Question about Many Worlds branching in Quantum Mechanics

[Mentz114]

Of course it does - its in axiom 2.
[clipped to save space]

Thank you, Bill. Just checking.

 

[atyy]

Do you know who Murray Gell-Mann is? Most would take what he says a bit more seriously and think through it a bit more carefully, asking for clarification on any point not clear.

Well, then take negative probabilities seriously: https://arxiv.org/abs/1106.0767

 

[PaleMoon]

when you compute the probability (<a + <b)(a> + b>) = aa* + bb* + ab* + a*b
the last sum may be negative. calling it a negative probability is a question of words
but it is not an exotic thing.

 

[stevendaryl]

I saw that. What I meant is that MWI was proposed to resolve the some of the collapse issue.

I personally don't see it as much of a solution, but that's me. After all, some collapse is reversible.

I don't understand the connection with the last sentence with the topic. Does reversibility of collapse say something about MW, or does it say something about human's fickleness in embracing and rejecting interpretations of quantum mechanics?

 

[bhobba]

Well, then take negative probabilities seriously: https://arxiv.org/abs/1106.0767

I had a conversation with my probability professor about that one. It occurs as an intermediate calculational result in a number of advanced areas - but is thought to be just that - results of calculations that can't be interpreted as usual. I gave him Feynman's famous article on it and explained when it first occurred in QM it turned out it was not really negative probabilities but positive probabilities of antimatter - although I do not think Gell-Mann is talking about something that easily resolved - not that it didn't take a while to sort it out as far as antimatter and the Klein-Gordon equation was concerned.

Decoherent Histories is not a fully developed interpretation - its quite ambitious in its aims so maybe what's going on with negative probabilities will eventually be clearer.

Thanks
Bill

 

[bhobba]

but it is not an exotic thing.

That was my probability professors view.

But here I tend to agree with ATTY - it may be something deeper going on here.

Thanks
Bill

 

[stevendaryl]

I'm not sure if it's the same idea as Feynman's, but I like the following sequence of pictures to show how something like negative probabilities can come into play.

Suppose that you have a string loosely hung between two points. I'm ignoring gravity, so the string has no tendency to droop down. We can possibly model the shape of the string as a random process giving the vertical position $y$ as a function of the horizontal position $x$, as shown in Figure A above. The shape of the string is almost arbitrary, except for the fact that the ends are fixed and the function $y(x)$ must be continuous. (We might also want to constrain the length of the string, and maybe make the string a little stiff so that $\frac{dy}{dx}$ tends to be small, but that's unnecessary details for what I'm going to say.) We could describe the possibilities for the shape of the string by giving a function $P(y,x)$, which is the probability that at horizontal position $x$ the vertical position of the string is $y$. If the shape of string is sufficiently close to being straight, as in Figure A, then there will be only one value of $y$ for each value of $x$, so we would demand that $\int P(y,x) dy = 1$. For each $x$, there is a probability of 1 that the string is at some vertical value $y$.

Now, Figure B shows a less well-behaved shape of the string. The string is allowed to double-back on itself. So it's no longer true that for each $x$ there is exactly one value of $y$ to find the string. As shown in the figure, there are points, such as the location of the light vertical line, where there are 3 values of $y$ where the string can be located, for the same value of $x$. So if we're trying to describe the string's shape using probabilities, we can no longer use a probability $P(y,x)$ that has to obey $\int P(y,x) dy = 1$ for each value of $x$.

Probably the most elegant approach would be to treat $x$ and $y$ symmetrically, and consider them both random variables as a function of a path variable, $s$. But for the purposes of motivating "negative probabilities", we can take the approach shown in Figure C. Rather than having a positive probability of finding the string at height $y$, we introduce a counting function $C(y,x)$ that can be positive (to indicate the presence of a section of the string with $\frac{dx}{ds} > 0$, where $x$ is the horizontal location, increasing to the right, and $s$ is the path parameter, which increases monotonically along the string) or negative (to indicate $\frac{dx}{ds} < 0$, so it's a section of a string that has doubled back). In the figure, at the light vertical line, there are two positive sections of the string and one negative section. So counting orientation, it adds up to 1 string. So our constraint on the probabilistic function $C(x,y)$ would be that for each value of $x$, $\int C(x,y) dy = 1$, allowing both positive and negative values for $C$.

If you think of the $x$ axis as being time, rather than a spatial dimension, then the string represents a worldline of a particle. The points where the string doubles back can be interpreted as particle/antiparticle pair creation, making three particles temporarily, two of which then annihilate each other.

 

[PaleMoon]

i appreciate mwi but not the w for world. the notion of probability would disappear in certain branches.
shooting a coin would obey the 0 1 0 1 0 1 0 1 0 1 0 1 0 1 ... rule in some worlds. qm mechanics would not be random in others and so on.

