## What is Many Worlds? What isn't Many Worlds?

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### What is Many Worlds? What isn't Many Worlds?

So there seems to be an unbelievable amount of confusion about what many worlds is and what many worlds is NOT. Here is one example

Quantum mechanics is a deterministic theory. In fact, you can call it the "most deterministic theory", as it has a method to transform every non-deterministic theory in a deterministic one*. Just split the world in multiple parts, one for each possible "outcome" of anything which might appear random. This directly leads to many worlds and can make every theory deterministic.

This is not many worlds, and not how many worlds works. This implies you can get the probabilities by simply counting up the worlds- which doesn't work. So I'll outline a bit of history:

Standard quantum mechanics rests on a set of axioms. One group of axioms tells us that we represent our objects as wavefunctions, another group tells us how the wavefunction evolves (according to the famous Schroedinger equation).

The final axiom relates to measurement- but its a very important one. Its how you connect the theory to experiment! This is where the probabilities creep into quantum mechanics. A wavefunction can be in multiple positions, momenta, or energies, but we only seem to measure one. So, Born, and later Von Neumann showed us that we can interpret the magnitude squared of amplitudes as probabilities. This is known as the Born rule.

This isn't problematic for the theory mathematically, but it is physically. We expect quantum mechanics to be the theory underlying macroscopic behavior- but we have to add a special rule for measurements where particles behave differently.

Enter Hugh Everett- http://www.univer.omsk.su/omsk/Sci/Ever ... r1957.html who said 'well, what happens if we just get rid of the measurement process and the Born rule.' In this case, the entire universe is a giant wavefunction. Everett (and DeWitt and Hartle, among others) noticed two things- 1. large portions of the wavefunction evolve independently of other portions of the wavefunction. 2. In the limit of infinite measurements,every one of these independent branches of the wavefunction obeys the Born rule. So Everett interpreted this to mean that each independent branch of the universe exists as a separate 'universe' AND that he had successfully derived the Born rule.

This is fantastic work- but there is a huge problem. When you DON'T have infinite measurements, then there are lots and lots of universes where the Born rule fails. In fact, there are orders of magnitudes MORE universes where the Born doesn't WORK. This is obviously a huge problem because the Born rule has been empirically verified over and over again.

So- can we fix the problem? Maybe. Deutsch said 'wait! We don't need to show that most universes obey the Born rule. Instead, we can use a bit of game/decision theory to show that a rational observer will play the game AS IF the Born rule were valid' http://uk.arxiv.org/PS_cache/quant-ph/p ... 2157v2.pdf (not the original Deutsch paper but an easier to follow demonstration of the same thing by Wallace).

Unfortunately, this runs into problems to- the first is philosophical. Is quantum mechanics to be a theory of how the world works, or a theory about our beliefs? Deustch's ideas tangle these two things up.

The second problem is more technical- Deutsch's proof rests on some specific definitions of rational behavior. See this: http://arxiv.org/PS_cache/arxiv/pdf/090 ... 0624v1.pdf for a summary of some of the criticisms.

So- many worlds is an attempt to kick away the Born rule/measurement postulates. Its a noble attempt- but when we do that WE NEED A WAY TO GET NUMBERS OUT OF THE THEORY. If we can't, we have built a mathematical edifice, but not a physical theory. Everett thought he had a way to recover the Born rule- and indeed he has the tantalizing start of one. Unfortunately, his proof fails for an finite number of measurements, and no one has been able to recover a way to get numbers out of the theory.

This is why no one actually uses many worlds to do calculations. Literally no one. Most use Copenhagen, and some use a consistent histories or path integral formulation. Even the believers of the interpretation can't use it- because there is no method to make actual predictions with the interpretation. So while its a tantalizing collection of ideas, right now the many worlds interpretation is not a physical theory.
SU3SU2U1

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### Re: What is Many Worlds? What isn't Many Worlds?

I'm certainly no expert on the matter, but, whilst MWI certainly doesn't immediately work as an attempt to derive the Born rule, if such a rule is inserted manually it's still no worse off than the various other interpretations. So, why isn't more common to take as axiomatic that, at any point where the worldline splits into many, they split according to the relevant Born rule (the square of the amplitude gives the probability of passing into that universe (and, with some basic conditional probability, squared amplitude must be conserved at any splitting of the worldline)).
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eSOANEM

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### Re: What is Many Worlds? What isn't Many Worlds?

eSOANEM wrote:I'm certainly no expert on the matter, but, whilst MWI certainly doesn't immediately work as an attempt to derive the Born rule, if such a rule is inserted manually it's still no worse off than the various other interpretations. So, why isn't more common to take as axiomatic that, at any point where the worldline splits into many, they split according to the relevant Born rule (the square of the amplitude gives the probability of passing into that universe (and, with some basic conditional probability, squared amplitude must be conserved at any splitting of the worldline)).

There still seems to be a problem because that sort of Born rule no longer makes total sense - if the wavefunction of the universe just evolves according to the Schrödinger equation, "splitting" isn't fundamental - the Schödinger equation already provides exact rules on how everything happens. Which would mean any Born-rule-thing compatible with the universe just being "normal" all the time has to be a high-level concept, a derivative of the other rules. But what if the other rules derive a contradiction of the Born rule instead? We don't know unless we have a proof, and so maybe using an ad-hoc Born rule actually gives you an inconsistent system.
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### Re: What is Many Worlds? What isn't Many Worlds?

So, why isn't more common to take as axiomatic that, at any point where the worldline splits into many, they split according to the relevant Born rule (the square of the amplitude gives the probability of passing into that universe (and, with some basic conditional probability, squared amplitude must be conserved at any splitting of the worldline)).

The whole point of many worlds is to say "the wavefunction evolves unitarily according to Schroedinger's equation ALWAYS". If you try to impose the born rule in the manner you are suggesting, you'll have to monkey with this.

There are serious problems with what it means to "pass into that universe." Remember, there is a 'you' in each universe. What does it mean for one 'you' to have more reality. To recover the probabilities, you can say 'ok, if my relative probabilities are 1/3 and 2/3 maybe what I want is for the wavefunction to divide into 3 worlds, of which 2 are the same'. But now, you are no longer using Schroedinger to evolve the wave function! The whole point of the paradigm is to always use Schroedinger- if we have to add on all this weird branching stuff, we've simply reformulated the measurement problem in a particularly unwieldy way.

