Many Worlds Interpretations and probabilities

[bhobba]

In layman's terms how is Gleason's idea different to Deutsch's?

Probability theory is based on the Kolmogerov axioms. Decision theory is the mathematical study of optimal strategies that makes use of probability theory as part of its tools.

Gleason simply uses probability theory. You can actually prove for dimensions greater than 2 there is only one way of defining probability on a Hilbert space - the Born Rule, but a careful study of the proof shows an assumption - the probability does not depend on the basis - this is called non contextuality. There are others as well such as all superposition's at least in principle are possible, but non-contextuality is the main one. It turns out you can actually prove in MW that its non-contextual - the proof is in the appendix of Walllce's book. It too may be contentious like the decision theory approach, but I think it's fine and haven't seen anyone criticize it.

Thanks
Bill

 

[name123]

Probability theory is based on the Kolmogerov axioms. Decision theory is the mathematical study of optimal strategies that makes use of probability theory as part of its tools.

Gleason simply uses probability theory. You can actually prove for dimensions greater than 2 there is only one way of defining probability on a Hilbert space - the Born Rule, but a careful study of the proof shows an assumption - the probability does not depend on the basis - this is called non contextuality. There are others as well such as all superposition's at least in principle are possible, but non-contextuality is the main one. It turns out you can actually prove in MW that its non-contextual - the proof is in the appendix of Walllce's book. It too may be contentious like the decision theory approach, but I think it's fine and haven't seen anyone criticize it.

Thanks
Bill

Thanks but how does the probability link to the worlds? Are there more worlds for more probable outcomes?

 

[bhobba]

Thanks but how does the probability link to the worlds? Are there more worlds for more probable outcomes?

No.

It's the probability of if you picked a random person from a world what world would they be experiencing.

Thanks
Bill

 

[name123]

No.

It's the probability of if you picked a random person from a world what world would they be experiencing.

Thanks
Bill

So imagine there are n worlds, and you picked a person at random, why would it be more probable that you picked them from one world than another, why wouldn't it be 1/n for each world?

 

[Nugatory]

So imagine there are n worlds, and you picked a person at random, why would it be more probable that you picked them from one world than another, why wouldn't it be 1/n for each world?

And there's the problem...
Consider performing a measurement on a system in the state $|\Psi\rangle=\frac{\sqrt{3}}{2}|\psi_0\rangle+\frac{1}{2}|\psi_1\rangle$. There are two possible outcomes so MWI interprets this as splitting into two worlds and by the 1/n line of thinking we end up with a 50% chance of each - but according to the Born rule and experiment the actual odds are 75/25. If MWI predicted that we got three worlds in which $\psi_0$ was realized and one in which $\psi_1$ was realized (or 300,000 and 100,000, or ...) we'd be OK, but so far no one has been able to show that that behavior does emerge. Instead we have to put the Born rule in by hand, basically asserting that some of the worlds are more probable than others - and your original question back in #1 seems to show a justifiable dissatisfaction with that approach.

 

[bhobba]

So imagine there are n worlds, and you picked a person at random, why would it be more probable that you picked them from one world than another, why wouldn't it be 1/n for each world?

Consider it done once then the person returned. You do it over and over and you find when you ask them what world they experienced you find some more probable than others - it follows from the basic principles of QM as I explained. It just part of the weirdness of MW and a reason myself and others don't adhere to it (plus others). However its mathematical elegance is breathtaking if you are into that sort of thing - as I am - but not enough to counter its weirdness.

Another way of looking at it is to imagine you take one person, pop them in a random world - what is the probability they will experience a certain world. It may seem 1/n but that is not what the theory says. MW uses decision theory to figure that out and you get the Born Rule - but the whole business is weird - really weird. However I don't think its logically inconsistent.

Thanks
Bill

 

[vanhees71]

How can you prove in how many worlds the universe (or whatever) splits? As far as I know, the "number of splittings" is not an observable in MWI. For me MWI is pure phantasy without any real physical meaning.

 

[bhobba]

How can you prove in how many worlds the universe (or whatever) splits? As far as I know, the "number of splittings" is not an observable in MWI. For me MWI is pure phantasy without any real physical meaning.

Same here. But its beautiful mathematically. I have a lot of sympathy for those that adhere to it.

