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Attached is the draft of a paper entitled,Derivation of the Born rule from outcome counting and a solution to the quantitative problem of the MWI, (MWI = multiple worlds interpretation), which I am considering for submission toLes Annales de la Fondation Louis de Broglie. I would welcome any input that any of you may have.

Here is the abstract to my paper:

This paper has undergone many drafts, and the current version is quite a bit shorter than previous versions. This is primarily because of the space limitations (6000 words) imposed by the journal; although I should say that the process of boiling it down to its essentials has proven to be a fruitful exercise in and of itself. One consequence, however, is that there are several arguments that I present only in skeleton form, and some that I have skipped entirely. So I would be especially interested in critiques of my presentation itself -- i.e., whether I have presented my arguments in a way that is ABSTRACT: The "quantitative problem" of the MWI is to justify the interpretation of the Born rule measure [itex]|a_{n}|^{2}[/itex] -- the squared norm of the amplitude associated with the [itex]n^{th}[/itex] out of [itex]N[/itex] possible results -- as a probability. The essential difficulty is that the basic framework of the MWI would seem to suggest an alternative probability rule, outcome counting, which is that each separate outcome should be equally likely. In this paper, a model is proposed that replaces the Born rule with outcome counting as the fundamental probability rule at the fine-grained level, and yet recovers the Born rule as a coarse-grained approximation. This model is proposed, not only as a solution to the quantitative problem, but also as a novel derivation of the Born rule.understandable. Of course, I am also interested in critiques of the arguments themselves!

There are a number of PF threads that I have been involved in that directly or indirectly pertain to my paper -- in particular, to philosophical issues surrounding "outcome counting." See, eg:

"My paper on the Born rule ..." ("my" = Patrick Van Esch's :-) )

https://www.physicsforums.com/showthread.php?t=95585

"Are world counts incoherent?"

https://www.physicsforums.com/showthread.php?t=101339

Attempts to make the Born rule emerge explicitly from

outcome counting

https://www.physicsforums.com/showthread.php?t=101982

Outcome counting, the action principle, and GR

https://www.physicsforums.com/showthread.php?t=113224

QM and action principles

https://www.physicsforums.com/showthread.php?t=112257

A democracy of spacetimes?

https://www.physicsforums.com/showthread.php?t=112556

GR and analytic continuation

https://www.physicsforums.com/showthread.php?t=112254

Many of my ideas have taken shape during some very interesting discussions in zapper's group undernetphysics -- see especially message #1382.

What is the significance of my paper? Twofold.

1.Philosophical. My theory was initially motivated by the conviction that the concept of probabilitymustbe based, in one way or another, on outcome counting. To me, it deserves the status of asymmetry principle, on a par with, say, the principle of relativity. Of course, I realize that not everyone shares this conviction; it's really a matter of taste more than anything else. But at the very least, the fact that it may befeasibleto devise a theory based on outcome counting should, I think, have an impact on philosophical discussions of "what probability really is."

I might add that the question: "whence the Born rule?" has received renewed attention of late. There have been a number of attempts to derive the Born rule over the years, two notable attempts being Gleason’s theorem and -- more recently -- Deutsch-Wallace decision theory. Deutsch-Wallace decision theory and related issues regarding the interpretation of probability in the MWI will be the focus of a conference in 2007, slated to coincide with the 50th anniversary of the publication of Everett's relative state formulation paper that formed the basis of the MWI:

http://www.fqxi.org/aw-saunders2.html

A very good argument can be put forward that these so-called “derivations” of the Born rule are not in fact derivations, because they contain hidden assumptions thatassumethe Born rule, so that their arguments are inherently based on circular reasoning. I argue that my theory does NOT contain such hidden assumptions, and is therefore a genuine derivation.

2.Physical. I argue that my theory could --potentially-- shed some light on what a theory of quantum gravity (QG) might look like. Now, it would be way too bold for me to say that my theory is itself a theory of QG. Rather, my theory starts with the assumption that “a theory of QG exists,” and I assume that this theory has certain characteristics. In particular, I assume a basic framework that looks a lot like (my understanding of) a loop quantum gravity (LQG) theory of QG, including the notion of an emergent spacetime that we can use as an approximation to the underlying theory. I then postulate certain features of this emergent spacetime, and use these features to derive quantum statistics in the form of the Born rule.

