On 2/14/2020 6:35 AM, Bruno Marchal wrote:

On 13 Feb 2020, at 23:59, Bruce Kellett <bhkellet...@gmail.com <mailto:bhkellet...@gmail.com>> wrote:

On Tue, Feb 11, 2020 at 11:16 PM Bruno Marchal <marc...@ulb.ac.be <mailto:marc...@ulb.ac.be>> wrote:

    On 7 Feb 2020, at 12:07, Bruce Kellett <bhkellet...@gmail.com
    <mailto:bhkellet...@gmail.com>> wrote:

    I don't think you have fully come to terms with Kent's argument.
    How do you determine the measure on the observed outcomes? The
    argument that such 'outlier' sequences are of small measure
    fails at the first hurdle, because all sequences have equal
    measure -- all are equally likely. In fact, all occur with unit
    probability in MWI.

    Each individual sequence of head/tail would also occur with
    probability, in the corresponding WM scenario, and in the coin
    tossing experience.

    In the MWI, what you describe is what has motivated the
    introduction of a frequency operator, and that is the right thing
    to do in QM.


I remembered reading something about such a "frequency operator" but couldn't find the reference.

I have given it. It is in Graham paper in the selected papers by DeWitt and Graham on the MW (Princeton, 1973).



I see it was in a paper by David Albert, who writes:

"Here's an idea: suppose we measure the x-spin of each of an infinite ensemble of electrons, where each of the electrons in the ensemble is initially prepared in the state (alpha|x-up> + beta|x-down>). Then it can easily be shown that in the limit as the number of measurements already performed goes to infinity, the state of the world approaches an eigenstate of the frequency of (say) up-results, with eigenvalue |alpha|^2. And note that the limit we are dealing with here is a perfectly concrete flat-footed limit of a sequence of vectors in Hilbert space, not a limit of probabilities of the sort that we are used to dealing with in applications of the probabilistic law of large numbers. And the though has occurred to a number of investigators over the years that perhaps all it *means* to say that the probability of an up-result in a measurement of the x-spin of an electron in the state (alpha|x-up> + beta|x-down>) is |alpha|^2 is that if an infinite ensemble of such experiments were to be performed, the state of the world would with certainty approach an eigenstate of the frequency of (say) up-results, with eigenvalue |alpha|^2.

Yes, that is the idea. I think it was shown (with some rigour) first by Paulette Février (a student of De Broglie), but unfortunately, her master (De Broglie) came back to the hidden variable theory the “onde pilote”), and the work by Paulette Février has remained forgotten.



But the business of parlaying this thought into a fully worked-out account of probability in the Everett picture quickly runs into very familiar and very discouraging sorts of trouble. One doesn't know (for example) about finite runs of experiments,

That is not correct, or correct for most practical use of probability.


and one doesn't know what to say about the fact that the world is after all very unlikely ever to be in an eigenstate of my undertaking to carry out any particular measurement of anything.”

That does not make sense to me.



Such reflections as those of David Albert here are probably why this particular line of thinking has never gone anywhere.

The frequency operator approach has been refined by different people, and generalised for non sharp partial measurement of subsystem.

Now, a quite similar idea has been developed by Finkelstein, and it shows how to derive relativity from quantum logic, but I have never completely understood. Selesnick (an expert in quantum logic) wrote an entire book on this idea by Finkelstein, and make the square law derivation (Born Rule) already in the first pages of the first chapter, then the math get a bit too much high for a classical logician, but I progress in it. Selesnick has written important paper in Quantum logic which can be used to show that the physics that I extract from the “dream of number” contains a quantum nor (I don’t bother you with a precise technical rendering of this theorem, and to be sure some lemma still needs some consolidation).

I am not sure why you say that such line of thinking never gone anywhere, except that you dislike both Everett  MWI, and the simplest (conceptually) arithmetical MWI.

I might later make a post on how Finkelstein derived the Born rule (in the simplest case of sharp measurement). But don’t hesitate to take a look on Graham paper.

Usually, though, I prefer to mention Gleason theorem (or even Kochen & Specker) to justify the necessity of the MWI together with the square law. It is not important, you can define Everett by MW+born rule, as with mechanism, we have to derive the whole formalism for what is at the start clearly an infinite set of histories/computations.

Just to be clear, are you OK with P(W) = 1/2 in the WM-duplicatipon, when “W” refers to the first person experience?

Bruno

What if P(W) = 0.499999 ?  We can't expect perfection in duplication machines.

Brent

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