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Everett and Born.

  1. May 3, 2005 #1


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    Hello all,

    I submitted an article (to the Royal Society) concerning the possibility (or rather the impossibility) of the Born rule to emerge out of the unitary part of quantum theory, as is one of the "programme points" in MWI.

    An almost last version (I messed up some stuff on my server so that I didn't put up the most recent version and for the moment I don't have access so I'll fix this later) of the paper can be found at:


    Feel free to discuss it. There are still a lot of typos and so on which I already corrected, but the main content is there.
    BTW, I don't know if it will be accepted, we'll see. But I would like to upload it to the arxiv, only I don't have endorsement for quant-ph. So if someone can endorse me, that would be nice.

    thanks and cheers,
    Last edited: May 3, 2005
  2. jcsd
  3. May 6, 2005 #2
    Hey Patrick,

    Have you managed to put the updated version on your server yet? I would be interested in seeing what changes you have made.

    As you already know, I think that your "alternate projection postulate" (APP) is a very interesting idea. It addresses what I see as a conceptual difficulty of standard MWI: namely, that if we define a "physical measure" m_phys as corresponding to a simple enumeration of the "number" of worlds, then it is easy to come up with situations such that the measure of worlds in which the observer concludes the Born rule to be false is significant. The only way to address this difficulty, afaict, is to adopt your APP; that is, to set the probability measure m_prob equal to the physical measure m_phys.

    As you state in your paper, a simple replacement of the Born-rule generated measure with the APP gives rise to a theory that is internally consistent, yet makes predictions that are just plain wrong. What I am interested in is how to set up a workable theory founded on the APP -- that is, a theory that makes the same predictions as QM. In your paper, you make the point that when we assume the probability measure to be prescribed by the Born rule, then we are in fact making an independent assumption. What I would like to do is to point out yet another assumption that is made by standard MWI.

    So here's my line of reasoning. It seems clear to me that when a world "splits" into N branches, this is understood to mean that the *observer* state evolves from one physical state into N physically distinct states. When an observer couples to a system under observation (say, a spin 1/2 particle), then we start out by enumerating the number of eigenstates of the particle (say, N = 2). We then go on (in the standard MWI) to assume that the observer-state is represented by the same number of eigenstates -- eg, N = 2. That is, we assume that there is a one-to-one correspondence between the number of eigenstates of the observer and the number of eigenstates of the system under observation. So this is the assumption that I wish to challenge.

    It seems to me that if we modify the MWI via the APP, and we *further* modify it by considering the notion that the number of observer-states and the number of observed-states are *not* necessarily in one-to-one correspondence, then we might be able to come up with a formulation that does, in fact, make the same predictions as standard QM.

    Sorry if I'm sounding like a broken record here. (Note to lurkers: I have been developing this idea via email with Patrick, so I'm sorta repeating myself in this post.) I'm halfway thinking of writing up a short follow-up to your paper, perhaps entitled "Enumeration of states in the Everett relative state formulation" or something like that.

  4. May 7, 2005 #3


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    That's what many people try to do: find a "natural" counting scheme that ends up being equivalent to the Born rule. What I tried to argue is that if ever you find such a scheme, it will be for a sufficiently specific value of "natural" :-), that you could call it an extra postulate.
    Nevertheless (we also had some e-mail exchange about that) I think that even _before_ you consider counting stuff, there is a more fundamental issue to be put forward: namely why having to choose ONE branch/term/state at all ?

    The only thing you get out of unitary QM is that after "measurement" the observer's physical construction (body) ends up in entangled states with the rest of the universe. I think it is already an extra postulate to say that the observer will only be aware of ONE of these terms in a probabilistic way.
    Of course, EVEN after this is postulated, then you can find different ways to assign probabilities to each of the possibilities, and I wanted simply to indicate that at least 2 radically different ways of this assignment were possible: Born rule and the APP. The logical existence of at least two assignment schemes then means that not one can be logically DEDUCED: you will again have to postulate which one, independently. Now you can find minimalistic ("natural") postulates which do this, or you can bluntly say: by postulate, it's the Born rule.
    As is rather well known, requiring "non-contextuality" is already sufficient (using Gleason's theorem).

