Hi Richard,
On 01 Apr 2014, at 18:11, Richard Ruquist wrote:
Bruno, I have a problem with the Gleason Theorem because it appears
to me to be saying that every possible quantum state is realized
with equal probability at first, but the frequency at which each
universe reoccurs is given by the FPI probabilities that are
measured in controlled quantum experiments.
If what I just said is true, I'm sure you can see my the source of
my skepticism. So please correct my understanding of the Gleason
Theorem.
Richard
Gleason theorem, on the contrary, shows that for Hilbert space with
dimension 3, the measure, assumed to be totally additive, made on
quantum propositions (i.e. closed subspaces of the Hilbert space) is
given by the trace of some density operator, and this leads to the
Born rules or its generalizations on mixed states. It does not use
either the FPI nor the MWI, and somehow rules out the uniform
probabilities for quantum states.
The original proof of Gleason is not easy, but a more elementary proof
(which remains not that simple) has been found by Cooke, Keane and
Moran, and can be found in the (very good) book by Richard Hugues (you
can find a PDF on the net).
A constructive (and readable, and free!) proof has also been given by
Richman and Bridges:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.42.9076&rep=rep1&type=pdf
Bruno
On Tue, Apr 1, 2014 at 11:18 AM, Bruno Marchal <marc...@ulb.ac.be>
wrote:
On 01 Apr 2014, at 03:33, meekerdb wrote:
On 3/31/2014 6:00 PM, LizR wrote:
On 1 April 2014 06:04, meekerdb <meeke...@verizon.net> wrote:
The price is not having a unified 'self' - which many people would
consider a big price since all observation and record keeping
which is used to empirically test theories assumes this unity. If
you observe X and you want to use that as empircal test of a
theory it isn't helpful if your theory of the instruments says
they also recorded not-X.
(I suspect some people would consider it a big price not to have a
unified self for other reasons, too!)
I can't see how it's worse for your theory to say that your
instruments "will record X and not X" as opposed to saying they
"will record X or not X, but we don't know which".
That's before the fact. I didn't write "will". MWI is a theory
that says when you read your instrument and it says X, it's only
one of an infinite set some of which say X and some say not-X.
The former explanation says there will be apparent but explicable
randomness, the latter says there will be intrinsic and
inexplicable randomness.
But is it explicable. Bruno is careful to refer to "uncertainty"
or "indeterminancy". Those are not necessarily probabilities
unless they can be quantified to satisfy Kolomogorov's axioms - and
it's not clear to me that they can. The axioms require that the
set of "everything" have measure 1. But in this case "everything"
is ill defined and uncountably infinite.
It might be definable though, like "the consciousness of the
universal machine". It is the least Turing emulable entity having
some "futures" in the arithmetical reality. It is the first person
mental state in front of the maximal FPI.
In common applications of QM one assumes isolation and considers
only a small (at least finite) set of possible results - which
works FAPP.
But in Everett and comp we "know" why, without having to invoke a
mysterious pseudo-God-like selection, apparently. ISTM.
Bruno
http://iridia.ulb.ac.be/~marchal/
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