 

[DrChinese]

the notion of probability would disappear in certain branches.

Most definitely not. Observers in every branch would swear there is random probability in quantum outcomes.

By the way: the coin tossing result 0 1 0 1 0 1 0 1 0 1 0 1 0 1 is as equally likely as 1 1 1 1 1 1 1 1 1 1 1 1 1 1. About once in every 16,000 series.

 

[PaleMoon]

in the branch in which every toss give a 1 the observers would have no idea of what would be a random toss.
spin would always be up.
remember that in the everett construction there are automats with memories. this could not be possible in several branches. a memory has to contain all the possible values from 00000000... to 11111111...
this would not be possible if all the spins are up
a memory has to be close to our physical branch
Several branches would not even contain atoms

 

[PeterDonis]

in the branch in which every toss give a 1 the observers would have no idea of what would be a random toss.

Sure they would. However, in such a branch, based on those results, the observers would conclude that the state of the system was "spin up", instead of a superposition of up and down, precisely because the results they observed were not a random mixture of 1 and 0, but all 1.

remember that in the everett construction there are automats with memories. this could not be possible in several branches. a memory has to contain all the possible values from 00000000... to 11111111...
this would not be possible if all the spins are up
a memory has to be close to our physical branch
Several branches would not even contain atoms

This all looks like word salad to me. I think you are confused about what the MWI actually says.

 

[PaleMoon]

i have everett's thesis at home.
we can discuss about these automats and their properties if you want.

 

[bhobba]

i have everett's thesis at home.
we can discuss about these automats and their properties if you want.

Everett is probably a bit dated these days.

I personally got Wallace's textbook to learn about MW:
https://www.amazon.com/dp/0198707541/?tag=pfamazon01-20

It has been discussed quite a few times on this forum. Due to the strong relation to decoherent histories you might find the following useful:
http://quantum.phys.cmu.edu/CQT/index.html

Thanks
Bill

 

[PaleMoon]

the book in the first lind is too expansive for me
dr chinese replied that 01010101010101... has the same probability than 111111111...
this is only true for us. we cannot get conclusions for other branches
i cannot see how statistics could be done in many other words or histories.
could they see that they belong to rare branches?

 

[bhobba]

i cannot see how statistics could be done in many other words or histories.
could they see that they belong to rare branches?

I don't see why not.

Some people have talked about this rare branch stuff and I must say I do not really get it. Its not in Wallace as far as I can recall.

Most certainly though you can read Griffiths text which he has kindly made available for free. Its not MW but as I said, as explained by Murray Gell-Mann, decoherent histories and MW in his view is more of a semantic than an actual difference. If I haven't posted it before here is Murray's video:


Thanks
Bill

 

[PaleMoon]

in fact the problem does not concern only rare branches.
in our branch bell's inequalities are violated.
what about this violation in all the branches? i recall that all possible outputs are supposed to exist.

i know that it is not your preferred interpretation. it is hard to have answers from defenders

 

[stevendaryl]

Some people have talked about this rare branch stuff and I must say I do not really get it.

I'm not sure what there is to "get". In some possible "worlds" the observed frequencies for repeated experiments will not equal the predictions of quantum mechanics.

 

[DrChinese]

could they see that they belong to rare branches?

Of course, every branch is equally rare. The thing you refer to is that quantum outcomes themselves appear to violate statistical predictions of QM. Accordingly, an observer in such a world might conclude spin up is the outcome of EVERY measurement rather than being a 50-50 proposition, as we observe. (Or maybe they see it as 60-40.)

I agree there are a few branches as you describe - as stevendaryl also says. Out of the many times greater branches that yield normal statistics. So what? Certainly, in any experimental situation, you might be part of an environment that gives a "biased" answer as compared to some other environment.

Honestly, that part of MWI doesn't bother me as it does you. My question is: where are the other branches? Are they accessible?

 

[PaleMoon]

if they are orthogonal to us they are not accessible. (orthogonal meaning with no possible transition)

 

[Mentz114]

if they are orthogonal to us they are not accessible. (orthogonal meaning with no possible transition)

That is a tautology and tells us nothing. It is an assumption that there is no interaction. This fits the facts so any other assumption obviously would be false.

