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## my paper on the Born rule...

Hi,
A while ago I discussed here about a paper I wrote, which you can find on the arxiv: quant-ph/0505059
 Proponents of the Everett interpretation of Quantum Theory have made efforts to show that to an observer in a branch, everything happens as if the projection postulate were true without postulating it. In this paper, we will indicate that it is only possible to deduce this rule if one introduces another postulate that is logically equivalent to introducing the projection postulate as an extra assumption. We do this by examining the consequences of changing the projection postulate into an alternative one, while keeping the unitary part of quantum theory, and indicate that this is a consistent (although strange) physical theory.
I submitted it to the Royal Society, and I received a notification of rejection, with the following comments from the referees, might be of interest for those who participated in the discussion. The emphasis is mine.
First referee:
 The paper critically assesses the attempt (Proc Roy Soc Lond 1999) by David Deutsch (followed up by various authors in later work) to derive the Born rule within the Everett interpretation via considerations of decision theory. The author interprets Deutsch as claiming that QM - whether or not the Everett interpretation is assumed - may be decomposed into a unitary part and a "projection postulate" part. He then proposes an "alternative projection postulate" (APP) which, he argues, is compatible with the unitary part of QM but which does not entail the Born rule. He claims that, since his APP is a counterexample, Deutsch's proposal and any variants of it must be rejected. A very similar project was undertaken by Barnum et al in a paper in Proc Roy Soc Lond in 2000. The author's APP has some mild technical advantages over Barnum et al's proposal, but these do not (in my view) merit a separate paper, especially since neither Barnum et al nor the author are proposing a viable alternative to the PP but simply making a logical point. More importantly, the post-2000 literature on Deutsch's argument has not attempted to criticise the details of Barnum et al's counterexample. Rather, it has claimed that Barnum et al, treating measurement as a black-box process, misread Deutsch. Deutsch sets out to analyse measurement as one more physical process (realised within unitary dynamics - as such, any rival proposal to the Born rule which is couched (as is the author's) in terms of measurement observables taken as primitive will not be relevant within the context of the Everett interpretation. It is fair to say that this point was somewhat obscure in Deutsch's 1999 paper, but it has been made explicitly in subsequent discussions, including some (by Wallace and Greaves) which the author cites. However, the author does not engage with this issue but continues to work in the Barnum et al tradition without further discussion. In conclusion: if the Barnum et al framework is valid then the author's paper does not seem to add sufficiently to their existing criticisms of Deutsch to justify publication. And if it is not valid, then it is at best unclear how the author's paper relates to Deutsch.
The second referee:
 The paper reviews an alternative projection postulate (APP) and contrasts it with the standard projection postulate (PP). Under the APP probabilities are uniform, instead of being proportional to the relative measure of vector components. APP is shown to be consistent with unitary symmetry and with measurements being defined in terms of projection operators, and it agrees with PP regarding results predicted with certainty. The paper also does a decent job of describing some of the strange empirical results that APP predicts. The main point, that we must rely on empirical data to favor PP over APP, is worth making. The paper, however, purports to do more than this. The abstract and introduction claim to deal a blow to the Everett programme, by showing that "there is no hope of deriving the PP directly from the rest of the machinery of quantum theory." Beyond the review of APP described above, however, the paper itself says very little about this subject. The introduction ends by promising "we will then examine where exactly it is in disagreement with Deutsch's reasonable assumptions,' or with Gleason's theorem." But the section at the end of the paper that is supposed to make good on this promise consists of only thirteen lines -- far too little to provide much exact examination. Worse, the paper does not mention or discuss any of the many other approaches that have been suggested for deriving the PP from the rest of quantum theory, within the Everett programme. The paper claims "APP is in fact the most natural probability rule that goes with the Everett interpretation: on each branching' of an observer due to a measurement, all