There is an interpretation that attempts to do what you say (many minds). In that interpretation 'you' have an infinite number of minds, and as the wavefunction evolves, the infinite set of minds becomes sub-divided. This set of minds gives you a way to play games with what it means for one branch to be 'more-real' than another. The problem is there is a huge apparatus (infinite sets of minds) sitting on top of normal quantum mechanics to make this work. And there is the philosophically unappealing mind-body dualism.

In both cases, why not just stick with standard copenhagen?
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### Re: What is Many Worlds? What isn't Many Worlds?

Hmm... That kind of makes sense I guess. Glad I've got that sorted. It's always bugged me.
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### Re: What is Many Worlds? What isn't Many Worlds?

Isn't it misleading to say that there are multiple universes in the MWI? I mean, yes it's part of the name, but it's just really a more spread-out wave-function with different parts acting differently isn't it? The amplitude squared aspect almost makes it seem like we should think about the wave as surfaces outlining volumes in a 3D graph rather than lines outlining areas in a 2D graph... but the most natural way to do that (spinning the wave function about the amplitude/new dimension plane) would make a factor of pi appear out of nowhere.
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WarDaft

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### Re: What is Many Worlds? What isn't Many Worlds?

Maybe I should have made my explanation a bit longer and less sloppy. I just wanted to point out the general idea there, that multiple "worlds" can exist in parallel.

>> WE NEED A WAY TO GET NUMBERS OUT OF THE THEORY
Keep in mind that this is a problem for most interpretations. The Kopenhagen interpretation just says "well, the probability is the amplitude squared". It is not so unnatural, as the sum/integral of squared amplitudes is conserved (this is a bit sloppy again) in the evolution of wave functions, but it is still an additional thing which is added in the interpretation.

"the many worlds interpretation is not a physical theory."
Correct. It is an interpretation of quantum physics (that is the theory), like everything else.

>> The whole point of many worlds is to say "the wavefunction evolves unitarily according to Schroedinger's equation ALWAYS".
That is the one-line description of many worlds .
And you can read off my point in the quote: With the MWI, the universe is deterministic. Always. With every fundamental theory based on quantum mechanics, even with theories which have not been formulated yet.

[quote=WarDaft]but it's just really a more spread-out wave-function with different parts acting differently isn't it?[/quote]
That is right, and the reason why I don't like the name "many worlds". It is one single world with an incredibly complicated wave function.
However, many quantum mechanical calculations have to deal with complicated wave functions (simulations of proton-proton collisions and similar things) - they just look at systems which are much smaller than the universe in order to have a chance to calculate it.
mfb

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### Re: What is Many Worlds? What isn't Many Worlds?

Keep in mind that this is a problem for most interpretations. The Kopenhagen interpretation just says "well, the probability is the amplitude squared". It is not so unnatural, as the sum of squared amplitudes is conserved (this is a bit sloppy again) in the evolution of wave functions, but it is still an additional thing which is added in the interpretation.

Right- the Copenhagen interpretation has a measurement postulate that tells us exactly how to get numbers out. Many worlds does NOT. Many worlds is the only interpretation that lacks this essential feature.

Correct. It is an interpretation of quantum physics (that is the theory), like everything else.

All of the 'interpretations' of quantum mechanics are actually very different formulations of quantum mechanics. Most even have testable differences (GRW/objective collapse interpretations have been effectively falsified by the various macroscopic superposition experiments).

For some reason, many physicists seem to have this idea that interpretations are just 'stories' grafted on to the same set of axioms. This is simply not the case. Interpretations are different sets of axioms that (hopefully) lead to the same predictions.

And you can read off my point in the quote: With the MWI, the universe is deterministic. Always.

Yes, but it doesn't make predictions, as I go on to explain. Many worlds is fully deterministic and absolutely useless. As the papers I reference try to describe- the current best attempts to get a prediction out of the theory require game theoretic descriptions of the rational behavior of the people running the experiments, and they come up a bit short.
SU3SU2U1

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### Re: What is Many Worlds? What isn't Many Worlds?

>> Yes, but it doesn't make predictions, as I go on to explain.
And this is a downside of MWI compared to what?

>> the Copenhagen interpretation has a measurement postulate that tells us exactly how to get numbers out.
"The probability that you (and your measurement result) are in some part of the wave function is proportional to the squared amplitude of this part".
This is a postulate, it does not follow from the interpretation, and it gives you Born probabilities.
In all interpretations, you calculate numbers by taking the amplitude squared. The Copenhagen interpretation (oh, with C in english) is just a more intuitive way to see this as "probability".
mfb

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### Re: What is Many Worlds? What isn't Many Worlds?

And this is a downside of MWI compared to what?

Copenhagen, consistent histories, Bohm, ensemble, objective collapse, many minds and pretty much any other interpretation of quantum mechanics. Many worlds is the only one (to my knowledge) that is formally incomplete.

mfb wrote:The probability that you (and your measurement result) are in some part of the wave function is proportional to the squared amplitude of this part".

Define "you" in this context? You, as a collection of particles, are in ALL the branches of the wavefunction. To get what you want you have to be saying that some branches are 'more real' than others- in what way?

I think you are slowly working yourself towards a 'many-minds' interpretation, which is fine, but is mind-body dualism REALLY optimal compared to a measurement postulate?

Also, as a technical matter, when you add this postulate you are no longer talking about many worlds- many worlds has no such postulate, and the people who advocate many worlds are proud that it doesn't (and generally convinced some way forward on Everett's proof will be found).

Finally- here is the simplest extension to the Copenhagen interpretation that removes the formal measurement problem- modify the postulates so that quantum mechanics is a theory of ensembles. The wavefunction represents an ensemble of systems, NOT an individual system, and it makes ensemble predictions instead of individual predictions. You get probability just fine, measurement can be well defined, and all we lose is the ability to talk about a single system. This is called an ensemble interpretation.

Also- while people rag on the measurement postulate for having a vaguely defined notion of measurement- remember that IN PRACTICE our definitions are all circular- we start with classical hamiltonians and then quantize them.
SU3SU2U1

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### Re: What is Many Worlds? What isn't Many Worlds?

>> I think you are slowly working yourself towards a 'many-minds' interpretation
I am not changing my mind, just formulating its opinion.

>> To get what you want you have to be saying that some branches are 'more real' than others- in what way?
In the same way a 80%-probability is more likely to happen ("more real"?) than a 20%-probability. Don't think in an integer number of different branches, as the wave function is continuous.