Again, it is important that my postulates do not "sneak in" the Born rule. So what are these features of emergent spacetime that I postulate? A central assumption is that each of the "all possible paths" of the Feynman path integral (FPI) (which "live" in the emergent spacetime) has its counterpart in each of the quantum spacetimes of the underlying theory. Probably the next most significant assumption that I make has to do with how particle trajectories are modeled in each of the quantum spacetimes of the underlying theory. How do these work ... well, it's all in my paper.

Let me address the terms of the guidelines:

See the abstract above. Tom Mattson said:1. The opening post must contain an abstract stating the results obtained and how the new theory is at variance with currently accepted theories.

My proposal fits into the latter category. The great majority of the paper is concerned with demonstrating empirical equivalence to standard quantum mechanics, at least in the approximation. As it now stands, I cannot suggest experiments to distinguish between my theory and standard QM; some new ingredient would need to be added for that to happen. That doesn't make me feel too bad though; string theory has about a zillion people working on it, and still no testable predictions. I'm just one lone guy doing this in my spare time . Tom Mattson said:2. The opening post must contain a section that either cites experiments that have been done that decide between the new and old theories, or it must propose experiments that could be done to decide between the two. If the submission contains a theory that is empirically equivalent to an existing theory, then this section may be substituted with a section that demonstrates the empirical equivalence and that compares and contrasts the insights gained from the submitted and existing theories.

As for the potential insights -- see the discussion above.

I should point out that I know of two independent proposals that, like mine, assume outcome counting as the fundamental probability rule within an MWI context. These are the “mangled worlds” theory of Robin Hanson, and an independent proposal by Mike Weissman. Robin and Mike have each participated in some of the PF threads listed above. Tom Mattson said:3. All references to relevant prior work must be documented in the opening post.

Robin Hanson's mangled worlds proposal, published in Foundations of Physics:

http://xxx.lanl.gov/abs/quant-ph/0108070

http://xxx.lanl.gov/abs/quant-ph/0303114

Robin, btw, has presented his work at the Perimeter Institute ;-).

Mike's paper, published in Foundations of Physics Letters:

http://xxx.lanl.gov/abs/quant-ph/9906127

The primary focus of my paper is a derivation of the Born rule. As I said earlier, some parts of the derivation are presented in outline format only, due to space limitations. I would be interested in whether you think I should be more explicit for this introductory paper of mine (versus, say, being more explicit in a followup paper). Tom Mattson said:4. Quantitative predictions must be derived, wherever appropriate, and mathematical expressions and equations must be presented legibly, using LaTeX whenever necessary. For instructions and sample code see this thread. This should be done in the opening post.

(Oh yea -- I use LaTeX, with figures drawn in Powerpoint.)

I make no such claims. Tom Mattson said:5. New theories must not be already strongly inconsistent with the results of prior experiments.

6. If a new theory is strongly inconsistent with prior experiments, but the theorist is insisting that the experiments were either misconducted or misinterpreted by the scientific community, then the thread will be rejected. Instead the theorist should rebut the contradicting scientists in an appropriate journal.

Well I hope not … but that’s why I’m here! Tom Mattson said:7. Theories containing obvious mathematical or logical errors will not be accepted.

8. Threads which contain obvious misrepresentations or gross misunderstanding of basic accepted science, especially when used in attempt to compare one's personal theory to currently accepted science, will not be accepted.

If my paper exhibits any gross misunderstandings of basic science, it probably involves LQG, with which I am less familiar than, say, the FPI (Feynman path integral).

Well I might add that for anyone interested, some of the older (and much much much longer!) drafts of my theory are archived for public viewing in my yahoo! briefcase. But I do not offer them as a formal part of the current submission. Tom Mattson said:9. External links will be permitted only for lengthy derivations and for diagrams. Any other expository text pertaining to the submitted theory must be posted at Physics Forums. Please note that this is a temporary Guideline that will remain in place only while we work on enlarging the maximum allowable attachment size in the IR Forum. Once that happens, we will require that all material pertaining to the theory be either posted at Physics Forums or attached to the thread.

David Strayhorn

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# Derivation of the Born rule

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