of its alternative `worlds' receive and equal probability." However, many authors do not accept that equal probability per world is the most natural. Furthermore, many other authors do accept an equal probability rule, but then try to derive the PP from it, instead of the APP. For example, the review article at http://plato.stanford.edu/entries/qm-manyworlds/ says "Another idea for obtaining a probability law out of the formalism is to state, by analogy to the frequency interpretation of classical probability, that the probability of an outcome is proportional to the number of worlds with this outcome. This proposal immediately yields predictions that are different from what we observe in experiments. Some authors, arguing that counting is the only sensible way to introduce probability, consider this to be a fatal difficulty for the MWI, e.g., Belifante 1975. Graham 1973 suggested that the counting of worlds does yield correct probabilities if one takes into account detailed splitting of the worlds in realistic experiments, but other authors have criticized the MWI because of the failure of Graham's claim. Weissman 1999 has proposed a modification of quantum theory with additional non-linear decoherence (and hence even more worlds than standard MWI), which can lead asymptotically to worlds of equal mean measure for different outcomes." (Hanson 2003, which you incorrectly cite as discussing the Deutsch proof, is another such attempt.) I cannot recommend the paper for publication as it is, but I can hold out hope that the author could make an acceptable revision. Such a revision could simply be a review of the APP, including its implications. Such a review should mention many of the previous authors who have considered such a posulate. Alternatively, a revision could critique some of the attempts to derive PP from quantum theory. To accomplish this second goal, the author must first choose a set of previous papers that it is responding to. (It may not be feasible to respond to all previous papers on this topic.) Second, the author must explain exactly where there purported demonstration is claimed to fail. That is, at what point does a key assumption of theirs go beyond the basic machinery of quantum theory. Third, the author must explain why this key assumption is no more plausible than simply assuming the PP directly. This is what it would take to successfully show that such attempts to derive the PP from the machinery of quantum theory has failed. Here are two minor comments. The paper switches its notation from from APP to AQT and PP to SQT, for no apparent reason. It would make more sense to stick with one notation. Also, as there may be other alternatives proposed someday, it might be better to call APP a "uniform projection postulate" (UPP). Finally, the title should more specifically refer to this alternative projection posulate, however named.
On a personal note, although this paper was a bit outside of my field and thus "for fun", in my field too, I had several rejections of similar kind, which always make me think that the referee has missed the point I was trying to make (which must be due to the way I wrote it up, somehow).
The only point I tried to make was a logical one, as seems to be recognized by the first referee only, but then he seems to miss the point that in the end of the day, we want a theory that spits out results that are given by the PP, whether or not we take that "as primitive". So I don't see why considering the PP "as primitive" makes the reasoning "not relevant". The second referee seems to have understood this (that we have to rely on empirical data to endorse the PP), but he seems to have missed the point I was making a logical claim, and seems to concentrate on the minor remark when I said that "this APP seems to be the most natural probability rule going with MWI".
The very argument that some have tried to MODIFY QM introducing non-linear decoherence is *exactly what I claim*: that you need an extra hypothesis with unitary QM if you want to derive the PP. Finally, the proposition of revision, namely to limit myself to the consequences of the APP, take away the essential point of the paper which simply stated: since two different probability rules, the APP, and the PP, are both compatible with unitary QM, you cannot derive the PP logically from unitary QM without introducing an extra hypothesis.
The only truely valid critique I find here, is the one of the first referee who finds that my paper is not sufficiently different from Barnum's paper (something I ignored) - which is of course a valid reason of rejection (which I emphasised in red).
Most other points seem to miss the issue of the paper, I have the impression, and focus on details which are not relevant to the main point made. This often happens to me when I receive referee reports. Do others also have this impression, or am I such a terrible author ?

