|Cdead> + b|Clive> where <Cdead|Clive> = 0 and environment in state |E>, the state of the composite system being a product state. Final state of the system is then a|Cdead>|Edead> + b|Clive>|Elive>, with <Edead|Elive>≈0 FAPP. But while these cross terms <Edead|Elive> are vanishingly small, they can never become exactly 0, otherwise the unitarity is violated.

This has nothing to do with unitarity. Consider the cat basis {|d>,|a>} and the input environment basis {|e1>,|e2>} and construct a separable state. Then map the product basis (|e1>|d>, |e1>|a>,|e2>|d>,|e2>|a>) to a new basis with 4 orthogonal environment states, in this order: (|d>|e'1>,|a>|e'2>,|d>|e'3>,|a>|e'4>). That is a unitary map, because you map an orthonormal basis to another orthonormal basis. The input and the output space are different, because the size of the environment increases. This can be understood as increasing entropy in the environment, which is necessary for what we do here. This is why I said that the result is much more related to infinite dimensional state spaces than unitarity. Calculate the density matrix of the input and output states. The interference terms vanish completely.

Cheers,

Jazz

 

[kith]

That's like saying "**** science"
Science is about finding the truth, not stopping when **** gets complicated.

No, science is about models which match observations. You can never tell how far these models are away from the real thing. We don't know if our universe is something big or something small in an absolute sense, we only know it is big wrt to us.

 

[Jazzdude]

Now, as a result of decoherence, terms corresponding to inconsistent states (where observer's left eye sees X and right eye sees Y) will have vanishingly small amplitude and the remaining terms can be divided into two non-overlapping sets: one set has observer and the environment consistent with the result X, another with result Y. We can call these two sets of terms "branches". Each branch then describes infinitely many observers, all of them having seen the same result (either X or Y) but differing in all other aspects.

No. Decoherence can resolve interference of superpositions, but not different local already stabilized results. So an observed difference between two eyes is not resolvable by any decoherence mechanism. It's not that simple.

Cheers,

Jazz

 

[jartsa]

No. Decoherence can resolve interference of superpositions, but not different local already stabilized results. So an observed difference between two eyes is not resolvable by any decoherence mechanism. It's not that simple.

Cheers,

Jazz

When experiment is prepared so that both eyes will observe same unknown polarization, then the eyes are being tuned so that the eyes that reseive same kind of light are propelled into aproximately same direction in Hilbert space.

Right?

 

[lukesfn]

Right
Counting branches (as 1+1=2) is the wrong approach, nothing will help there. It is like the lottery example: if I just count "I will not win" and "I will win" I don't get a relevant result - I just get a list of all options.

Sorry to interject, but I think the lottery example is not posed in a fair way. If you pic one number from say, 1 million possible out comes in the lottery, then there are 999,999 possible out comes where you don't win, and only 1 where you do. Put into MWI perspective where each outcome is a branch you could simply count the branches to work out the probability.

If only 2 branches are created for a a non 50/50 probability, then you seem to be saying that you would most likely find your self being the most likely you, not the less likely you. But if for every more likely you, there is guaranteed to be a less likely you, how can the less likely you actually be less likely. I would think by definition, they must be equally likely, which is where everything falls apart.

However, I personally don't have any problem adding as many branches as necessary to reproduce probabilities. If you already believe there are infinite branches, then it won't even change the total count;)

 

[kith]

I think also the Many Many Worlds phenomenon is relevant here: the number of worlds depends on how you decompose the whole universe into subsystems (see Many Worlds proved inconsistent? - Physics Forums for example). Explantions of the emergence of probabilities have to be consistent with this.

 

[tom.stoer]

The problem seems to be that we are not discussing the Many-World-Interprertation but a collection of interpretations of Many-Worlds-Interpretations.

There is a rather simple statement in the Kopenhagen Interpretation (or variants): when measuring a quantity A which is represented by an observable A acting on Hilbert space states
1) a state $|\psi\rangle$ collapses to an eigenstate $|\psi\rangle \,\longrightarrow\,|a\rangle$ with eigenvalue a (we don't know how and why)
2) the probability observing the eigenvalue $a$ is determined by $|\langle a|\psi\rangle|^2$
So these interpretations talk about states, probabilities and collapse; they can exactly say how to calculate probabilities and how to define a collapse (expressed in terms of the standard formalism)

What I am still missing is the same clarity of statements for MWI. MWI is about branches (or whatever) so it should be possible to write down the definition of a branch (again in terms of the standard formalism)

 

[kith]

What I am still missing is the same clarity of statements for MWI. MWI is about branches (or whatever) so it should be possible to write down the definition of a branch (again in terms of the standard formalism)

A simple definition is to take the von Neumann measurement scheme and instead of singling out one term in the final sum by collapse, you interpret each term as a branch. The physical justification to talk about such a sum in the first place is given by decoherence.