>> Many worlds is the only one (to my knowledge) that is formally incomplete.
What is missing? There is spacetime with a wave function inside. Done.
Know the wave function at any specific time, and you can compute the function for any time in the past and the future (given a theory of everything, and ignoring issues like the fact that gravity modifies the spacetime).

You are asking questions which you cannot define as questions about the physical state of the universe. This is fine - but why do you expect answers from physics then?
mfb

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### Re: What is Many Worlds? What isn't Many Worlds?

mfb wrote:What is missing? There is spacetime with a wave function inside. Done.
Know the wave function at any specific time, and you can compute the function for any time in the past and the future (given a theory of everything, and ignoring issues like the fact that gravity modifies the spacetime).

You are asking questions which you cannot define as questions about the physical state of the universe. This is fine - but why do you expect answers from physics then?

You need a way to go from having a wave function to predicting the outcome of an experiment. A wave function is not itself observable, it is the Born rule that connects the wave function to observable things. If you've just got a theory that tells you how some unobservable object behaves, then you haven't got a fully formed scientific theory. Having a wave function plus the Born rule gives you a nice theory, but in many worlds you don't have the Born rule (in a satisfactory manner). As discussed above, MWI suffers from the additional problem that it may or may not be possible to add the Born rule as a postulate and get a consistent theory, not to mention adding it as a postulate removes the very feature that makes MW appealing in the first place!

Unrelated note: I really want to thank SU3SU2U1 for taking the time to make this thread and provide a good overview of what many worlds is and is not. It is time for the misconceptions about this subject to be put to rest, and he/she is likely the most qualified person here to do it, so we should be grateful.
JWalker

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### Re: What is Many Worlds? What isn't Many Worlds?

I am not changing my mind, just formulating its opinion.

Right, but it sounds like your opinion is many-minds and requires a mind-body dualism of sorts. Thats fine, its a perfectly valid interpretation, but the wave function is not all there is. You are also making very strong assumptions about conscious vs unconscious observers.

In the same way a 80%-probability is more likely to happen ("more real"?) than a 20%-probability. Don't think in an integer number of different branches, as the wave function is continuous.

Except BOTH THINGS HAPPEN. If you have an electron prepared so that its 4/5 spin up and 1/5 spin down and you measure it then the new wave function is root(4/5)'you entangled with a spin up electron'+root(1/5)'you entangled with a spin down electron'. Both branches exist. Both versions of 'you' exist. You want to assert that there is MORE you in the first branch than the second. What does that mean? How do you want to formalize it?
SU3SU2U1

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### Re: What is Many Worlds? What isn't Many Worlds?

JWalker wrote:You need a way to go from having a wave function to predicting the outcome of an experiment.

The outcome is easy: All possible outcomes (when viewed with Copenhagen) are realized, with their appropriate amplitude. This is unintuitive like hell, and it is not useful to use this model if you want to write Monte Carlo generators, sure.

I think you try to force conciousness in physical terms here. But what is conciousness in physical terms?
What does it even mean that your experiment will (with Copenhagen) measure A 80% of the time and B 20% of the time?
It has a clear meaning when you look at the past and track the outcomes, but for the future?

>> but in many worlds you don't have the Born rule (in a satisfactory manner)
Only if you have a problem with things which are "more real" (thanks for the expression) than others. Let things be "real with the Born probability", and it is fine.
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### Re: What is Many Worlds? What isn't Many Worlds?

mfb wrote:What does it even mean that your experiment will (with Copenhagen) measure A 80% of the time and B 20% of the time?

It means that if you prepare N systems the same way, then in the limit of large N, 0.8N will have outcome A and 0.2 will have outcome B.

Only if you have a problem with things which are "more real" (thanks for the expression) than others. Let things be "real with the Born probability", and it is fine.

But you are dancing around what you mean by more real- after all, the 'you' that measures outcome A will have JUST as strong a claim to being real as the 'you' that measures outcome B. You want to say the outcome A observer is somehow more real, but why?

What I'm asking for here is an actual mathematical definition of what you mean by probability that 'you' end up in a given world. I'm arguing you can't do it without an extension to your theory that requires some sort of continuum of minds.

In short- the baggage associated with making many-worlds work is as large or worse than the baggage of just using an ensemble interpretation. Or, if you insist on no collapse, something like Bohm's interpretation.
SU3SU2U1

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### Re: What is Many Worlds? What isn't Many Worlds?

mfb wrote:The outcome is easy: All possible outcomes (when viewed with Copenhagen) are realized, with their appropriate amplitude. This is unintuitive like hell, and it is not useful to use this model if you want to write Monte Carlo generators, sure.

I think you try to force conciousness in physical terms here. But what is conciousness in physical terms?
What does it even mean that your experiment will (with Copenhagen) measure A 80% of the time and B 20% of the time?
It has a clear meaning when you look at the past and track the outcomes, but for the future?

>> but in many worlds you don't have the Born rule (in a satisfactory manner)
Only if you have a problem with things which are "more real" (thanks for the expression) than others. Let things be "real with the Born probability", and it is fine.

You can't just have a universe evolving according to just the Schrodinger equation, because the Schrodinger equation alone does not give you the quantum behavior that we observe. In fact, the Schrodinger equation is not really quantum mechanical at all; you can write down a Schrodinger equation for classical systems as well. The weird quantum stuff does not come from the Schrodinger equation, you need something more, and this is why we have collapse in the Copenhagen interpretation. MWI needs to be able to produce the quantum behavior that we observe, and that means you need something just as weird as the Copenhagen collapse in order to make correct predictions. Without that, the theory either makes no predictions or makes incorrect predictions. MWI has this, but as a consequence you cannot just assume things are real with the Born probability, that needs to be derived from the MWI framework. Either it can be, and MWI is fine, or it cannot be, and MWI has a big problem. We don't know which is the case currently.
Last edited by JWalker on Sat Mar 24, 2012 1:14 am UTC, edited 2 times in total.
JWalker

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### Re: What is Many Worlds? What isn't Many Worlds?

I've weighed in this a few other times on seperate threads, so I'll hold off getting too involved for now, as it'd probably just lead to you repeating things you've said elsewhere. I suspect I'm underqualified compared to mfb to make a good argument anyway.