Thanks
Bill

 

[LeandroMdO]

Gleason proves what is the correct probability measure in Hilbert space, but it doesn't explain why probability emerges at all in a purely unitary theory. Yes, you can assign these real numbers to pieces of the state and they add up to 1, but that doesn't mean that frequencies of outcomes of repeated experiments will converge to these numbers. The question might not even make sense, or even if classical worlds arise from the state, they could arise with uncontrollable, wholly nondeterministic frequencies that don't converge to any probability distribution. Clearly more arguments are necessary.

 

[bhobba]

Gleason proves what is the correct probability measure in Hilbert space, but it doesn't explain why probability emerges at all in a purely unitary theory. Yes, you can assign these real numbers to pieces of the state and they add up to 1, but that doesn't mean that frequencies of outcomes of repeated experiments will converge to these numbers. The question might not even make sense, or even if classical worlds arise from the state, they could arise with uncontrollable, wholly nondeterministic frequencies that don't converge to any probability distribution. Clearly more arguments are necessary.

One of the assumptions of MW is like I said if you are randomly put in a world there is a different probability of what world you experience. Its weird, counter intuitive and downright maddening - but that's the interpretation.

The reason I prefer Gleason to the usual decision theory approach is its often criticised as being circular - I have been through it and I don't think it is, but I don't want to argue about it. MW already is far too weird for me.

Thanks
Bill

 

[LeandroMdO]

One of the assumptions of MW is like I said if you are randomly put in a world there is a different probability of what world you experience. Its weird, counter intuitive and downright maddening - but that's the interpretation.

Right, but that's not purely unitary at all. It has an extraneous axiom. At that point you might as well just assume Born's rule and be done with it.

The reason I prefer Gleason to the usual decision theory approach is its often criticised as being circular - I have been through it and I don't think it is, but I don't want to argue about it. MW already is far too weird for me.

I agree that Gleason inspired approaches are better. Decision theoretic arguments make too many "rationality" assumptions that have no business being in a physics theory.

 

[name123]

Consider it done once then the person returned. You do it over and over and you find when you ask them what world they experienced you find some more probable than others - it follows from the basic principles of QM as I explained. It just part of the weirdness of MW and a reason myself and others dot adhere to it (plus others). However its mathematical elegance is breathtaking if you are into that sort of thing - as I am - but not enough to counter its weirdness.

Another way of looking at it is to imagine you take one person, pop them in a random world - what is the probability they will experience a certain world. It may seem 1/n but that is not what the theory says. MW uses decision theory to figure that out and you get the Born Rule - but the whole business is weird - really weird. However I don't think its logically inconsistent.

Thanks
Bill

I'm not conversant with decision theory, but I assume you are not suggesting that the MWI theorists suggest a person use the Born rule to make a decision as to which is most likely, and then because it works out, the MWI theorists say "there you go it works". Because that wouldn't explain why given n worlds there wouldn't be a probability of 1/n of experiencing one.

I don't think saying that the theory explains it by making up a variable linked to each world ( the Probability of Being Experienced variable for example) and assigning it a probability value works. If there were n worlds and there was a copy of me in each, so n of me, then for any given world that copy of me was 1 out of the n copies. Which then raises the question of what is being said to have more probability of being that 1 out of n?

Is the reason that multiple versions of the same world isn't used rather than an assigned probability value, that it would require something to have done the maths to get the whole number of required worlds to allow for the required ratio?

Also are the motives behind the theory to avoid "spooky action at a distance", and given the history of science being in search of a cause behind an effect, reaching the conclusion that there exist certain effects which have no physical cause (in the sense of a physical cause for why it was that effect rather than another)?

 

[bhobba]

I don't think saying that the theory explains it by making up a variable, the Probability of Being Experienced variable for example and assigning it a probability value works.

MW adherents would argue that.

You can think of it as greater than n worlds each of equal probability, but MW adherents don't think of it that way - they simply accept the weirdness.

Thanks
Bill

 

[name123]

MW adherents would argue that.

You can think of it as greater than n worlds each of equal probability, but MW adherents don't think of it that way - they simply accept the weirdness.

Thanks
Bill

But as I went on to write,

If there were n worlds and there was a copy of me in each, so n of me, then for any given world that copy of me was 1 out of the n copies. Which then raises the question of what is being said to have more probability of being that 1 out of n?