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 All attempt to derive the PP from unitary theory is condemed to failure. It is a simple mathematical (and physical) question. The information contained into a nonunitary evolutor is more rich that informaiton contained into a unitary evolutor. 'More' cannot be derived from 'less'. It took near 100 years that physicists understood that measurement problem CANNOT be solved via QM of closed systems. During 50 years or so had an intensive research in open systems and decoherence. Finally decoherence is in a dead way. I wait that in some 100 or 200 years physicists will understand that the old unitary Schrödinger equation is an approximation to realistic nonunitary evolutions. In fact, in some other fields this is known for decades... See page 17 of Nobel Lecture, 8 December, 1977 http://nobelprize.org/chemistry/laur...ne-lecture.pdf The measurement process is an irreversible process generating entropy. QM conserves entropy and is reversible, therefore QM cannot explain the PP without invoking it adittionally. But then one is invoking a postulate that violates others postulates of the axiomatic structure, doing QM both incomplete and internally inconsistent.

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 Quote by vanesch Hi, A while ago I discussed here about a paper I wrote, which you can find on the arxiv: quant-ph/0505059 I submitted it to the Royal Society, and I received a notification of rejection, with the following comments from the referees, might be of interest for those who participated in the discussion. The emphasis is mine. First referee: The second referee: On a personal note, although this paper was a bit outside of my field and thus "for fun", in my field too, I had several rejections of similar kind, which always make me think that the referee has missed the point I was trying to make (which must be due to the way I wrote it up, somehow). The only point I tried to make was a logical one, as seems to be recognized by the first referee only, but then he seems to miss the point that in the end of the day, we want a theory that spits out results that are given by the PP, whether or not we take that "as primitive". So I don't see why considering the PP "as primitive" makes the reasoning "not relevant". The second referee seems to have understood this (that we have to rely on empirical data to endorse the PP), but he seems to have missed the point I was making a logical claim, and seems to concentrate on the minor remark when I said that "this APP seems to be the most natural probability rule going with MWI". The very argument that some have tried to MODIFY QM introducing non-linear decoherence is *exactly what I claim*: that you need an extra hypothesis with unitary QM if you want to derive the PP. Finally, the proposition of revision, namely to limit myself to the consequences of the APP, take away the essential point of the paper which simply stated: since two different probability rules, the APP, and the PP, are both compatible with unitary QM, you cannot derive the PP logically from unitary QM without introducing an extra hypothesis. The only truely valid critique I find here, is the one of the first referee who finds that my paper is not sufficiently different from Barnum's paper (something I ignored) - which is of course a valid reason of rejection (which I emphasised in red). Most other points seem to miss the issue of the paper, I have the impression, and focus on details which are not relevant to the main point made. This often happens to me when I receive referee reports. Do others also have this impression, or am I such a terrible author ?
I always find that I have to ask an "outsider" to read my manuscript before I submit it. This is because what I find to be rather obvious, is really isn't. Authors have a clear idea in their heads what they're writing. Other people don't. So we tend to write things as if the reader already has an insight into our punch line. If you find that most people seem to miss the main point you're trying to make, chances are that you are not emphasizing it in the clearest fashion. This has happened even to the best of us.

I find that the most effective means to emphasize the main points I'm trying to get across is by clearly numbering them. I've been known on here to list the points one at a time:

(i) Point 1

(ii) Point 2

.. etc. Unfortunately, if you're writing to PRL, or trying to save publication costs, those take a lot of valuable spaces, so I also have listed them in line. As a referee, I also find them to be easier to focus on. I can easily go back and look at them again while I'm reading the rest of the paper to keep reminding myself that these are the points the authors are trying to make. It is no secret that most of us start a paper by reading the abstract, intro, and conclusion first (well, I certainly do). So you have to keep in mind that you literally have to reveal to the reader in the most direct way the message you are trying to get across in those sections of the paper.

Zz.

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## my paper on the Born rule...

 Quote by ZapperZ I always find that I have to ask an "outsider" to read my manuscript before I submit it. This is because what I find to be rather obvious, is really isn't. Authors have a clear idea in their heads what they're writing. Other people don't. So we tend to write things as if the reader already has an insight into our punch line.
This reminds me of a simple paper I wrote once with a student, about how the signal generating process should be included in a reliable simulation of the behaviour of the front end electronics of a neutron detector, because assuming that the detector just "sent out a delta-pulse" was giving results which deviated by a factor of 2 from observations, while including the signal formation did predict this factor 2. I found this maybe worth publishing - even though not big news - because other papers omitted exactly that: they only took into account the electronics, and supposed a deltafunction for the signal coming from the detector (which might have been justified in their application, no matter - but it was not mentioned in their papers).
So I carefully described the setup, and gave an explicit calculation of how the signal was generated in the detector, to show that this was the relevant part which allowed us to explain the discrepancy of a factor of 2. My point being that it was necessary to include this part in the description.
I got a rather nasty referee report, in which he explained me that I must be pretty naive to think that I was the first one explaining how signals were generated in radiation detectors

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 Quote by vanesch This reminds me of a simple paper I wrote once with a student, about how the signal generating process should be included in a reliable simulation of the behaviour of the front end electronics of a neutron detector, because assuming that the detector just "sent out a delta-pulse" was giving results which deviated by a factor of 2 from observations, while including the signal formation did predict this factor 2. I found this maybe worth publishing - even though not big news - because other papers omitted exactly that: they only took into account the electronics, and supposed a deltafunction for the signal coming from the detector (which might have been justified in their application, no matter - but it was not mentioned in their papers). So I carefully described the setup, and gave an explicit calculation of how the signal was generated in the detector, to show that this was the relevant part which allowed us to explain the discrepancy of a factor of 2. My point being that it was necessary to include this part in the description. I got a rather nasty referee report, in which he explained me that I must be pretty naive to think that I was the first one explaining how signals were generated in radiation detectors
I think it highly depends on WHERE you sent that in. If you sent it to, let's say, PRL, then I'd say you might get something like that. However, journals like EJP, or AJP, routinely publish pedagogy and techniques, especially when it is something relevant in physics education, be it at the undergraduate or graduate level.