    But, before you can require "non-contextuality", you have already to say that you ARE going to assign probabilities to individual terms in the wave function (otherwise there is nothing you can require non-contextuality of !).

    So all this means that unitary QM, all by itself, does not generate anything like a natural probabilistic quantum theory, as was the original goal of Everett.

    I think I see more or less what you're aiming at, which is a kind of "phase space" argument.
    But first I'd like to point out that we do not start by enumerating the number of eigenstates of the particle, but the number of eigenspaces with distinct measurement outcomes (the distinct eigenvalues). The original system's state can then be constructed by a unique sum of one single eigenvector out of each eigenspace, say v_n, to make up the whole state, and the observer will entangle with each of these individual eigenvectors.

    psi_sys = sum_n a_n v_n (n runs over the different eigenspaces of the system).

    Now, you seem to think somehow that you might have "more" states of the observer associated with one v_n than with another, giving it more weight. But I don't think that that is tenable.

    Indeed, BEFORE the measurement, the observer was in a specific state psi_obs0. This can be a very complex state, even unknown (part of a mixture), but it is in a certain state, even if we don't know which one. Now, due to linearity of the time evolution operator, with ONE v_n, this psi_obs0 will evolve into ONE specific psi_obsn ; there's no choice, it is fixed by the evolution operator corresponding to the measurement action. So I don't see how you can have "more" psi_obsn than psi_obsm. There's exactly one for each.
    However, you could say that psi_obs0 is drawn from an ensemble, and that they populate differently the phase space around psi_obsn than around psi_obsm if we let them all evolve through the time evolution operator.
    But that also has a serious problem: this would then mean that the probabilities of outcomes of measurement are determined, not by the state of the system, but by the mixture of states of the observer before measurement. That cannot generate a Born rule which only depends on the system states.

    Well, I see obstacles, as I outlined above. That doesn't mean that it cannot be done, but I see difficulties on how to approach things that way.
    Nevertheless, in all this, no matter how "natural" it may sound in the end, I'm pretty convinced that EXTRA postulates are somehow necessary and that it doesn't follow logically from unitary QM.
    That's what I wanted to show in my paper.

  5. May 7, 2005 #4


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  6. May 7, 2005 #5
    I agree with you that this is an extra postulate.

    I'm thinking that the time evolution operator that we use to calculate the evolution of the observer in my proposed new scheme is *not* the same operator that we use in standard QM to calculate the evolution of the observed system. Given your comment above, perhaps this new operator would have to be nonlinear.

    As I mentioned to you previously in emails, classical GR (classical in the sense that it does not invoke QM) can give rise to situations in which a system in a single definite state has multiple distinct solutions to the time dependent evolution. I'm glad you brought up the point about the operator being linear, because it occurs to me now that the equations of GR are nonlinear. Perhaps that is one way to explain why multiple distinct evolutions can arise in GR.

    So: in standard MWI, we use the standard linear operator to calculate the evolution of the observed system; in this new scheme, we use a novel, GR-derived, nonlinear operator to calculate the evolution of the observer. If we can demonstrate that each scheme makes the same predictions, then we have demonstrated a "duality" of sorts between QM and GR!

    I agree with you that that would not work; and the scheme that I have in mind does not work like that. Rather I would prefer to have psi_obs0 correspond to a single definite state which then has multiple possible evolutions.


    PS - got your updated pdf, thanks ;)
  7. May 7, 2005 #6
    Yes, I understand the difficulty here. I guess I'd just point out (what I'm sure has already occurred to you) that this is a difficulty that is inherent to *any* probabilistic theory, not just "weird" ones like QM.

    And I can't think of any "solution" to the issue that you raise here. I merely accept the notion (at least provisionally) that it makes sense somehow to talk about "probabilities," acknowledge that there is a fundamental issue that I have not addressed, and move on. ;)

  8. May 7, 2005 #7


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    Ok, but then we leave "standard QM". All bets are off then. I know vaguely of some attempts to solve the measurement problem by small non-linearities, but I don't know the details. However, what these attempts (I think) try to do, is to induce a genuine collapse. EPR situations put stringent constraints on these possibilities.