So what would their answer be as to what they are suggesting has more probability of being that 1 copy out of n copies?

 

[bhobba]

So what would their answer be as to what they are suggesting has more probability of being that 1 copy out of n copies?

No answer - nature is just like that. Science doesn't answer all questions you know.

Thanks
Bill

 

[name123]

No answer - nature is just like that. Science doesn't answer all questions you know.

Thanks
Bill

But the question was about what they were suggesting about nature. They cannot very well say "nature is just like that" if they do not even know what they are suggesting nature is like. If that isn't clear then the question

So what would their answer be as to what they are suggesting has more probability of being that 1 copy out of n copies?

could be re-phrased as: What in nature are they suggesting has more probability of being that 1 copy out of n copies, such that nature could be like their suggestion?

 

[bhobba]

But the question was about what they were suggesting about nature. They cannot very well say "nature is just like that" if they do not even know what they are suggesting nature is like.

They know exactly what they are suggesting. Nature is weird. So?

Nature does not have to conform to your intuition. Symmetry implies if you have 1/n outcomes then there is equal probability of being popped in any particular world. But maybe nature doesn't follow that symmetry.

Thanks
Bill

 

[LeandroMdO]

They know exactly what they are suggesting.

To this day I have never seen a precise mathematical formulation of many worlds quantum mechanics. If something can only be described in dozens of pages of prose rather than a handful of axioms, it's probably not "exactly" understood.

 

[bhobba]

To this day I have never seen a precise mathematical formulation of many worlds quantum mechanics. If something can only be described in dozens of pages of prose rather than a handful of axioms, it's probably not "exactly" understood.

Read Wallaces book:
https://www.amazon.com/dp/0198707541/?tag=pfamazon01-20

Its very theorem, proof, theorem, proof in its style.

But the emphasis has shifted from observations to Histories.

Thanks
Bill

 

[name123]

They know exactly what they are suggesting. Nature is weird. So?

Nature does not have to conform to your intuition. Symmetry implies if you have 1/n outcomes then there is equal probability of being popped in any particular world. But maybe nature doesn't follow that symmetry.

Thanks
Bill

I was assuming that I was being thought of as the configuration of chemicals that made up the human form I experience having at the time of the world splitting event. And that it wasn't a case of which world would that configuration appear in, it appears in all of them. So the probability of the configuration appearing in a world is 1. Which then leaves the question, of what, if not the configuration of chemicals, are they assigning a probability of <1 of appearing in a world to?

 

[LeandroMdO]

Read Wallaces book:
https://www.amazon.com/dp/0198707541/?tag=pfamazon01-20

Its very theorem, proof, theorem, proof in its style.

But the emphasis has shifted from observations to Histories.

Thanks
Bill

Okay. What are the axioms?

 

[bhobba]

Okay. What are the axioms?

As far as I can see simply the Schrödinger Equation.

Thanks
Bill

 

[Quantumental]

As far as I can see simply the Schrödinger Equation.

I feel like you are always talking with 2 tongues, on one hand you dismiss MWI because it's too weird, then you go onto say: nature is weird, therefore MWI doesn't have to explain itself, yet I don't believe it. Sometimes I am wondering if you are a hardcore MWI proponent that just troll for fun.

Re: axioms, you yourself literally say that the only way to accept something like Gleason or Decision Theory is to presume (another axiom) that nature just 'behaves like that'. That's not science.

Let's look at the most devoted MWI supporter out there: David Deutsch. Mr. Deutsch postulates (axiom alert) that there is an infinite amount of universes from start, he calls these 'fungible' meaning that there is no difference between them at all, there is literally no fact of the matter which of these infinite ones you are, yet there is an infinite of them. So in his reading of Everett if there is a 30% probability of something occurring, now 30% of the infinite worlds had this outcome while the 70% others had the other outcome, but if there is no fact of the matter of which world is which (fungibility) then how can it just magically split into a 3/7 formation? This is riddled in axioms and incoherence.

David Wallace, Simon Saunders, Alastair Wilson etc. has advocated for a slightly different approach where you start out with the wavefunction and all the worlds within it, but then they diverge instead of splitting, so in the aforementioned example there was initially 2 worlds, let's call them world A and world B, then A had the 30% outcome and 70% world had the other outcome. This is at least coherent, but begs the question: what made A evolve into the 30% outcome? What mechanism/property of that world from t0 determined this? Correct me if I am wrong, but this is also an axiom.