Having said that, as you've mentioned to at least a few people that what they think is MWI is in fact many-minds, do you have any links to a good explanation of the details of that to contrast with the MWI papers you've linked? (Ideally something more detailed than wiki, which may or may not offer a satisfactory assessment of it. Alternatively, a paper that directly compares MWI with many minds. [I'd look myself, but you seem more qualified to tell papers that the authors are correctly interpreting the theories from those using a 'wrong' interpretation.])

Is there any argument to be made that there exists MWI as a theory (Everetts thing), and a seperate MWI that is strictly an interpretation of results offered by the copenhagen framework? Or is there a reason why adopting the latter necessarily throws you into Everetts camp (not necessarily a bad thing)?

Dopefish

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### Re: What is Many Worlds? What isn't Many Worlds?

Dopefish wrote:Having said that, as you've mentioned to at least a few people that what they think is MWI is in fact many-minds, do you have any links to a good explanation of the details of that to contrast with the MWI papers you've linked?

The original work is Albert and Loewer's "Interpreting the Many Worlds Interpretation." Unfortunately, I don't know of a version thats not behind a pay wall or I would have linked to it.

You can also follows some of the literature by back-tracking from the papers linked above (the Wallace paper has a nice discussion of some of the discussion revolving around many worlds). The point is that its tricky to recover the probabilities in many worlds without more structure than simply the wavefunction.

In general, I don't think that academic papers you'll find will make the same mistake that commenters on web boards do- the larger issue is that no one seems to teach interpretations of quantum mechanics, and the courses generally teach an ensemble interpretation. This means that most physics undergrads have this idea of many worlds that is built up out of a fusion of textbook quantum and pop-sci many worlds. I'm honestly consistently surprised how few undergrads with an interest don't dig into the original interpretation literature- its pretty easy to follow, and remember you may never have easy access to journal articles again!

Is there any argument to be made that there exists MWI as a theory (Everetts thing), and a seperate MWI that is strictly an interpretation of results offered by the copenhagen framework?

The no-collapse theories are in theory experimentally different from the collapse-interpretations. Deutsch wrote an interesting article in the book Quantum Concepts of Space and Time outlining some possible tests you could do (with magical technology that doesn't yet exist). The key point is that collapse is fundamentally irreversible while unitary evolution isn't. If you can easily get macroscopic objects into super-positions, you can make various experiments to test the theories.

Also- to many world supporters- why is many worlds preferable to no-collapse interpretations like Bohm's?

Also, just so I'm not beating up on many worlds too badly- there are still really good on going attempts to get the probabilities right. Mike Weissman has a really nice paper where he shows adding some non-linear terms to the wavefunction allows you to recover the Born probabilities by simple counting in a many worlds framework. Its a fun paper http://xxx.lanl.gov/pdf/quant-ph/9906127.pdf Of course, you lose the linear schroedinger equation for something much more complicated.
SU3SU2U1

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### Re: What is Many Worlds? What isn't Many Worlds?

SU3SU2U1 wrote:I'm honestly consistently surprised how few undergrads with an interest don't dig into the original interpretation literature- its pretty easy to follow, and remember you may never have easy access to journal articles again!

I think the majority of experiances undergrads (which includes me for another month or two) have with journal articles is relatively modern papers on material intended for experts (e.g. not undergrads), which they're only reading since a prof/supervisor made them. As such, we get the idea that all papers are all highly technical stuff that we can't easily follow. I have to admit, the first time I read a paper I could follow (I think it might actually have been QM interpretation themed), I was really surprised, since it wasn't super dry/technical and I didn't really think published stuff could be.

Couple that perception with the fact we're usually preoccupied with assignments, and I'm not surprised theres very little recreational paper reading done by undergrads.

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### Re: What is Many Worlds? What isn't Many Worlds?

Just for fun... (and let's ignore relativity and pretend that we can talk about a universal notion of time):

There are exactly 2^^16 (two tetrated to the sixteen) universes. At every instant in time, these universes are distributed with density given by the square magnitude of the wave function. As time goes forward, this continues to remain true because after every infinitesimal increment of time (perhaps every planck-time), each one of these universes automatically re-samples itself according to the new wave function at that time. That is, each universe independently draws a new sample at random from the distribution established the wave function, and then the contents of that universe are instantaneously replaced with that of the new sample drawn. Of course, observers do not subjectively experience this very rapid repeated replacement. Indeed, all the copies of you reading this message in universes that are essentially the same as the one you're in right now, all certainly remember existing for much longer, because when you all winked into existence during the last resampling less than a planck-time ago, all of you did so with complete with all of the memories in your brains.

And of course, almost all of the observers in these universes should observe (or rather, remember observing) a history consistent with the Born probabilities, during the all brief times that they exist.
lightvector

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### Re: What is Many Worlds? What isn't Many Worlds?

I think the majority of experiances undergrads (which includes me for another month or two) have with journal articles is relatively modern papers on material intended for experts (e.g. not undergrads), which they're only reading since a prof/supervisor made them

Aren't you curious to go back to some of the historical papers? As an undergrad, one of my most formative moments was reading Einstein's papers and realizing that I could actually understand them.
SU3SU2U1

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### Re: What is Many Worlds? What isn't Many Worlds?

SU3SU2U1 wrote:The no-collapse theories are in theory experimentally different from the collapse-interpretations. Deutsch wrote an interesting article in the book Quantum Concepts of Space and Time outlining some possible tests you could do (with magical technology that doesn't yet exist). The key point is that collapse is fundamentally irreversible while unitary evolution isn't. If you can easily get macroscopic objects into super-positions, you can make various experiments to test the theories.

This only works if you have a very specific definition of when collapse happens. I know people who would say collapse is a real process, and have also said that if you put photon source, a beamsplitter, and a photon detector in a perfect box, you would get a photon detector in a superposition of detected and not detected. That is, "causes collapse" isn't an inherent property of systems, in whatever this interpretation is. This seems to be pretty common, and is always used to make the same predictions as MW + ad-hoc Born rule would.

EDIT:
Also- to many world supporters- why is many worlds preferable to no-collapse interpretations like Bohm's?

You know what's nice? Causality.

Actually, I just finished reading Everett's paper you linked and his argument for the Born probabilities seems more general than he lets it be. He defines probability only as the limit of a large number of trials, which is not all that it is (see: Bayesian probability). So I think as long as you buy that the probability of a state should be the sum of the probabilities of independent eigenstates (or whatever symmetries that statement can be decomposed into), the derivation of the Born rule in the paper works just fine.
Last edited by Charlie! on Sat Mar 24, 2012 11:39 am UTC, edited 1 time in total.
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### Re: What is Many Worlds? What isn't Many Worlds?

