# Can Outcome Counting Replace the Born Rule in Quantum Mechanics?

• straycat
In summary: Well, one feature is that my emergent spacetime is "continuous," in the sense that there is no sharp divide between the "quantum" and "classical" realms. In other words, the Born rule applies to both particles and waves in my emergent spacetime. Another feature is that my emergent spacetime is "non-Euclidean," in the sense that it has a "different" geometry from the geometry of classical spacetime. In fact, I claim that this geometry is based on a new form of symmetry called "geometric symmetry breaking."In summary, my paper is about a model that replaces the Born rule with outcome counting as the fundamental probability rule at the fine
straycat
Hello all,

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 to Les 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:

ABSTRACT: The "quantitative problem" of the MWI is to justify the interpretation of the Born rule measure $|a_{n}|^{2}$ -- the squared norm of the amplitude associated with the $n^{th}$ out of $N$ 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.

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 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 :-) )

"Are world counts incoherent?"

Attempts to make the Born rule emerge explicitly from
outcome counting

Outcome counting, the action principle, and GR

QM and action principles

A democracy of spacetimes?

GR and analytic continuation

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 probability must be based, in one way or another, on outcome counting. To me, it deserves the status of a symmetry 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 be feasible to 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 that assume the 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:

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.

See the abstract above.

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.

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 .

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

Tom Mattson said:
3. All references to relevant prior work must be documented in the opening post.

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.

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

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.

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).

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

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.

I make no such claims.

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.

Well I hope not … but that’s why I’m here!

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).

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.

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.

David Strayhorn

#### Attachments

• Born2006Sep.pdf
339.4 KB · Views: 416
journal submission: slow review process

On 5 Oct 06 I submitted my paper, along with a short companion, to The Annales de la Fondation Louis de Broglie http://www.ensmp.fr/aflb/AFLB-Web/en-annales-index.htm , at about the same time that I originally posted this thread - more than seven months ago. Having heard nothing from the journal, I emailed to ask the status of my submission, and received the reply:

We have well received your two manuscripts 6 months ago and you have received an acknowledgment of receipt dated 16/10/07 [sic: 16/10/06] but unfortunately we have met some difficulties to find a referee that is the reason why it lasts more time.

Seven months, and still haven't found a referree. Has anyone ever heard of such a thing happening?

I know that my papers are dense (as evidenced by the lack of response I've gotten on this forum! ) -- so should I conclude that is the reason why AFLB cannot find a referree? OTOH, the journal website lists the current issue as 2006. Should I conclude the journal is about to go belly-up? Or that the journal is simply slow to do things in general?

David

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straycat said:
more than seven months ago

oops -- should have said 6, not 7.

I should have posted this a long time ago: there is an online version of my paper, posted at philica:

"Derivation of the Born rule from outcome counting and a solution to the quantitative problem of the multiple worlds interpretation"
http://www.philica.com/display_article.php?article_id=28

I have a second paper archived at philica as well:

"An illustration of the quantitative problem of the multiple worlds interpretation of quantum mechanics and the motivation for outcome counting"
http://www.philica.com/display_article.php?article_id=27

I would suggest to anyone trying to wade through my ideas :shy: to start with the latter (id=27) paper. Its purpose really is to provide the justification for launching into the longer and more complex paper (id=28).

On a different note: the Perimeter Institute will be hosting a conference this september on the Everett interpretation, ie the multiple worlds interpretation:
http://www.perimeterinstitute.ca/en/Events/Many_Worlds_at_50/Many_Worlds_at_50/

I'm going to see if I can go to that!

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paper accepted!

Good news! A revised version of my second paper, "An illustration of the quantitative problem of the multiple worlds interpretation of quantum mechanics and the motivation for outcome counting," has been accepted for publication in Annales de la Fondation Louis de Broglie.

BTW I had the opportunity to attend the Perimeter Institute conference on the many-worlds interpretation last fall. I thought it was great that they allowed a non-professional like me (and one or two others) to attend. It gave me an opportunity to meet some of the leaders in this particular field of research and to place my thoughts and ideas in the context of the state of the field -- iow, to relate what I'm doing to what everyone else is doing. This is the sort of thing that physics outsiders often fail to do, and I am determined not to make the same mistake. Aided by what I learned at PI, I have done a complete overhaul of my above two manuscripts into a new one, "Egalitarianism offers a coherent alternative to decision theory as a solution to the problem of probability of the many worlds interpretation." It is currently under review at Studies in History and Philosophy of Modern Physics.