    Ah, I'm not knowledgeable enough of GR to comment seriously, but I didn't know that that was the case. I thought that GR was deterministic.

    Honestly, if you have different evolutions for the "system" and for the observer ; and if you have linear evolution for the system, and non-linear evolution for the observer, you render things so ugly that you can just as well stay with Copenhagen, no ?

    But instead of critisizing, I should maybe look again at your stuff...

  9. May 7, 2005 #8
    I'm not really aware of such attempts -- perhaps I ran across them at some point, it does ring a distant bell -- but I doubt it's similar to what I'm doing. Actually, I do recall that Bohm's original pilot wave papers mentioned that you could add some small linearities to the equations that were too small to detect by the then-contemporary standards, but I think he was just playing a "what if" game.

    I also recall that Mendell Sachs makes a big issue out of the fact that QM is linear and GR is nonlinear, although I never did understand where he was going with it.

    I am specifically referring to some papers published about 10-15 years ago by Kip Thorne, in which he analysed the trajectory of a billiard ball in a GR model that allows closed timelike curves. He found that there were multiple distinct solutions to the billiard ball's trajectory, each one of which was internally self-consistent. And it was *not* because the initial conditions were insufficiently specified. I can dig up the reference if you would like.

    I have actually not seen anyone in authority say explicitly that "GR is not deterministic" -- but it seems to me that Thorne has demonstrated by example that that is the case. But in his paper he had different fish to fry, which perhaps explains why the conclusion that GR is "nondeterministic" did not get publicized per se as a result of his papers.

    Naw, it's not ugly once you get used to it ;). You just have to work out the kinks. For example: I'm not saying to use one operator for the observer and another operator for the observed. What I'm really trying to say is to use one operator for one formulation, and another operator for a different formulation. Not the same thing at all!

    An appropriate mixture of the two is always appreciated (I think by anyone, not just me) ;).

  10. May 7, 2005 #9
    This discussion has me wondering about how exactly the Schrodinger equation tells us the number of distinct eigenvalues that we need when expressing the state of a system under observation. Or maybe the Schrodinger equation doesn't tell us at all; rather, as you say above, we have to enumerate the number of eigenvalues independently, *before* we use the Schrodinger equation.

    Let's imagine, purely hypothetically, that we are in the initial stages of discovery of quantum theory. Imagine that the results of the 2-slit experiment are well understood, but that spin has not yet been observed. Someone comes along and builds the first stern-gerlach apparatus and is about to shoot some charged particles through it. Question: Would we at this point be able to *predict* the phenomenon of spin? IOW, would we predict that the deflection of the charged particle would result in a continuous distribution, or would we be able to predict a quantization in the deflection?

    I guess what I'm asking is whether the phenomenon of "spin" is an "extra" assumption of QM, or whether it arises from the basic formalism of QM itself.

    So for example: particles of spin 1/2, 1, ... will have 2, 3, ... distinct eigenvalues corresponding to a spin measurement. My basic question is whether the number of eigenvalues arises from the Schrodinger equation, or whether alternatively the number of eigenvalues is an extra assumption that we have to throw in. Right now I'm thinking that it is the latter; that is, when we write the initial state of the particle in its pre-measurement state, we *assume* that it can be written as a superposition of, say, three (for the case of spin 1) distinct eigenvectors. In which case the predictive power of the Schrodinger equation is in a way diminished, no?

  11. May 21, 2005 #10
    Hey Patrick and lurkers,

    I have posted an initial draft of a paper that I sorta intend as a follow-up to Patrick's paper. It contains some of the ideas that I've posted about in this thread, with a few things pared down to try to keep it simple. (Although a few ideas are fleshed out in greater detail, too.) I would be interested in anyone's comments. Patrick, perhaps I can be the first to reference your paper!

    It can be downloaded from my briefcase:


    from the folder "Probability interpretation of the MWI."

    Any and all comments are welcome.