>> It means that if you prepare N systems the same way, then in the limit of large N, 0.8N will have outcome A and 0.2 will have outcome B.
How to you prepare N universes?
Looking at 1000 experiments you did in the past is not the problem, as I said.
Building the same machine 1000 times does not help, it just gives you other values for probabilities (like 0.8^1000, 1000*0.8^999*0.2 and so on) for a larger experiment.

JWalker wrote:You can't just have a universe evolving according to just the Schrodinger equation, because the Schrodinger equation alone does not give you the quantum behavior that we observe.

Why not? Assume that the world is a single, very complex wave function (read: "many worlds"). Every part of it which is something like a human brain will make observations according to its part in the universe. There are humans which have a history far away from Born probabilities. Turns out that the xkcd board which we use is in a universe which is not so far away from them.
If you think of "you" as a human in a certain point in time, the probability that "you" are in some part of the wave function is its amplitude squared. But what does that mean, '"you" as a human in a certain point in time'? You want a framework to answer this question. Give me a framework to formulate this question!

SU3SU2U1 wrote:If you can easily get macroscopic objects into super-positions, you can make various experiments to test the theories.

Well, there is some progress, with ~50µm as the upper limit IIRC. But which experiment would rule out MWI or Copenhagen? Or would they just rule out other ideas?
Schroedingers cat as a real experiment would give some additional problems to define "measurement" - is the isolated cat able to observe its status? If the wavefunction does not collapse (and therefore cats can be in superpositions), why should it do when we open the box? We are just parts of a larger box called "universe".
What about a virus on a 50µm-object in superposition of different oscillation states?
And why the heck should we need living objects* to talk about physical properties?

*please, no discussion what is required to call an object "living"

>> why is many worlds preferable to no-collapse interpretations like Bohm's?
As far as I (and the wikipedia articles) know, it has some problems with special relativity, and may need special initial conditions (to get the Born probabilities!). And it adds particles to the theory, in addition to the wave functions.
mfb

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### Re: What is Many Worlds? What isn't Many Worlds?

SU3SU2U1 wrote:Aren't you curious to go back to some of the historical papers? As an undergrad, one of my most formative moments was reading Einstein's papers and realizing that I could actually understand them.

Well yes, but that curiousity doesn't translate well into actually doing so. Now, I have read a few things unlike many of my peers who tend to take a 'leave school at school' stance, but it's difficult to find the time to jump through the appropriate hoops to track down the papers from a university computer when at any given moment theres 5ish assignments to be done. If I feel as if I have free time to spend not doing assignments, I'm more inclined to satisfy my academic curiousities by watching video lectures on material not covered in my courses, which I can easily do from home, unlike accessing papers.

The above isn't really a good excuse since I realise going to a university library computer doesn't take a huge amount of time, but doing so is a fair deviation from the typical undergrad lifestyle*. I would think that as a grad student where you likely have an office(/research group workspace) with a computer with journal access, acquiring papers for recreational reading becomes much more trivial, and less of a chore. I agree we (interested undergrads) probably should be more actively acquiring these things, but I'm not terribly surprised we don't.

*=Based on my experiances with physics folks, where we cling to our textbooks as we try to figure out how math works since we're not allowed to use maple (or similar). Undergrads of other varieties might find themselves spending more time on university computers. (Also, my year is tiny; I am one of a whopping 3 honours physics students. My sample size may be lacking.)

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### Re: What is Many Worlds? What isn't Many Worlds?

Charlie! wrote:You know what's nice? Causality.

Yes, but Bell puts strict limit on what we can mean by 'causal' in all quantum theories.

Actually, I just finished reading Everett's paper you linked and his argument for the Born probabilities seems more general than he lets it be. He defines probability only as the limit of a large number of trials, which is not all that it is (see: Bayesian probability).

Please continue reading the literature- which has nice discussions on why Everett isn't the end of the story.

Quantum probability as Bayesian is also really tempting no matter what your interpretation (after all, after a measurement you know for sure what value you got, so 'collapse' is completely logical in this context). Chris Fuchs has done interesting work on this http://perimeterinstitute.ca/personal/cfuchs/ see the second power point under talks,and his lecture notes.
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### Re: What is Many Worlds? What isn't Many Worlds?

mfb wrote:How do you prepare N universes?

You don't have to- the point is that with an ensemble interpretation we say that the wavefunction represents an ensemble of universes,and thats how we define the probabilities. What we DO limit is our ability to ask certain questions about the universe as described by quantum mechanics.

Building the same machine 1000 times does not help, it just gives you other values for probabilities (like 0.8^1000, 1000*0.8^999*0.2 and so on) for a larger experiment.

? This is how you probability works. The sample average approaches the ensemble average in large N limits. Are you seriously trying to argue that ensemble probabilities aren't well defined? Remember, we are talking formalism and definitions. Whether or not you can prepare infinitely many identical systems is no more relevant than whether or not you can flip a coin infinitely many times.

If you think of "you" as a human in a certain point in time, the probability that "you" are in some part of the wave function is its amplitude squared.

Except, empirically its not. The probability that I'm located in this one segment of the many worlds framework, typing on this board is exactly 1. If I measure the spin of an electron, there is a me that will see spin up and he occupies the world where he saw spin up with probability 1. There is a second me that will see spin down, and he occupies the world where he saw spin down with probability 1. This is true REGARDLESS OF THE RELATIVE AMPLITUDES. Thats important.

What you seem to be wanting to say is that BEFORE coming entangled with the electron, the me in this world can predict with certain probabilities which branch he will end up in when he does become entangled. Even though there is a spin down world with probability 1 and a spin up world with probability 1, there is somehow different amounts of continuity with the 'pre-entanglement' world. If we do this all the action is in the process of entanglement- but this gets into literally the exact same dicey area as the idea of a measurement. You haven't solved the problem, you've simply reformulated it.

But which experiment would rule out MWI or Copenhagen? Or would they just rule out other ideas?

Unfortunately, I don't think Deutsch's paper is online anywhere. The idea is something like a delayed choice quantum eraser (which must already constrain somewhat what we mean by measurement), using a quantum computer to make and erase measurements.

As far as I (and the wikipedia articles) know, it has some problems with special relativity, and may need special initial conditions (to get the Born probabilities!). And it adds particles to the theory, in addition to the wave functions.