I don't know what you submitted that paper to, but honestly, where you send your manuscript is almost as important as what you wrote.

Zz.

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 Quote by ZapperZ I don't know what you submitted that paper to, but honestly, where you send your manuscript is almost as important as what you wrote. Zz.
It was Nuclear Instruments and Methods, quite appropriate, I'd think

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 Quote by vanesch It was Nuclear Instruments and Methods, quite appropriate, I'd think
Well, I'm not sure about that.

NIM is supposed to be a journal on new techniques, or an improvement of a technique. Your paper, from your description, is simply clarifying the missing piece that isn't commonly mentioned. In other words, there's nothing new or a new extension on an existing technique. If this is the case, then the referee is correct in asking you if you think that what you're describing is not known.

I still think AJP or EJP might have been more suitable. You could emphasize the point that what you're describing is important and often omitted in the details of the experiment being reported in many papers that use the same technique. Such a paper would have been appropriate for those two journals.

Zz.

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 Quote by ZapperZ Well, I'm not sure about that. NIM is supposed to be a journal on new techniques, or an improvement of a technique. Your paper, from your description, is simply clarifying the missing piece that isn't commonly mentioned. In other words, there's nothing new or a new extension on an existing technique.
This is an interesting comment ! Nobody ever made that, and it explains several other problems I had with NIM ; indeed, each time I erred more on the explanatory part than the "here's a new method" part, I got rebiffed (or one asked me to remove or reduce the explanatory part and to emphasize the practical application). It is true that amongst my collegues, I'm by far the most "explanation" oriented.

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 Quote by vanesch This is an interesting comment ! Nobody ever made that, and it explains several other problems I had with NIM ; indeed, each time I erred more on the explanatory part than the "here's a new method" part, I got rebiffed (or one asked me to remove or reduce the explanatory part and to emphasize the practical application). It is true that amongst my collegues, I'm by far the most "explanation" oriented.
I tend to be quite verbose too in some of the things I write. But as a referee, when I pick up a paper that I'm reviewing, I would like to be hit right off the bat with the punch line. Tell me in no uncertain terms what you are trying to say here, and why it is important. I tend to pay attention to statements such as these:

"To be best of our knowledge, this is the first report on.... "

"This results contradicts an earlier report...."

"This is the most accurate result so far on.... "

"This is a new result..... "

etc. These should be either in the abstract, or somewhere in the intro or the 1st 2 paragraph. If not, I will lose track of what you're trying to say, or why it is important. (Ironically, I've just finished reposting in my Journal an article I wrote a while back titled "It may be interesting, but is it important?") :)

If you write a paper in such a way that the referee has to put an effort to find the point you are making, or why it is important, then you are just making it more difficult for that referee to recommend your paper to be published. It is that simple.

Zz.

 Quote by vanesch Hi, A while ago I discussed here about a paper I wrote, which you can find on the arxiv: quant-ph/0505059 I submitted it to the Royal Society, and I received a notification of rejection ...
Drat! I have my follow-up to your paper nearly ready for submission. Every weekend for the past several weeks now, I've told myself I'm going to make the final revisions and send it out, and then I run across something else that I need to incorporate. Like the Weissman paper, for instance ... In fact, I should probably make it at least evident that I'm aware of Weissman, Deutsch, Barnum, Hanson, and all the other authors mentioned in the reviews.

So Patrick, do you think you're going to resubmit? I hope you do - I think (obviously) that it is a very important topic. I'll try to throw out my comments on the reviewers' comments on this thread, as I go through the literature (may be a slow process ...)

BTW, does it normally take that long for review? Hasn't it been, what, 5 months?

David

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 Quote by straycat ... In fact, I should probably make it at least evident that I'm aware of Weissman, Deutsch, Barnum, Hanson, and all the other authors mentioned in the reviews. So Patrick, do you think you're going to resubmit? I hope you do - I think (obviously) that it is a very important topic.
First I'll check out the Barnum paper. If (according to referee 1) my paper contains the same argument as his, well I conclude that 1) I'm in not a bad company (just 5 years late ) and 2) I won't resubmit.

