The closest there is to a proof of Born's rule is Gleason's theorem. Born's
rule in one sense make a lot of sense, but as yet there is no airtight
proof of it. I will try to respond more later.
LC
On Tuesday, April 12, 2022 at 3:41:14 PM UTC-5 geka...@gmail.com wrote:
> On Friday, April 8, 2022 at 3:19:08 AM UTC+3 Lawrence Crowell wrote:
>
>> This is an appeal to some sort of imperative that demands the Born Rule
>> because the counterfactual lack this certainty. This is a sort of "It must
>> be true" type of argument.
>
>
> Thanks for the comments! I wonder though, do you agree with my criticisms
> of previous proposals for deriving the Born Rule, or are you undecided? I
> will challenge you (and you all) on this matter, later in this message.
>
> First, a correction: I have not referred to *counterfactuals* (I think
> that you meant "alternatives") but now that you mention them, I may have
> implied one:
> "If QM were not a workable theory, *we would have no direct, experimental
> clue that it is a fundamental theory in physics*".
> (Not the typical use of a counterfactual, which is in an "if..." clause,
> as in "*If I was a rich man*...".)
>
> What I say is not exactly
> > "It must be true"
> but rather
> "Although I cannot be certain, it seems to be in my interests to form this
> assessment now, when I decide how to act in the present situation".
>
> If you find this argument too loose: I have pointed out that it is the
> same kind of argument that a judge uses to form a decision based on the
> evidence, or an engineer uses, to trust the theory of real numbers, for her
> project.
>
> My aim has been to complete *Everett's argument,* which I outline next.
> Imagine that we repeat the same trial N times, and we record the ratio
> {statistical "frequency") r of one among the possible outcomes
> (eigenstates). Conventional QM assigns a probability R for this outcome, so
> we need an explanation why r SEEMS to approach R in the long run (though we
> know that in very many worlds it will not be so!). Everett noted that, for
> any positive real ε (however small), the measure of all "outlier"
> sequences, that is: for which r is outside
> [R-ε, R+ε],
> is small, with limit zero as N increases to infinity. However, *a problem
> remains:* why "small measure" or "vanishing measure" have any
> significance in the interpretation of QM? *My proposal answers this
> question,* finding an argument about "small measure" within the reasoned
> assessment that QM is a workable theory.
>
> *Here is my challenge to you.* I ask you if you agree with either of the
> following two proposals (for deriving the Born Rule in a MWI).
>
> First, Deutsch's (1999) proposal, here in a simplified version. Imagine a
> simulated tossing of a fair coin, using a qubit instead of a coin, with
> which you either win or lose one dollar. If this bet has a definite, single
> value to you (presumably, by some kind of intuitive averaging over possible
> futures) it will necessarily be zero, for symmetry reasons. Caveat: Deutsch
> points out that we do not derive probability strictly speaking. I accept
> the reasoning, but not the premise: I am uncomfortable with averaging my
> future selves, and there is no direct rationale why I SHOULD do so. So, *what
> do you think?*
>
> Second (and last), proposals such as Zurek's are of the following pattern
> (here I reuse the previous example): I am uncertain about the outcome, and
> I expect the theory to give me some clue, which will be probability -- what
> else? For symmetry reasons, the probability here must be 1/2. My objection
> is that there is no randomisation in MWI (no shuffling, stirring, or God
> playing dice) so that the use of probability is not rationally justified.
> Again* I ask for your opinion.*
>
> Clarification. Instead of probability proper, I derive the following. With
> regard to any given application, an Everettian agent may expect "with moral
> certainty" (remember the judge and the engineer!) that statistical
> frequency in the long run will be as close to the Born probability as one
> needs it to be (in the particular application). Some people may think
> "po-tah-toes, pot-eight-os", but at some level of thinking *this* is the
> crucial issue. In particular, a serious consequence for decision theory
> results from failing to find any rationale for probability proper!
>
> George K.
>
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