On 2/14/2020 6:35 AM, Bruno Marchal wrote:
On 13 Feb 2020, at 23:59, Bruce Kellett <[email protected]
<mailto:[email protected]>> wrote:
On Tue, Feb 11, 2020 at 11:16 PM Bruno Marchal <[email protected]
<mailto:[email protected]>> wrote:
On 7 Feb 2020, at 12:07, Bruce Kellett <[email protected]
<mailto:[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 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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