  12. Jun 21, 2005 #11
    Hello again,

    I have updated my MWI paper to include an *explicit* demonstration of how Patrick's aesthetically pleasing "alternate projection postulate" (APP) can be made into a workable theory of Nature -- that is, how quantum statistics may arise.

    It's in my briefcase next to my previous MWI manuscripts:
    folder: Probability interpretation of the MWI
    file: MMWI-3

    Comments welcome!

  13. Jun 23, 2005 #12
  14. Jun 26, 2005 #13
    I see that Wallace is presenting a "proof" of the Born probability rule that is similar to Deutsch's. Patrick, in his paper (which is now arXived), makes the argument -- which I would tend to agree with -- that Deutch's "reasonable assumptions" are essentially equivalent to an *assumption* of the Born rule.

  15. Jun 27, 2005 #14
    In the last few sections of the paper, Wallace makes a pretty extensive argument why we should take Deutsch's assumption (or a different one, "equivalence", that will also work) as a rationality principle. See for example p. 21 of the paper, or sections 5.2 and 5.3 of this one: any sort of probability assignment (like the APP) that doesn't lead to the Born rule, if I'm understanding this correctly, has to arbitrarily pick a precise preferred basis beyond just the approximate one provided by decoherence, and this leads to many other problems. (I think Patrick's paper mentioned something like this, too.)

    Have both of you looked at this, by the way?
    Last edited: Jun 27, 2005
  16. Jun 27, 2005 #15
    I haven't read Deutsch's stuff directly -- I really just know about it indirectly through Patrick's paper. But it looks like something I'd like to go through at some point.

    Actually, I've been having an email discussion with the author (Robin Hanson). His "mangled worlds" scheme is similar to mine and Patrick's in the sense that we are all using the APP. Mangled worlds is also similar to my scheme in this sense: in mangled worlds, instead of assigning one branch (world) to each measurement outcome, you start out with, say, N branches associated with each outcome (due to lots of misc. decoherence events that we typically do not consider for the sake of simplicity). Next, the branches with relatively smaller Born measure are more likely to get "mangled" (destroyed) compared to the ones with large Born measure. Thus, in mangled worlds, you end up having the number of worlds associated with a given measurement outcome being proportional to the Born measure of that outcome. In my scheme, I just assume (in the simplest version) that the number of worlds is proportional to the Born measure from the outset. So my scheme and mangled worlds end up at the same place, so to speak, but via slightly different ways.

  17. Jun 27, 2005 #16


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    There is a logical error in this approach (in the mathematical sense of "logical", not in the common sense which would say that it is bull**** :smile:) IMHO. I think everybody agrees that we have to have the Born rule to make QM work, so the point is not to argue that the Born rule must be correct.
    The point is that if you do not want to *postulate* the Born rule (for reasons very well explained by Hanson), you will have to DERIVE it from unitary quantum theory. But then you cannot use the argument "and this leads to many other problems", to avoid the negation of the Born rule! Because that is an argument to introduce the *postulate* (namely: if you don't introduce this postulate, you can run into many other problems !).
    It is what I tried to call 'reasonable assumptions' in my paper: namely, a kind of deus ex machina which says: "hey, look, if you're a reasonable guy, you HAVE to take the Born rule, otherwise you get something crazy out, and BOUM you have the Born rule". But that is not a PROOF of the derivation of the Born rule from unitary QM. It only indicates that the theory WITHOUT that rule makes crazy predictions, and hence is not a good theory.

    Unitary QM has no natural probability concept build in it, so you cannot REQUIRE anything "reasonable" that that probability concept should obey. Hanson makes about the same "leap" when he *requires* some stability of the approximate decoherence basis (which leads him, if I understand that well, to discard "small size terms" because they continuously mix). But "neglecting small values of the measure" somehow automatically turns that measure into a probability measure, so this, to me, is again equivalent to postulating the Born rule.
    That's not a bad thing to do, because we might somehow be able to find a more "natural" formulation of the Born postulate, so this might be esthetically pleasing (and maybe even give us deeper insight). But logically, I think you do nothing else but postulate the Born rule in disguise.

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