 

[PaleMoon]

you are right i should have said that they become rapidly orthogonal.
during decoherence, a state v><v evolves as ∑ p_i(t) v_i(t) ><v_i(t)
when t = 0 it is
(∑ p_i) v><v = v><v
and during decoherence the v_i(t) ><v_i(t) rapidly tend to be orthogonal (but never are exacty orthogonal)
it is not cleat to me why the output should be exactly on the limit

 

[PaleMoon]

during decoherence, a state v><v evolves as ∑ p_i(t) v_i(t) ><v_i(t)

i wonder if the probabilities p_i depend on time or if they are constant and equal to the transition probability
from v> to the i-th eigenvector of the measured operator

 

[Mentz114]

you are right i should have said that they become rapidly orthogonal.
during decoherence, a state v><v evolves as ∑ p_i(t) v_i(t) ><v_i(t)
when t = 0 it is
(∑ p_i) v><v = v><v
and during decoherence the v_i(t) ><v_i(t) rapidly tend to be orthogonal (but never are exacty orthogonal)
it is not cleat to me why the output should be exactly on the limit

i wonder if the probabilities p_i depend on time or if they are constant and equal to the transition probability
from v> to the i-th eigenvector of the measured operator

All I can remember about this is the basic model where a state $|s_n\rangle$ becomes coupled to the environment $|e_n(t)\rangle$ to give $\psi_n(t)=|s_n\rangle|e_n(t)\rangle$. If the $|e_n(t)\rangle$ are independent random vectors of increasing dimension, then their correlation is zero (orthogonality) in the limit. Clearly you can think of the limit as a few million degrees of freedom without everything falling apart.

The density matrix contains off-diagonal terms that have products $|e_i(t)\rangle \langle e_j(t)|$ which go rapidly to zero if $i\ne j$. So only the diagonal elements are preserved.

 

[bhobba]

I'm not sure what there is to "get". In some possible "worlds" the observed frequencies for repeated experiments will not equal the predictions of quantum mechanics.

That's what I do not get. By construction it MUST equal the predictions of QM.

After decohenece each element of the mixed state is a new world. This is exactly as QM predicts - by it's very definition. But I have only read Wallace - in a discussion a while ago evidently in some views it is possible. For example in Decoherent histories it is claimed in the paper by Gell-Mann and Hartle you can even have histories with negative probabilities. I trust what guys like that say - but haven't verified it personally.

Thanks
Bill

 

[PeterDonis]

By construction it MUST equal the predictions of QM.

No, it mustn't, at least not if you're expecting all of the observed frequencies to match the predictions of QM.

Consider a simple example: we have a source of qubits and a spin measuring device that each qubit passes through. The source emits qubits in a state which gives equal probabilities for spin up and spin down for the measurement result. We run 10 qubits through the device. We record a "1" for spin up and a "0" for spin down for each measurement.

According to the MWI, there are now 1024 different worlds, in each of which the sequence of 10 measurement results is a different sequence of 10 binary digits (0 or 1). Many of these sequences of 0s and 1s give relative frequencies of spin up and spin down that are different from 50-50 (at the extremes, there will be one sequence of all 0s and one sequence of all 1s). That is what I think @stevendaryl meant by results not matching the predictions of QM.

Btw, the fact that this will be the case according to the MWI is one of the key issues I personally see with the MWI. I understand that various physicists (including, IIRC, Bryce DeWitt) have published papers arguing that this isn't really a problem, but their arguments look like black magic to me and I've never been convinced by them. But however that may be, the fact that the MWI has the implications I've described above should be uncontroversial.

 

[bhobba]

That is what I think @stevendaryl meant by results not matching the predictions of QM.

Yes - Wallace deals with that and explains why it must be like that. Of course in some sequence of observations there are very low probability sequence outcomes - but that is in the formalism itself. If it s 50-50 and the observation the the count of say the number of 1's then of course the probability will be small for all 1's. That's the central limit theorem.

Btw, the fact that this will be the case according to the MWI is one of the key issues I personally see with the MWI. I understand that various physicists (including, IIRC, Bryce DeWitt) have published papers arguing that this isn't really a problem, but their arguments look like black magic to me and I've never been convinced by them. But however that may be, the fact that the MWI has the implications I've described above should be uncontroversial.

Wallace actually argues such a view is inconsistent - I will need to dig up the book to find the page.

Thanks
Bill

 

[PeterDonis]

If it s 50-50 and the observation the the count of say the number of 1's then of course the probability will be small for all 1's. That's the central limit theorem.

"Probability" has a different meaning here, though: it means "fraction of worlds". (In more general cases where there is a continuous observable, that has to become "measure of worlds", which introduces additional issues about how to define the measure.) But an observer in a particular world has no way of knowing what fraction of worlds have the same fraction of 0s/1s as his, so this "fraction of worlds" probability is unobservable. We only know it in the scenario I described because we declared by fiat what the state emitted by the source was.

But in a real experiment, we have to infer the state emitted by the source from observed frequencies of measurement results; so in a world of all 1s, we would infer that the source was emitting all spin up qubits, not an equal amplitude superposition of up and down qubits. In other words, the MWI predicts that worlds will exist in which the natural method of inferring the source state from observed frequencies does not work; yet in practice it always does work. That is an issue that I don't think is addressed adequately.

