Many-Worlds, Deriving the Born Rule?

[stevendaryl]

I always thought of Bayesian "level of confidence" as a probability. It's a subjective notion of probability, but it's a probability--it obeys the usual rules of probability.

Are you making the distinction that a "probability" refers to a nondeterministic event that has yet to happen, while a "level of confidence" can refer to any uncertain fact (whether about the future, the past, the laws of physics, etc.)?

 

[bhobba]

And if we consider the inside perspective of the frog why isn't his incomplete knowledge -namely about which world he is in- sufficient to justify the use of probabilities?

What MW's does is not specify which world you experience. Its not part of the theory. All you can do is assign a rational level of confidence to which world will be experienced. With that level of confidence you can do Bayesian hypothesis testing. Probabilities become a derived concept in that view.

This has all been discussed in the longer thread. MFB did many posts there explaining it eg:
https://www.physicsforums.com/showthread.php?p=4487215#post4487215

At first I didn't get it, but then I did and kicked myself for not seeing it earlier.

Thanks
Bill

 

[tom.stoer]

It concerns introducing a measure - how that is interpreted it's silent about.

OK, I agree. Gleason's theorem says nothing about the interpretation. That's why any interpretation, probability, confidence level etc. has to be motivated by something else - not by Gleason's theorem. yes, I think we agree on that.

I am specifically claiming it's NOT interpreted as a probability - but as a Bayesian level of confidence of which world you are in.

I still can agree.

But then please explain WHY we should introduce a concept of Bayesian hypothesis testing in a fully deterministic theory?

I agree that - as soon as you are convinced to introduce a Bayesian level of confidence - Gleason's theorem tells you how to do that. But looking at the bare formalism (= Hilbert space plus unitary time evolution; w/o the Born rule, w/o any collapse interpretation, w/o knowledge about the history of QM!), why should I introduce a Bayesian level of confidence. Why should I do that?

w/o any further explanation it seems that the MWI has to introduce or derive the Born rule simply b/c we know that it works in the collapse interpretation. But the MWI should be able to explain this w/o referring to anything else. So I am asking for this specific reason.

 

[bhobba]

I always thought of Bayesian "level of confidence" as a probability. It's a subjective notion of probability, but it's a probability--it obeys the usual rules of probability.

You can view it that way. But its a different view reflecting simply a lack of information.

Thanks
Bill

 

[stevendaryl]

But then please explain WHY we should introduce a concept of Bayesian hypothesis testing in a fully deterministic theory?

If it is sensible for us to introduce a notion of probability to describe the apparent nondeterminism in our historical record of past events, then how could it cease to be a sensible thing to do once there are alternative worlds where things turned out differently? The subjective notion of probability isn't affected by the existence of multiple worlds.

The evolution of the whole many-worlds may be deterministic, but the perspective of an observer in a single branch is certainly not deterministic. Past history in that branch doesn't determine the future.

 

[bhobba]

But then please explain WHY we should introduce a concept of Bayesian hypothesis testing in a fully deterministic theory?

Because that's all you can do. The theory doesn't provide any way of determining what world will be experienced. All you can do as a rational agent is figure out a rational level of confidence.

Its like throwing a coin. We have not enough information that will allow us to determine what side will come up. We can say its inherently probabilistic or you can say its deterministic and we simply lack the information. MWI adherents prefer the second alternative.

Here is another way of looking at it. People often say in MW why are some outcomes more likely than others - we have a feeling all should be equally likely ie we have the same level of confidence in any world. But that in fact is naive - Gleason's Theorem shows if you do that you are singling out one basis over another because only the Born rule is basis independent.

Its an ASSUMPTION these guys make - that we make a rational 'guess' since that's all we can do. If you don't think that resolves the issue how probabilities are introduced - fine - but that's the argument they use.

Thanks
Bill

 

[bhobba]

If it is sensible for us to introduce a notion of probability to describe the apparent nondeterminism in our historical record of past events, then how could it cease to be a sensible thing to do once there are alternative worlds where things turned out differently?

Ahhhh. Now we are getting to if their whole view is sensible. Is it sensible to have a theory that doesn't detail this key element ie what world will we experience; and we have to resort to this subtle 'hocus pocus' to get sensible predictions.