David

straycat said:
Good news! A revised version of my second paper, "An illustration of the quantitative problem of the multiple worlds interpretation of quantum mechanics and the motivation for outcome counting," has been accepted for publication in Annales de la Fondation Louis de Broglie.

Dude!

Congratulations! Ans many happy returns!

Thanks Carl! Here is the referee's original report. I sent a revised draft that was accepted.

"A Referee report on D. Strayhorn's paper: "An Illustration of..."

The paper discusses a variation of the "Many World Interpretation"
(MWI) of quantum mechanics. As such, the paper does not belong to
the narrow definition of physics but to its (philosophical)
interpretation. In my opinion, this kind of papers should be welcome
in a physical journal, provided it is restricted to a small percentage
of the entire publication. Thus, I think that the paper can be accepted.
However, I have some remarks and I wish that the Author consider them.
Maybe he will decide to revise some points of his paper.

1. By its definition, the Author's idea of "Outcome Counting" has no
theoretical value in the sense of making a reliable prediction. At
most, it can be regarded as a tool depending on phenomenological
results of earlier experiments. As such, this idea cannot be
incorporated in theoretical physics. Thus, it may only be considered
as just one interpretation of physics. In particular, the Author's
suggestion that Outcome Counting be elevated "to the level of a
symmetry principle" like the principle of relativity (see the middle
of p. 7) cannot be accepted.

2. The Author discusses a hypothetical spin-1/2 experiment and claims
that the Born probability interpretation is inconsistent with the
idea "that each possible experimental outcome should be equally
likely" (see the middle of p. 2). He also remarks that "from an
empirical perspective, the success of the Born rule is undisputed"
(see p. 2, end of the first paragraph).

Referring to this matter, I wish to show a case where the Born
statistical approach agrees with the "equally likely" idea of
all possible experimental results. The example discusses a continuous
variable.

Consider a double slit experiment and examine the results on a film.
Looking not very closely at the film, one sees an interference
pattern of a continuously varying gray level. The gray level
represents the Born statistical interpretation. On the other hand,
a close observation of the film reveals many isolated black points
obtained from the collision of each photon with the film. This is
the "equally likely" approach.

Therefore, in a case of an experiment measuring a continuous
variable, the two approaches are consistent with each other. Hence, the
discrete spin results may deserve an appropriate treatment of infinities
related to a Dirac delta function."

## 1. What is the Born rule and why is it important in science?

The Born rule, also known as the Born interpretation, is a fundamental principle in quantum mechanics that relates the mathematical description of a quantum system to its physical properties. It is important because it allows scientists to make predictions about the behavior of quantum systems and has been confirmed by numerous experiments.

## 2. How was the Born rule derived?

The Born rule was derived by German physicist Max Born in 1926. He used the mathematical framework of quantum mechanics developed by Werner Heisenberg and Erwin Schrödinger to show that the square of the wave function of a quantum system represents the probability of finding the system in a particular state.

## 3. What are the assumptions made in the derivation of the Born rule?

The derivation of the Born rule relies on two main assumptions: the wave function of a quantum system is a complete description of its state, and the wave function evolves according to the Schrödinger equation. These assumptions are supported by experimental evidence and have been confirmed by the success of quantum mechanics in predicting the behavior of particles.

## 4. Can the Born rule be applied to all quantum systems?

Yes, the Born rule can be applied to all quantum systems, regardless of their complexity. It has been successfully used to predict the behavior of particles ranging from atoms to subatomic particles. However, it should be noted that the Born rule is a probabilistic interpretation and cannot predict the exact outcome of a measurement, only the probability of obtaining a particular result.

## 5. Are there any alternative interpretations or derivations of the Born rule?

Yes, there are alternative interpretations and derivations of the Born rule, such as the many-worlds interpretation and the consistent histories approach. These interpretations offer different ways of understanding the meaning and implications of the Born rule, but they all ultimately rely on the same mathematical framework of quantum mechanics.

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