It doesn't actually have problems with special relativity. You DO need a preferred frame, but its not experimentally observable, so just like Lorentz ether theory, it works just fine. There are even extensions to field theories. (http://arxiv.org/abs/quant-ph/0303156)

It does add particles to the theory- but thats a GOOD thing. Now you have what you wanted all along- a physically realistic definition of measurement (when particles hit the detector).
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### Re: What is Many Worlds? What isn't Many Worlds?

SU3SU2U1 wrote:What I'm asking for here is an actual mathematical definition of what you mean by probability that 'you' end up in a given world. I'm arguing you can't do it without an extension to your theory that requires some sort of continuum of minds.
Shouldn't there be a continuum of everything from quarks up in MWI? And so macroscopic objects (like people) being made of a lot of quarks, naturally take on a continuous form as well? If quarks could 'think' then they shouldn't experience anything fundamentally different from we do in such a continuum - it's only that we aggregate information over time in our macroscopic domains.

One huge advantage of the MWI, is that it has at least an exponentially larger number of distinct observers, and so if we consider the space of all possible universes, it contains far more observers than simple linear time lines. So if we can find that the majority of observers tend to agree with us as to the general principles of the universe in MWI, the anthropic principle turns this into a hugely persuasive argument that we have a good description of our reality.
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WarDaft

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### Re: What is Many Worlds? What isn't Many Worlds?

WarDaft wrote:Shouldn't there be a continuum of everything from quarks up in MWI?

I mean a continuum of minds that is separate from the wavefunction itself, formally. Thats why I kept mentioning the worrying mind-body dualism of many minds.

So if we can find that the majority of observers tend to agree with us as to the general principles of the universe in MWI, the anthropic principle turns this into a hugely persuasive argument that we have a good description of our reality.

But, as is shown in the papers I linked to, the overwhelming majority of observers DO NOT agree with us. They don't get anything like Born probabilities. Thats why you have to play these games with some branches being 'more real' than other branches- you have to make the observers who agree with us count more, or else the argument you are trying to make kills many worlds.
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### Re: What is Many Worlds? What isn't Many Worlds?

lightvector wrote:Just for fun... (and let's ignore relativity and pretend that we can talk about a universal notion of time):
Spoiler:
There are exactly 2^^16 (two tetrated to the sixteen) universes. At every instant in time, these universes are distributed with density given by the square magnitude of the wave function. As time goes forward, this continues to remain true because after every infinitesimal increment of time (perhaps every planck-time), each one of these universes automatically re-samples itself according to the new wave function at that time. That is, each universe independently draws a new sample at random from the distribution established the wave function, and then the contents of that universe are instantaneously replaced with that of the new sample drawn. Of course, observers do not subjectively experience this very rapid repeated replacement. Indeed, all the copies of you reading this message in universes that are essentially the same as the one you're in right now, all certainly remember existing for much longer, because when you all winked into existence during the last resampling less than a planck-time ago, all of you did so with complete with all of the memories in your brains.

And of course, almost all of the observers in these universes should observe (or rather, remember observing) a history consistent with the Born probabilities, during the all brief times that they exist.

(Spoiled out for length, but I do like that example )

You've pointed out one of the problems it seems. As much as MWI makes it seem right in mathematically, it does not seem right in reality (testable). The "last plankdaism" you suggested is similar to "last Thursdayism". The maths might suggest that it is a possibility to view the universe as coming into existence last Thursday but would you go with that theory? MWI seems to be pushing the jump from last Thursday to next Thursday. It's the same implication. The impossible probability.
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### Re: What is Many Worlds? What isn't Many Worlds?

But, as is shown in the papers I linked to, the overwhelming majority of observers DO NOT agree with us. They don't get anything like Born probabilities. Thats why you have to play these games with some branches being 'more real' than other branches- you have to make the observers who agree with us count more, or else the argument you are trying to make kills many worlds.
Oh, I know, I'm just saying, it's a reason to prefer MWI if we could.
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### Re: What is Many Worlds? What isn't Many Worlds?

mfb wrote:Why not? Assume that the world is a single, very complex wave function (read: "many worlds"). Every part of it which is something like a human brain will make observations according to its part in the universe. There are humans which have a history far away from Born probabilities. Turns out that the xkcd board which we use is in a universe which is not so far away from them.
If you think of "you" as a human in a certain point in time, the probability that "you" are in some part of the wave function is its amplitude squared. But what does that mean, '"you" as a human in a certain point in time'? You want a framework to answer this question. Give me a framework to formulate this question!

The short answer is that the Schrodinger equation by itself does not rule out hidden variable theories. In other words, you cannot derive Bell's theorem from just the Schrodinger equation. You absolutely need something weird (collapse, etc.) to give you the observed quantum behavior. It might be useful to examine some naive hidden variable theories and see how they also produce Schrodinger-like equations that do not give quantum behavior.

First off, we can take a view similar to nonequilibrium statistical mechanics and say "Well, particles have well defined positions and momenta, but there is fundamental randomness to their motion." If we do this (for a free particle, for example), we wind up with an equation that describes the probability distribution:
\frac{\partial p(\vec{r},t)}{\partial t}=D\nabla^2p(\vec{r},t)

Notice how similar this looks to the free particle Schrodinger equation:
\frac{\partial \psi(\vec{r},t)}{\partial t}=\frac{i\hbar}{2m}\nabla^2\psi(\vec{r},t)

We can do the same thing for motion in potentials or with interactions. This similarity rests on the fact that the Schrodinger equation is formally equivalent to a Fokker-Planck equation, which describes the probability distributions of objects experiencing random forces. In the Fokker-Planck case, the position and momentum of a particle are well defined at all times, but in the Quantum Mechanics case, they are not, yet both cases have the same Schrodinger style time evolution equation.

Secondly, we can do a bit of silly trickery with math*. Imagine we have a classical dynamical system
\frac{d \vec{q}(t)}{dt}=f(\vec{q})

Now we can do something silly and define a familiar operator \vec{p}=-i\frac{\partial}{\partial\vec{q}} and use this to define another operator
H=\frac{1}{2}\left(\vec{p}\cdot f(\vec{q})+f(\vec{q})\cdot\vec{p}\right)

This operator has some nice properties: H=H^\dagger and \dot g(\vec{q})=i\left[H,g(\vec{q})\right]. This is nice because now we can introduce a totally made up wave function \psi(\vec{q},t)=\left<\vec{q}|\psi(t)\right> and use these nice properties to arrive at an evolution equation for our made up wave function:
\frac{d}{dt}\left. |\psi(t)\right>=iH\left. |\psi(t)\right>

This is exactly the Schrodinger equation. Its even got the nice property that since H=H^\dagger the quantity \left<\psi(t)|\psi(t)\right> is conserved, and can be used to define a probability identical to the Born probabilities if we want.