 

[bhobba]

I feel like you are always talking with 2 tongues, on one hand you dismiss MWI because it's too weird, then you go onto say: nature is weird, therefore MWI doesn't have to explain itself, yet I don't believe it. Sometimes I am wondering if you are a hardcore MWI proponent that just troll for fun.

I fail to see how any of those things are an issue. I find MWI too weird, but MWI adherent's don't. I don't dismiss it either - I just don't adhere to it - there is a difference.

Re: axioms, you yourself literally say that the only way to accept something like Gleason or Decision Theory is to presume (another axiom) that nature just 'behaves like that'. That's not science.

You get a weird prediction from a theory that is in accord with observation and accept it. That is science. I am being unscientific in rejecting it as too weird - you shouldn't do that - but I do. It called personal prejudices.

Let's look at the most devoted MWI supporter out there: David Deutsch. Mr. Deutsch postulates (axiom alert) that there is an infinite amount of universes from start,

I don't know what Deutch said, I learned from Wallace.

The prediction of MW, as espoused by Wallace is an exponentially increasing number of worlds, it's very very large, but not infinite. Its not an axiom either. The probability of experiencing a world is not an axiom - it simply asks a question. I will find myself in some world. I don't know which. What is the probability I will experience a certain outcome. Its like tomorrow I will experience some weather - I don't know what it will be but what is the probability it will be wet, cloudy, sunny or whatever. The issue with MW is when you consider all the other worlds why is it a different probability in other worlds - that is very counter intuitive - in fact downright weird. But its what the theory says.

I would suggest getting the book the Emergent Multiverse and actually studying it.

Thanks
Bill

 

[LeandroMdO]

I would suggest getting the book the Emergent Multiverse and actually studying it.

But that's the thing. If MWI had a precise formulation I wouldn't need to do that. I could get a list of axioms that define the theory and work from there.

Just the Schrödinger equation doesn't do at all. It's not clear that there are probabilities when evolution is purely deterministic. Yes, a pure state gets hopelessly entangled with the environment over time, but that by itself doesn't entitle me to write down a density matrix, discard off-diagonal elements, and interpret the diagonal as a probability distribution. For that, I need a probabilistic interpretation which is precisely what I'm trying to prove.

Even assuming some uncertainty, it doesn't follow from Gleason's theorem or anything else that frequencies of outcomes of repeated experiments converge to the Born rule derived probabilities, or any probability distribution at all.

 

[bhobba]

But that's the thing. If MWI had a precise formulation I wouldn't need to do that. I could get a list of axioms that define the theory and work from there.

It just one axiom as far as a I can see.

Of course you have to assume a question like what is the likelihood I will find myself in a particular world is meaningful. I think its a matter of taste if you consider such an axiom or not. In classical mechanics we define the velocity as the derivative of position, but obviously an axiom is such a derivative exists is needed, but nobody ever explicitly states it because its so damn obvious,

Thanks
Bill

 

[LeandroMdO]

It just one axiom as far as a I can see.

Of course you have to assume a question like what is the likelihood I will find myself in a particular world is meaningful. I think its a matter of taste if you consider such an axiom or not.

Not just likelihood. You have to prove that it is a probability. Assigning a weight to the branch is not enough, because a weight is just a number. It can mean anything. It could mean that "the branches on the left have |a|² units of beauty". The identification with actual frequencies needs to be stated explicitly, preferably as a consequence of an easily accepted axiom.

Some manner of branch counting always seems to be implied. If you tell me that you have |a|² branches on one side and |b|² branches on the other it doesn't take much work for me to accept that the probability of finding myself in the former side is |a|², and all I really had to assume is that my experience corresponds to one branch (which is an axiom, but a much weaker one). But then proponents of many worlds turn around and say that the number of worlds isn't well-defined/meaningful, that branch-counting is misguided, and that probabilities come from someplace else. If that's so, the connection with frequencies must be justified. One cannot use a branch-counting intuition to establish something, and then discard branch counting.

In classical mechanics we define the velocity as the derivative of position, but obviously an axiom is such a derivative exists is needed, but nobody ever explicitly states it because its so damn obvious.