I think it can be guessed what my view is since I do not hold to the interpretation. But that doesn't detract from my admiration for its extreme elegance and beauty. That's why I went to the trouble of getting Wallace's book and reading it. It proved interesting for me in that I am pretty sure Wallace's argument is simply Gleason's in another guise.

Thanks
Bill

 

[stevendaryl]

Ahhhh. Now we are getting to if their whole view is sensible. Is it sensible to have a theory that doesn't detail this key element ie what world will we experience; and we have to resort to this subtle 'hocus pocus' to get sensible predictions.

I think that almost any probabilistic theory is in the same boat. It doesn't actually say ANYTHING about what we observe, it just says that the probability of getting relative frequencies that differ substantially from the computed probabilities goes to zero. But what is the meaning of that notion of probability (on possible histories)? It can't itself be a relative-frequency notion of probability. So what is it?

Whether we have many-worlds or not, to get testable predictions from our theories, we have to assume that our history is going to be typical (where "typical" is defined via a measure on possible histories).

 

[bhobba]

But what is the meaning of that notion of probability (on possible histories)? It can't itself be a relative-frequency notion of probability. So what is it?

I personally give it no meaning - simply saying we have assigned this thing called probability that obeys the Kolmogorov axioms to the outcomes.

You can of course do that in MWI if you like and argue along those lines, but that doesn't sit well with the deterministic nature of the theory that you introduce this notion of probability in an abstract but objective sense.

The view that seems to sit best with MWI is lack of information as to which world you are in so all you, as a rational agent, can do is figure out the likelihood, confidence, or whatever words you want to use, and use decision theory which, when you get right down to it, it's really probability in another guise. But its different foundation allows a different view.

In that other long thread I did a few posts illustrating the fundamental difference between the views eg:
http://en.wikipedia.org/wiki/Bayesian_statistics

'In general, Bayesian methods are characterized by the following concepts and procedures:

The use of random variables to model all sources of uncertainty in statistical models. This includes not just sources of true randomness, but also uncertainty resulting from lack of information.

The sequential use of the Bayes' formula: when more data become available after calculating a posterior distribution, the posterior becomes the next prior.

For the frequentist a hypothesis is a proposition (which must be either true or false), so that the frequentist probability of a hypothesis is either one or zero. In Bayesian statistics, a probability can be assigned to a hypothesis that can differ from 0 or 1 if the true value is uncertain.'

The key point to realize is Bayesian probabilities are slightly different to the usual concept of probabilities you may be more familiar with - its a degree of confidence. It is also important to realize this confidence may be the result of true randomness, but also can be the result of lack of information. In the MWI the idea is the different outcomes of observations is not true randomness, but lack of information, namely which world you are in. Its logically equivalent to the frequentest interpretation (which in this case would correspond to the Ensemble interpretation of QM) but there is no requirement to introduce these concepts.

Although MWI is a deterministic theory it gives a set of data, and as such, if one can introduce some kind of utility function expressing a confidence in what data you will get, then one can do hypothesis testing. It's not necessarily saying its not a deterministic theory just like when one does hypothesis testing on any results one obtains in experiments. Rather what one interprets the different outcomes not as true randomness, but as lack of information, namely exactly what world am I in.

Thanks
Bill

 

[tom.stoer]

Thanks for that summary; the picture becomes much clearer now.

 

[stevendaryl]

For the frequentist a hypothesis is a proposition (which must be either true or false), so that the frequentist probability of a hypothesis is either one or zero.

So in a deterministic universe, there can be no role for frequentist probability? That seems weird to me.

Whether or not a coin flip is truly random, we can still approximate it by a random process and probability theory gives us good predictive power. We can still compute relative frequencies and compare them to the ideal coin case of 50/50.

It also seems strange, because that notion of probability would only make sense if there were truly random (nondeterministic) events. But before quantum mechanics, most people believed that there were no such events. But frequentist probability was developed before quantum mechanics.

 

[bhobba]

So in a deterministic universe, there can be no role for frequentist probability? That seems weird to me.

Its logically equivalent to the frequentest interpretation (which in this case would correspond to the Ensemble interpretation of QM) but there is no requirement to introduce these concepts.

Obviously there is - see above from the post where I copied from that long thread.

Thanks
Bill

 

[stevendaryl]

Obviously there is - see above from the post where I copied from that long thread.

The ensemble interpretation is frequentist? Well, it seems to me that probability in an ensemble interpretation is subjective--you're actually in one member of the ensemble, but you don't know which.