Notice what we did here: we started with an entirely classical system and derived the fact that a wave function defined for it must obey the Schrodinger equation, even though we still have purely classical deterministic underlying system. This should really drive home the point that the cool stuff in Quantum mechanics doesn't come from the Schrodinger equation, it comes from the other features of the theory (like collapse in the Copenhagen interpretation). There is nothing inherently quantum about the Schrodinger equation.

*An important disclaimer here: people (t'Hooft in particular) like to try to use this same trickery to argue that hidden variable theories are not ruled out by current experimental results on Bell's inequality violations using the Superdeterminism loophole. This is not the mainstream view and I do not agree with that viewpoint. It is certainly tangent to the discussion in this thread, so lets try not to take the conversation that way.
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### Re: What is Many Worlds? What isn't Many Worlds?

This similarity rests on the fact that the Schrodinger equation is formally equivalent to a Fokker-Planck equation, which describes the probability distributions of objects experiencing random forces

I can't help but point out that Bohm would say this isn't merely a formal equivalence, its deeply physically significant- his interpretation suggests that in some sense the Schroedinger equation IS an equation for particles being buffeted by random forces (where random in this sense means related to the pilot wave).
SU3SU2U1

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### Re: What is Many Worlds? What isn't Many Worlds?

SU3SU2U1 wrote:
Actually, I just finished reading Everett's paper you linked and his argument for the Born probabilities seems more general than he lets it be. He defines probability only as the limit of a large number of trials, which is not all that it is (see: Bayesian probability).

Please continue reading the literature- which has nice discussions on why Everett isn't the end of the story.

The overall point is well taken - the proof in Everett's paper takes an unjustified metaphysical step to say that we should be assigning these things probabilities, and that the probabilities should depend on the amplitudes is also not well-supported in the paper. The unanswered question is still "where does a probability come from in this system?"

The author, however, might not know the origin of a probability if it came up and bit him in the face. For example section 4 A. When agents in a classical multiverse flip a bunch of coins and follow some probability-updating procedure, the author faults the ones who flipped all heads for coming to the wrong conclusion! If you flipped a coin but didn't look at it, that sort of interpretation of hypothesis testing would fault you for assigning a probability of 0.5 to heads, since the coin is already heads or tails with probability 1, and therefore a probability of 0.5 will be "wrong." Using probability is not about being magically right all the time, it's about doing the best that you can given your information.

Quantum probability as Bayesian is also really tempting no matter what your interpretation (after all, after a measurement you know for sure what value you got, so 'collapse' is completely logical in this context). Chris Fuchs has done interesting work on this http://perimeterinstitute.ca/personal/cfuchs/ see the second power point under talks,and his lecture notes.

Well. That was handwavey. Interesting, but suuuuuper handwavey.
Last edited by Charlie! on Mon Mar 26, 2012 4:20 am UTC, edited 1 time in total.
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### Re: What is Many Worlds? What isn't Many Worlds?

Is there a variant of MWI that says that the branches of the multiverse where the Born Rule fails simply don't "materialize", or is that just a bit too ad hoc?

I used to think the MWI was cool, until I heard about the Transactional interpretation. I'd love to know what people here think about the TIQM, especially SU3SU2U1 & JWalker.

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### Re: What is Many Worlds? What isn't Many Worlds?

SU3SU2U1 wrote:But, as is shown in the papers I linked to, the overwhelming majority of observers DO NOT agree with us. They don't get anything like Born probabilities.

Which would not be impossible after all. For all the possible parts of the universe, there would be minds thinking about the universe. One of them is you, living in a region with approximately Born probabilities.
Or at least in a region where your mind is in a state that it thinks this. Hooray Boltzmann brains .
Well, this is a bad direction. But note that it is similar to the special initial conditions required for the de-Broglie-Bohm mechanics.

Special reference frame: Ok, if that is not observable it looks like a simple mathematical trick.

@JWalker: Ok, imagine "Schroedinger equation plus wave functions for particles" everywhere where this is not explicitly said.
As far as I see, your classical things won't have stuff like interference (?).

>> Is there a variant of MWI that says that the branches of the multiverse where the Born Rule fails simply don't "materialize", or is that just a bit too ad hoc?
I saw some idea, but as far as I can tell it is far away from any calculations.
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### Re: What is Many Worlds? What isn't Many Worlds?

Of all the things you could tack on to quantum mechanics - nonlinear terms in Schrodinger, unobservable special frames, objective collapse, what have you - I can think of none so loathsome as a privileged role for "minds." It gives me the heebly jeeblies just to think about it.
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### Re: What is Many Worlds? What isn't Many Worlds?

PM 2Ring wrote:Is there a variant of MWI that says that the branches of the multiverse where the Born Rule fails simply don't "materialize", or is that just a bit too ad hoc?

The Born rule failing is not an inherent property of a state - even if you flip a coin 100 times and they're all heads, it could still be a fair coin. In fact, in a multiverse interpretation of coin flipping, the existence of the all-heads world is necessary to be able to say that the coin was fair.

The problem is that if you measure the state 3/5 |0> + 4/5 |1>, measuring 0 and 1 are both "an outcome," and therefore a super-naive symmetry argument would say you should assign them equal probabilities. And the question is, is there a way to be less naive that doesn't use any information not in the Schrodinger equation and maybe your initial conditions?
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### Re: What is Many Worlds? What isn't Many Worlds?

Just so we are all discussing the same many worlds argument, here is a summary of the problem-

Imagine you prepare an infinite number of electrons, in a state

\left|\psi \right> = c_+\left|+\right> + c_-\left|-\right>

Now, Everett and DeWitt told us- look the probability to get a sequence of measurements +,-,+... (represented as \left|+-++...\right> is going to be

\left<+-+...\right|\left.\psi\right>^2

And this quantity will be 0 for any infinite sequence of measurements that falls outside the Born probability. Look! We get born probabilities because those are the only amplitudes that exist!

Unfortunately, thats not true for any FINITE number of measurements, where no amplitude has gone to 0. No matter how skewed c+ and c- are, every measurement will result in a splitting into TWO worlds. That means that if we count observers MOST observers will have seen an equal number of spin up and spin downs. Very, very few observers will have observed born probabilities.