If not, well, I wouldn't really know where to submit. Maybe foundations of physics.

 BTW, does it normally take that long for review? Hasn't it been, what, 5 months? David
It's usually a bad sign when it takes that long But it is strongly dependent on the journal. Some journals first as one referee, and if that one doesn't give positive returns, they ask a second one for a second opinion. Others do it in parallel.

 Quote by vanesch First I'll check out the Barnum paper. If (according to referee 1) my paper contains the same argument as his, well I conclude that 1) I'm in not a bad company (just 5 years late ) and 2) I won't resubmit.
A review article might not be such a bad idea. You could review the motivation behind the APP, review the various attempts to implement it, and perhaps include your own contribution in a separate section.

 Quote by vanesch If not, well, I wouldn't really know where to submit. Maybe foundations of physics.
What exactly is the reputation of FoP? Is it a lesser tier than the Royal Society?

DS

Hey Patrick,

I've been trying to make sense of some of the comments made by your first referree:

 Quote by vanesch A very similar project was undertaken by Barnum et al in a paper in Proc Roy Soc Lond in 2000. The author's APP has some mild technical advantages over Barnum et al's proposal, but these do not (in my view) merit a separate paper, especially since neither Barnum et al nor the author are proposing a viable alternative to the PP but simply making a logical point. More importantly, the post-2000 literature on Deutsch's argument has not attempted to criticise the details of Barnum et al's counterexample. Rather, it has claimed that Barnum et al, treating measurement as a black-box process, misread Deutsch. Deutsch sets out to analyse measurement as one more physical process (realised within unitary dynamics - as such, any rival proposal to the Born rule which is couched (as is the author's) in terms of measurement observables taken as primitive will not be relevant within the context of the Everett interpretation. It is fair to say that this point was somewhat obscure in Deutsch's 1999 paper, but it has been made explicitly in subsequent discussions, including some (by Wallace and Greaves) which the author cites.
I looked at one of Greaves' papers, "Understanding Deutsch's probability in a deterministic multiverse" which is archived at the PhilSci archives at http://philsci-archive.pitt.edu/archive/00001742/ . Section 5.1 "Measurement neutrality" and section 5.2: "Measurement Neutrality versus Egalitarianism" really helped me to understand the above point. Basically, Greaves explains that one of the essential assumptions in Deutsch-Wallace decision theory is the postulate of "measurement neutrality," which is "the assumption that a rational agent should be indifferent between any two quantum games that agree on the state |phi> to be measured, measurement operator X and payoff function P, regardless of how X is to me measured on |phi>." afaict, this means that if we think of the measurement process as a "black box," then Deutsch assumes that a rational agent should in principle be indifferent to the details of the innards of this black box.

In sec 5.2, Greaves very clearly argues that measurement neutrality automatically *excludes* the APP (where the APP = egalitarianism) as a possible probability rule. Therefore, measurement neutrality, as innocuous as it may appear at first glance, is not so innocuous at all.

I've referenced Greaves (among others) in the revised introduction to my paper [1] on the probability interpretation of the MWI. I'm glad you posted your referree comments, Patrick -- they've helped me on my paper!

-- David

[1] To be submitted to Foundations of Physics Letters -- latest draft available at
http://briefcase.yahoo.com/straycat_md
Folder: Probability interpretation of the MWI
archived (slightly older) versions also at:
http://philsci-archive.pitt.edu/
http://www.sciprint.org/

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 Quote by straycat which is "the assumption that a rational agent should be indifferent between any two quantum games that agree on the state |phi> to be measured, measurement operator X and payoff function P, regardless of how X is to me measured on |phi>."
Yes, that's exactly the point. As I showed in my paper, that's NOT the case with the APP, (as I explicitly show with the example of X and Y where one is a refinement of the other) where the probabilities are dependent on context (on the other variables that are being measured).

 In sec 5.2, Greaves very clearly argues that measurement neutrality automatically *excludes* the APP (where the APP = egalitarianism) as a possible probability rule. Therefore, measurement neutrality, as innocuous as it may appear at first glance, is not so innocuous at all.
Ok, that's exactly my argument too. So I have some extra homework to make with this as reference.

Thanks for pointing that out!