It is explicitly stated: it's part of Newton's first law. The common wording of it does some extra work that excludes noninertial frames, but the meat of it is the idea that the state of motion doesn't change without reason. This excludes discontinuous changes in position and velocity (assuming no infinite forces, which is a much milder assumption).

It's also worth noting that even if it were obvious, the emergence of probabilities in MWI certainly isn't, or there'd be a consensus in its favor already; this conversation wouldn't happen because we'd both happily accept many worlds.

 

[bhobba]

Not just likelihood. You have to prove that it is a probability.

You mean to say, for example, you can't assume tomorrow there is a probability it will rain or not. As far as I can see that's all they are doing. You will have a number of copies after. You will not know about any of the others. What is the probability the copy you are experiencing gets a certain outcome. Most would say that's not an assumption even though it is.

Regarding Newtons first law by mentioning acceleration you are assuming position is twice differentiable - that is an implicit but unstated assumption. Should it be stated as an axiom? I think modern versions of classical mechanics based on sympletic geometry do just that - but is really only used by mathematicians of the pure variety. Such things are purely a matter of taste and semantics which I think is the silliest thing I know to argue about.

The classic example though would have to be good old Euclidean geometry. Would you consider Euclid's axioms valid? If so exactly what did Hilbert do when he gave his axioms? Again its a matter of taste as to what level of rigor you want.

Thanks
Bill

 

[bhobba]

Here is an article that I found that examines the isssue of axiomatics in what I consider a fair way - giving points for and against:
https://arxiv.org/pdf/gr-qc/9703089.pdf

The only issue I have is it was written in 1989.

I learned MW from Walllace's book which is very recent, written in 2012, so things have likely moved on a lot since 1989.

As far as I can see modern MWI is simply decoherent histories with each history being in a separate world. I think its likely if you have issues with one, you have the same issue with the other - not always of course but they seem to have so much in common it would be likely.

I will state my own preference - Decoherent Histories is way better than MW - but that is, as I try to explain to some, just a preference, it doesn't mean anything.

Thanks
Bill

 

[LeandroMdO]

You mean to say, for example, you can't assume tomorrow there is a probability it will rain or not.

In a sense that's right. The statement "there is an 80% chance of rain tomorrow" is a bit meaningless unless it is qualified. For instance, you can say that, historically, there has been rain at your location 8 out of 10 March 23rds, and you expect the trend to continue in the future. Or you may have a model of the climate, that when fed today's atmospheric data, produces rain in 24 hours in 8 out every 10 model runs.

If you did your job right, you will see your predictions of this type be confirmed by experiment which will then justify the shorthand "there's a 80% probability of rain tomorrow". But the precise meaning of probability in this context is clear: I have a frequentist definition that came from my use of a model that I can, in fact, run many times.

When it comes to many worlds, I don't see such a frequentist approach working unless there's a branch counting argument, which people such as Wallace explicitly advise against. It is then illegitimate to refer to the measure obtained from Gleason's theorem as a probability; it is best termed a weight. Then more work is required to find the physical meaning of these weights.

Regarding Newtons first law by mentioning acceleration you are assuming position is twice differentiable - that is an implicit but unstated assumption. Should it be stated as an axiom? I think modern versions of classical mechanics based on sympletic geometry do just that - but is really only used by mathematicians of the pure variety. Such things are purely a matter of taste and semantics which I think is the silliest thing I know to argue about.

I don't require the axioms to be stated with mathematical rigor, necessarily, but I want them to elucidate what assumptions are being made. I think it's clear, if implicit, that Newton's first law implies differentiability properties for classical paths. The issue with MWIs is that they don't say anything, explicitly or implicitly.

EDIT:

Here is an article that I found that examines the isssue of axiomatics in what I consider a fair way - giving points for and against:
https://arxiv.org/pdf/gr-qc/9703089.pdf

Yes, I'm familiar with that paper. I agree with him that "I'm uncomfortable with the idea of many worlds" is not a valid objection, and that the program is worthwhile. I also agree with him that it's still not clear what all these not obviously equivalent versions of MWI are saying. It's possible that things have moved on since then, but if we take a look at more recent papers, e.g.

https://arxiv.org/abs/1405.7907

and its rejoinder by A. Kent

https://arxiv.org/abs/1408.1944

The situation doesn't appear to be any better resolved.