Charlie! wrote:The author, however, might not know the origin of a probability if it came up and bit him in the face. For example section A. When agents in a classical multiverse flip a bunch of coins and follow some probability-updating procedure, the author faults the ones who flipped all heads for coming to the wrong conclusion!

Which section A? Also, keep in mind he is arguing mostly against arguments of the Deutsch/Wallace type- these argue about probabilities by talking about a rational observers game-theoretic expectations. I'm reasonably certain we can both agree that a rational Bayesian will conclude after flipping hundreds of heads, that a coin is biased. He certainly won't put much money on tails.
Last edited by SU3SU2U1 on Mon Mar 26, 2012 3:54 am UTC, edited 1 time in total.
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### Re: What is Many Worlds? What isn't Many Worlds?

mfb wrote: But note that it(only some branches have born) is similar to the special initial conditions required for the de-Broglie-Bohm mechanics.

No, its the inverse in Bohm. The majority of initial conditions have normal born probabilities, but a small subset do not. Thats why its not considered a major problem.

Is there a variant of MWI that says that the branches of the multiverse where the Born Rule fails simply don't "materialize", or is that just a bit too ad hoc?
I saw some idea, but as far as I can tell it is far away from any calculations.

Yes, there sort of are, but they require extra stuff. See, for instance http://arxiv.org/abs/hep-th/0606062 which uses discretization to kill the 'rogue' worlds.

There is also the variant I linked to above, where the non-Schroedinger evolution makes the distribution of universes follow a pattern that gets your Born by simple counting.
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### Re: What is Many Worlds? What isn't Many Worlds?

PM 2Ring wrote:Is there a variant of MWI that says that the branches of the multiverse where the Born Rule fails simply don't "materialize", or is that just a bit too ad hoc?

mfb wrote:I saw some idea, but as far as I can tell it is far away from any calculations.

Thanks, mfb. That looks interesting, but I agree that it does look a bit hand-wavey. (I've only read the page itself, not the articles it links to).

Charlie! wrote:The Born rule failing is not an inherent property of a state - even if you flip a coin 100 times and they're all heads, it could still be a fair coin. In fact, in a multiverse interpretation of coin flipping, the existence of the all-heads world is necessary to be able to say that the coin was fair.

The problem is that if you measure the state 3/5 |0> + 4/5 |1>, measuring 0 and 1 are both "an outcome," and therefore a super-naive symmetry argument would say you should assign them equal probabilities. And the question is, is there a way to be less naive that doesn't use any information not in the Schrodinger equation and maybe your initial conditions?

Yeah. Failing the Born rule isn't a property of any branch considered by itself, it's a property of the set of all branches from a given point, so any kind of branch pruning or mangling process can't just operate locally within a (potential) branch, it has to operate globally on all the branches at each branch point. To quote from mfb's mangled worlds link:

The big problem with the many worlds view is that no one has really shown how the usual linear rule in disguise can reproduce Born probability rule evolution. Many worlders who try to derive the Born rule from symmetry assumptions often forget that there is no room for "choosing" a probability rule to go with the many worlds view; if all evolution is the usual linear deterministic rule in disguise, then aside from unknown initial or boundary conditions, all experimentally verifiable probabilities must be calculable from within the theory. So what do theory calculations say? After a world splits a finite number of times into a large but finite number of branch worlds, the vast majority of those worlds will not have seen frequencies of outcomes near that given by the Born rule, but will instead have seen frequencies near an equal probability rule. If the probability of an outcome is the fraction of worlds that see an outcome, then the many worlds view seems to predict equal probabilities, not Born probabilities.

(Some philosophers say world counts are meaningless because exact world counts can depend sensitively on one's model and representation. But entropy, which is a state count, is similarly sensitive to the same sort of choices. The equal frequency prediction is robust to world count details, just as thermodynamic predictions are robust to entropy details.) That is, if the many worlds view is true, then you and I are right now together in some particular world. Because of previous measurement-like (i.e., "decoherence") processes, there are a googol or googolplex or more other worlds out there. Since others in our world have in the past done statistical tests of the Born rule, these many other worlds are in part distinguished by the results of those statistical tests. In some worlds, including our world, the tests were passed, while in other worlds the tests were failed. (And in far more worlds, the tests were never tried.)

[...]

The mangled worlds approach to quantum mechanics is a variation on many worlds that tries to resolve the Born rule problem by resorting only to familiar probability concepts, standard linear physical processes, and a finite number of worlds. The basic idea is that while we have identified physical "decoherence" processes that seem to describe measurements, since they produce decoupled wave components corresponding to different measurement outcomes, these components are in fact not exactly decoupled. And while the deviations from exact decoherence might be very small, the relative size of worlds can be even smaller.

As a result, inexact decoherence can allow large worlds to drive the evolution of very small worlds, "mangling" those worlds. Observers in mangled worlds may fail to exist, or may remember events from larger worlds. In either case, the only outcome frequencies that would be observed would be those from unmangled worlds. Thus worlds that fall below a certain size cutoff would become mangled, and so should not count when calculating probabilities as the fraction of worlds that see an outcome.

This mangling process allows us to ignore the smaller worlds, but this by itself is not enough to produce the Born probability rule. To get that we also need the cutoff in size between mangled and unmangled worlds to be in the right place. Specifically, we need the cutoff to be much nearer to the median measure world size than to the median world size. The median measure is the world size where half of all measure is held by worlds larger that this size, and half is held by worlds smaller than this size. The median world size is the size where half of all worlds are larger, and half are smaller.

EDIT:

SU3SU2U1 wrote:Yes, there sort of are, but they require extra stuff. See, for instance http://arxiv.org/abs/hep-th/0606062 which uses discretization to kill the 'rogue' worlds.

There is also the variant I linked to above, where the non-Schroedinger evolution makes the distribution of universes follow a pattern that gets your Born by simple counting.

I like the simple counting argument, although the fact that it require non-Schroedinger evolution is unfortunate.

I can't read the text of the discretization article, but I guess it somehow uses measure theory to make the rogue worlds dissolve into piles of Cantor dust. The mental image I have is that rogue worlds are like Julia sets outside the Mandelbrot set and Born-conforming worlds are like Julia sets inside the Mandelbrot set.
Last edited by PM 2Ring on Mon Mar 26, 2012 4:25 am UTC, edited 1 time in total.

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