 

[bhobba]

The ensemble interpretation is frequentist? Well, it seems to me that probability in an ensemble interpretation is subjective--you're actually in one member of the ensemble, but you don't know which.

In my ignorance ensemble interpretation - yes - of course that member comes into being after decoherence and the 64 million dollar question there how does that happen? Its not quite amenable to the same very elegant resolution as MWI where it's interpreted as lack of knowledge about what world you experience - the ignorance interpretation doesn't explain why you experience an outcome at all - that's trivial in MW.

However Ballentine doesn't agree with decoherence having any role in interpretations so it doesn't apply to his view - only the variant I hold to with decoherence - which he specifically rejects. Interesting isn't it?

Murray Gell-Mann has commented his Decoherent Histories is also very similar. It seems all interpretations that use decoherence are similar in a sense.

Thanks
Bill

 

[kurros]

It also seems strange, because that notion of probability would only make sense if there were truly random (nondeterministic) events. But before quantum mechanics, most people believed that there were no such events. But frequentist probability was developed before quantum mechanics.

I don't this is really the case. Early frequentist philosophers tried to make sense of the approach with concepts like "propensity", i.e. the idea that certain events had a "tendency" to happen with certain kinds of frequencies, which to me sounds like they thought some things were "really" random. Sure, Newton's laws were ruling the town, but I don't think that means everyone thought the universe was actually totally deterministic. Actually I have seen some arguments that event Newton's laws are not strictly deterministic since there are some systems whose futures you cannot uniquely predict (I think due to cusps or other catastrophes or some such in the configuration space).

In any case I have never heard a definition of frequentist probability that didn't sound totally circular, so even with quantum mechanics potentially providing "true" propensities I have a kind of hard time appreciating what that really means.

 

[bhobba]

Actually I have seen some arguments that event Newton's laws are not strictly deterministic

They are strictly deterministic. Its the butterfly effect - small inaccuracies in knowledge of initial conditions grow until they dominate and you can't predict anything after a sufficient length of time.

Since its impossible to know initial conditions with 100% accuracy (real numbers require knowledge to an infinite number of decimal places so in principle it not possible) in practice Newtonian physics is also probabilistic. But its important to understand the two different causes - QM is probabilistic at its foundations - Newtonian mechanics simply because we can never know enough for the fact it is deterministic to be of any value.

Thanks
Bill

 

[kurros]

They are strictly deterministic. Its the butterfly effect - small inaccuracies in knowledge of initial conditions grow until they dominate and you can't predict anything after a sufficient length of time.

Sure, that's the standard case. I was talking about weirder exotic things people argue about, e.g. http://www.jstor.org/stable/10.1086/594524

It is not clear that this has much to do with probability though.

 

[bhobba]

Sure, that's the standard case. I was talking about weirder exotic things people argue about

Yea - written by a guy in the Philosophy and History Of Science department - philosophy and me don't get along too well.

I really don't see much value in such 'exotic' things.

Each to their own I suppose.

Thanks
Bill

 

[Quantumental]

Bhobba: just got to ask: are you a proponent of MWI?
It seems you think it has no drawbacks whatsoever so I don't understand why you even contemplate other interpretations anymore?

 

[bhobba]

Bhobba: just got to ask: are you a proponent of MWI?
It seems you think it has no drawbacks whatsoever so I don't understand why you even contemplate other interpretations anymore?

Ahhhh. Now we are getting to if their whole view is sensible. Is it sensible to have a theory that doesn't detail this key element ie what world will we experience; and we have to resort to this subtle 'hocus pocus' to get sensible predictions.

I think it can be guessed what my view is since I do not hold to the interpretation. But that doesn't detract from my admiration for its extreme elegance and beauty. That's why I went to the trouble of getting Wallace's book and reading it. It proved interesting for me in that I am pretty sure Wallace's argument is simply Gleason's in another guise.

I thought the above was fairly clear.

I don't like this continually exponentially increasing number of new worlds - its far too extravagant for my tastes. But that doesn't detract from its great elegance.

Thanks
Bill

 

[kurros]

I don't like this continually exponentially increasing number of new worlds - its far too extravagant for my tastes.

I agree intuitively with this, but do you really think it is the case? I mean sure, pieces of the wavefunction become distinct at some probably even combinatoric rate, but you don't 'really' create new worlds right? They are all there to begin with, it is just that they start off all the same, and entangled with each other.