Juan wrote:

 All attempt to derive the PP from unitary theory is condemed to failure.
I cannot agree with that statement, altough I recognize a conceptual difficulty there.
For me, this problem is similar to the problem of irreversibility seen from the classical mechanics point of view. Non-unitary evolution might be a good approximation (maybe even *exact*!) when an interaction with a huge system (huge freedom) is involved.

My favorite example is the decay of atomic states: clearly the interaction of the discrete atomic system with the continuum system of electromagnetic radiation brings the decay. This decay is very conveniently represented by a "non hermitian" hamiltonian: this allows modeling of an atom (for the Stark effect e.g.) without including the whole field. This represents correctly the reality, altough the fundamental laws are unitary.

For many people, the interaction with a 'classical' or 'macroscopic' system is all that is needed to derive the PP. I think this is the most probable explanation for the PP. Landau considered this so obvious that it comes in the first chapters in his QM book.

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 Quote by lalbatros Juan wrote: I cannot agree with that statement, altough I recognize a conceptual difficulty there. For me, this problem is similar to the problem of irreversibility seen from the classical mechanics point of view.
The irreversibility in classical statistical mechanics comes about from the very specific initial condition, which is highly improbable.

 Non-unitary evolution might be a good approximation (maybe even *exact*!) when an interaction with a huge system (huge freedom) is involved.
I don't see how this can come about. The hamiltonian gives rise to a unitary operator, no matter how complicated the system. Especially the EM radiation field can always be considered as a discrete system with a huge number of degrees of freedom (it shouldn't make any difference if you put your system in a box with diameter one hundred billion lightyears or not, should it).

 My favorite example is the decay of atomic states: clearly the interaction of the discrete atomic system with the continuum system of electromagnetic radiation brings the decay. This decay is very conveniently represented by a "non hermitian" hamiltonian: this allows modeling of an atom (for the Stark effect e.g.) without including the whole field. This represents correctly the reality, altough the fundamental laws are unitary.
No, it is a shortcut, where you *apply* already the PP in its derivation.

 For many people, the interaction with a 'classical' or 'macroscopic' system is all that is needed to derive the PP. I think this is the most probable explanation for the PP. Landau considered this so obvious that it comes in the first chapters in his QM book.
This is the standard "explanation". But it is *postulated* and not *derived* from unitary QM. What qualifies a system as "macroscopic" and "classical" (without making circular reasoning ?) and why shouldn't it obey standard quantum theory ?
Or is there an upper limit to the number product hilbert spaces (number of particles) before the exponentiation of a hermitean operator suddenly doesn't become unitary anymore ?

 Hey Patrick et al, I'm posting an idea on this thread that has occurred to me on a potential consequence of the APP. Suppose that Alice is doing two Aspect-like experiments, one with Bob, and another simultaneously with Bert. Alice and Bob are 1 km apart, and Bert is 0.1 km farther away than Bob. Otherwise the experiments are the same, done at the same time. Bob and Bert flash the results of their measurements to Alice as soon as they get them. Before Alice receives these messages (which we suppose travel at the speed of light), Bob and Bert each exist in a superposition of the "B-- sees up"/"B-- sees down" state. Because of the general relativistic restriction on the speed of light, from the point of view of Alice, Bob's state will collapse prior to Bert's state. Pretty elementary. The point I wish to make is that relativity imposes a restriction on the order with which collapse occurs, from the point of view of Alice. So let's take this point and extrapolate. Suppose now that we have an observer Amandra who observes some variable X characteristic of a particle. But imagine that the value of X is not communicated to Amandra all at once, but rather in little chunks. That is, suppose that X_min is the lowest allowable value of X, and that it is quantized, ie it takes values in [X_min, X_min +1, X_min + 2, ...]. Imagine furthermore that Amandra's observation of X comes in a series of steps, like this: she observes either X = X_min, or X \in [X_min+1, X_min +2, ...]; if the latter, she next observes either X = X_min + 1 or X \in [X_min+2, X_min +3, ...]; if the latter, she next observes either X = X_min + 2, or X \in [X_min+3, X_min +4, ...]; and so on. If you draw the MWI-style world-splitting diagram to characterize this process **and apply the APP**, then it is apparent that lower possible values of X are *more probable* than higher values. In effect, X is MINIMIZED. We could equally well suppose that X might be maximized, if the information regarding the value of X were propagated to Amanda in the reverse order. So here's the Big Idea: the APP, perhaps, offers a mechanism by which Nature enforces various extremum laws. If X is the action, for instance, then we have the principle of least action. What d'ya think? David