Question regarding the Many-Worlds interpretation

[Jazzdude]

How do you define a "single event" in the real world, where decoherence happens everywhere all the time, and for all those events parallel?

You don't have to do that in a real world. Instead you can fabricate your own hypothetical setup and describe it mathematically. A single event would then be any reasonably isolated process with temporally asymptotic stable input and output states. If the Born rule emerges, that emergence must be reducible to some such system and can be studied there.

Cheers,

Jazz

 

[tom.stoer]

the ratio of the number of branches is meaningless unless we can say something definitive about their relative probabilities.

...

On the contrary I was just trying to show that branch counting is meaningless unless a) you can say something about relative measures of the branches (eg from symmetry considerations) and b) there is an unambiguous way to count the branches.

The problem is that as long as you are not able to say anything about branch counting, measures etc., you have gained nothing.

Let's come back to my experiment. It is clear that changing the superposition of the polarization and their Born probabilities must somhow affect the branching. What has the MWI to say about that?

 

[stevendaryl]

That problem only appears if you try to count events along a single line of history. That's not what is usually done. We have the alternative of counting over the branching ensemble of a single event. For that the born rule must be accurate, because all options are realized independently. So the scenario you describe is not an issue.

Cheers,

Jazz

I don't think it matters how you do the counting. The fact is that the only information we have is about what happens along a single line of history. That doesn't logically tell us anything about the Born rule unless we assume that our particular history is "typical" in some sense. But the notion of "typical" is theory-dependent.

 

[tom.stoer]

Simple counting does not work.

I fully agree. I started with a very simple counter example, but of course there are more sophisticated examples.

And the way to avoid that is the squared amplitude as measure for the branches.

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

 

[Jazzdude]

I don't think it matters how you do the counting. The fact is that the only information we have is about what happens along a single line of history. That doesn't logically tell us anything about the Born rule unless we assume that our particular history is "typical" in some sense. But the notion of "typical" is theory-dependent.

You don't have to make any such assumption if you can show that the Born rule holds for the ensemble of observations following from a single event. Because that already implies that we are in a "typical history" with overwhelming likelihood if enough such events are concatenated.

Cheers,

Jazz

 

[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 (Schrödinger'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)