 

[bhobba]

I agree intuitively with this, but do you really think it is the case? I mean sure, pieces of the wavefunction become distinct at some probably even combinatoric rate, but you don't 'really' create new worlds right? They are all there to begin with, it is just that they start off all the same, and entangled with each other.

That's all true (they are not entangled but they are there) - but with each 'observation' you get new copies of you and those copies increase at an alarming exponential rate. Its just a bit too weird for me. Its a personal view only.

Here is a link that goes a bit deeper into the issue of these 'new' worlds and exactly what it means as far as the energy needed to create them etc goes - issues not related to this thread regarding the Born rule:
http://www.askamathematician.com/20...y-and-matter-for-the-new-universes-come-from/

Although the energy of each world is 'diluted' in proportion it reduces at an alarming rate - to me that is also bit hard to swallow.

Thanks
Bill

 

[Quantumental]

That's all true (they are not entangled but they are there) - but with each 'observation' you get new copies of you and those copies increase at an alarming exponential rate. Its just a bit too weird for me. Its a personal view only.

Here is a link that goes a bit deeper into the issue of these 'new' worlds and exactly what it means as far as the energy needed to create them etc goes - issues not related to this thread regarding the Born rule:
http://www.askamathematician.com/20...y-and-matter-for-the-new-universes-come-from/

Although the energy of each world is 'diluted' in proportion it reduces at an alarming rate - to me that is also bit hard to swallow.

Thanks
Bill

So what you're saying is that basically you find no problem with MWI in terms of Born Rule, preferred basis or non-locality like most physicists and philosophers, but because it *feels* weird you don't believe it?

That's just denial though if you have no other reasons for rejecting

 

[tom.stoer]

Let's come back to the energy question:

Suppose I have an electron in a double-split experiment. Due to the plane wave and spherical wave ansatz we nearly have energy eigenstates. When measuring the position we measure its momentum and therefore energy, too. For the incoming electron I have energy

$E = \text{tr} H \rho$

For the detected electron I will find nearly the same amount of energy. This energy E is what I prepare in the beam and what I measure in the detector.

Now let's compare collapse and MW interpretation. In the collapse interpretation I find the electron at position x. All the energy E is contained in one "new world" to which the electron state collapsed. In the MW interpretation we have a set of branches, each containing one electron at a certain position. The energy measured in all these branches is nearly identical to the original one. So what an observer in one single branch measures is still

$E = \text{tr} H \rho$

He does not find some thing like

$E^\ast = \text{tr} H P \rho$

where P projects to his branch.

So my conclusion is that energy is conserved over all branches. Each single branch would contain only a small portion. The observer "in one branch" finds an electron position for "his branch", but an amount of energy like "over all branches". So even so other branches become invisible to the observer when it comes to the location of the electron, these other branches are visible when it comes to measurement of energy in one single branch.

This conclusion seems to be weird.

What's wrong with my conclusion?

 

[bhobba]

That's just denial though if you have no other reasons for rejecting

If so I am not the only one (see my next post - I think many simply believe its an unnecessarily confusing complication that can be avoided).

BTW I am not sure most physicists have the issues with it you think they do. For example it was MFB, who is a particle physicist, that changed my view on it as far as Born's rule goes.

Philosophers I can't comment on, but since they generally don't as a group agree on anything I am not sure it carries the same type of weight.

Thanks
Bill

 

[bhobba]

This conclusion seems to be weird. What's wrong with my conclusion?

As far as I can see there is nothing wrong with it. MW is beautiful, elegant, logically coherent, but weird.

We all choose the interpretation we prefer based on our personal criteria and I think many people find MW just a little 'too much'

I know Murray Gell-Mann holds that view, and I am pretty sure he is not alone:
http://olena.tumblr.com/post/25070169127/on-hugh-everetts-many-worlds-theory
'his interpretation is often described in terms of ‘many worlds,’ whereas we believe that 'many alternative histories of the universe' is what is really meant. … the many worlds are described as being 'all equally real,' whereas we believe it is less confusing to speak of ‘many histories, all treated alike by the theory except for their different probabilities.’ To use [this recommended] language is to address the familiar notion that a given system can have different possible histories, each with its own probability; it is not necessary to become queasy trying to conceive of many ‘parallel universes,’ all equally real.”

