On Tue, Feb 11, 2020 at 11:16 PM Bruno Marchal <[email protected]> wrote:
> On 7 Feb 2020, at 12:07, Bruce Kellett <[email protected]> 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 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. 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, 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." Such reflections as those of David Albert here are probably why this particular line of thinking has never gone anywhere. Bruce -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/CAFxXSLSXKtZXZ%3DoTx75A%2Bvb-oAmpDmXnTGo29Z3cYnssECj9ZA%40mail.gmail.com.