Wallace actually argues such a view is inconsistent - I will need to dig up the book to find the page.

I'd be interested in seeing his arguments.

 

[bhobba]

"Probability" has a different meaning here, though: it means "fraction of worlds"

Got it now - yes I see that point - that for some worlds the probability will be so small they will FAPP just peter out. Fine in probability theory - but we are talking about worlds here - yes that is a strange consequence. A physicist in this world of all 1's would likely reach different conclusions about the laws of nature - its a good thing it peters out - but of course never actually vanishes. Is it a problem - I suppose that's a personal opinion.

I'd be interested in seeing his arguments.

He examines a number of alternative ideas for assigning probabilities in section 5.8 - page 189 of the Emergent Multiverse.

Here is one I remember. Suppose you have an observational black box with lights representing each outcome. Let's say you adopt the rule all worlds are equally probable ie if you have N outcomes from the black box then the probability of each outcome is 1/N If you have two outcomes you get 2 worlds with 1/2 chance, 3 outcomes 1/3 and so on. Now let's form a compound observation inside our black box - with three outputs from two, two output observations. You do the first observation - and light 1 goes on - but for the second outcome we observe it with another two output observation and you display those lights. What you would get knowing the contents of the black boxes is probabilities of 1/2, 1/4, 1/4. But if you didn't know what was inside, and the rule says nothing about that, you would say it was 1/3, 1/3, 1/3.

He examines a lot of ideas like that and all show similar logical flaws. Admittedly he has not considered every possible scheme you could come up with, but he does look at a lot. This is suggestive that in the MW scheme the only one that is feasible is the Born Rule. Indeed Gleason more or less says that must be the case because he has a non-contextuality theorem which basically means there is no real out for the theorem. On page 196 he explains in words without using the explicit theorem why in his decision theory approach it must be non contextual - it would mean a rational agent preferring a given act in one situation to the same act in another situation.

To me this is the real issue with MW - at least as Wallace presents it - this rational agent thing they introduce and how a rational agent would act. Yes it would be irrational to prefer a given act in one situation to the same act in another situation (or would it?). But rational behavior is hardly something you can pin down exactly. Why would it be irrational to do so? That would seem the key point. Lots of unstated assumptions being smuggled in via back door IMHO. I would say its fatal actually - except he has proofs based on axioms he states from the start. Its those axioms that need challenging. They are very very reasonable - but that's the point - science is correspondence with experiment - not reasonableness. I love reasonable proofs of thing like Maxwell's equations - it fact giving reasonableness augments for physical theories is one of my favorite things. But Feynman said it straight:


These arguments mean nothing - only experiment matters.

Thanks
Bill

.

 

[Chris Miller]

Isn't the current thinking that the universe is infinite in time and mass also refutable via "reductio ad absurdum." Or are there really infinite me's inhabiting infinite observable universes exactly identical to this one?

 

[bhobba]

Isn't the current thinking that the universe is infinite in time and mass also refutable via "reductio ad absurdum." Or are there really infinite me's inhabiting infinite observable universes exactly identical to this one?

I am not sure what you mean, but we have all sorts of theories in accord with current observation about cosmology. What mass is, is well known, but technical. There is an advanced argument based on irreducible representations of the Poincare group. That obviously makes no sense at the B level - but what mass is, is not a mystery in our current theories.

As to infinite me's inhabiting infinite worlds I urge you to watch Murray Gell -Mann's video I posted about the Everett interpretation. Its 'meaning' of what a world is, is more subtle than presented in some literature. So let's just say at the B level these sorts of things are just conjectures on which further research is needed.

My personal favorite for what its worth is eternal inflation:
http://cds.cern.ch/record/485381/files/0101507.pdf

Although not a proper reference here I am reading book by Penrose right now - Fashon, Faith and Fantasy In The New Physics Of The Universe that looks at a lot of these things. The interesting thing about Penrose is he does not hide the math, explaining definitively non trivial stuff like analytical continuation, and equations are used freely. It's not like his good friend and collaborator Hawking who believed every equation he used cut book sales in half. I much prefer Penrose to other popularizes - but he does reach some weird conclusions - he is a literal Platonist - its surprising how many math types are - but then again as the great physicist Ken Wilson noted when he was invited to a dinner in his honor by the math department for being named a Putman Fellow at an early age (he entered Harvard at 16) they were all excellent mathematicians, but quite mad. Wilson was a two time Putman Fellow at 17 and 19, so was Feynman who didn't even prepare for it - people usually go through a special preparation program for that tough test:
https://en.wikipedia.org/wiki/William_Lowell_Putnam_Mathematical_Competition

Thanks
Bill