Its not denial - its just not as confusing. But of course what is confusing to one, isn't necessarily that to another.

Thanks
Bill

 

[Quantumental]

If so I am not the only one (see my next post - I think many simply believe its an unnecessarily confusing complication that can be avoided).

BTW I am not sure most physicists have the issues with it you think they do. For example it was MFB, who is a particle physicist, that changed my view on it as far as Born's rule goes.

Philosophers I can't comment on, but since they generally don't as a group agree on anything I am not sure it carries the same type of weight.

Thanks
Bill

I understand that you find it weird, but is that enough reason to say it's wrong in your opinion? Don't you also find it weird that you are on a rock in the middle of space and that the world is a sphere, but you don't fall off ? I reject it due to the issues I mentioned and because I am sure there is a deeper theory that will involve the holographic principle and things like the amplituhedron. I.E. our current theories are approximations of a deeper underlying theory, which will obviously also be weird, but more classical than QM.
If space time is no longer fundamental then non-locality isn't that hard to swallow anymore and then suddenly determinism is back into play etc. But obviously we are far from close to creating a theory out of these speculations.

As for Gell-Mann, his statements are diffuse, some places he definitely reject the MWI view in some statements. But i don't know him so I won't put words in his mouth.
One person who used to lean toward MWI and now reject it due to Born Rule si Steven Weinberg.As for the open problems, even David Wallace flat out admits that wf onticism in hilbert space simply isn't enough. In his Space Time State realism paper he goes through the problems with that view and propose his own view.
David Deutsch claim that MWI is 100% local in the Einsteinian sense and refers to the Heisenberg picture as justification. David Wallace and Simon Saunders on the other hand accept some non-locality (non-narratability)

Then there is the issue of divergent worlds vs branching worlds.
Some Alastair Wilson and Simon Saunders argue in favour of the worlds being divergent. So you (Bhobba) are forever stuck on this branch and there are infinite other worlds where all other outcomes occur, but you are never in those worlds. I.E. all worlds sprung into existence at once.
Then you have people like Deutsch, Papineau, Greaves etc. who claim the worlds do infact branch.
Wallace holds the view that there are no facts about it and it's all semantics, but that doesn't make sense. Reality has to be this or that way.

Also take for instance the proponents of Many Minds, they claim there is problems with not specifying an observer ( Zeh being one )

Then there are also those who propose Many Bohmian Worlds, physicists like Sebens, Boström, Wiseman all came up with this independently of each other as a response to the basis issue.So there is clear disagreement among the top developers of this modern Many Worlds view and no one seem to be sure of anything yet.
The basis problem and Born Rule is still one of the most debated subjects.

 

[bhobba]

I understand that you find it weird, but is that enough reason to say it's wrong in your opinion?

You misunderstand. No interpretation is wrong since none can be proven or disproven by experiment. Until that is the case all are equally valid. The choice of one over the other is purely based on its appeal to you.

because I am sure there is a deeper theory that will involve the holographic principle and things like the amplituhedron

The above being a prime example.

Actually I believe the amplituhedron is trying to tell us something important - but exactly what is unclear - only time will tell. Opinions are like bums - everyone has one - it doesn't make it correct - even mine - no especially mine.

Thanks
Bill

 

[kith]

So what an observer in one single branch measures is still

$E = \text{tr} H \rho$

He does not find some thing like

$E^\ast = \text{tr} H P \rho$

where P projects to his branch.

I can't follow.

If ρ is an energy eigenstate, there's no branching in an energy measurement and therefore no difference between E and E*.

If ρ is a superposition of energy eigenstates, the measurement outcome will be one of the corresponding eigenvalues. But in gneral, neither E nor E* is equal to such an eigenvalue.

 

[kith]

I reject it due to the issues I mentioned and because I am sure there is a deeper theory that will involve the holographic principle and things like the amplituhedron. I.E. our current theories are approximations of a deeper underlying theory, which will obviously also be weird, but more classical than QM. If space time is no longer fundamental then non-locality isn't that hard to swallow anymore and then suddenly determinism is back into play etc.

To me, this doesn't sound much different from the MWI. The MWI is deterministic and the cause of non-locality is that the wavefunction doesn't live in spacetime but in Hilbert space.

Or do you think that determinism may be restored to the experimental realm? If so, I disagree. I don't see anything hinting in this direction.