Further ruminations on Born's rule: I will
take the presentation in appendix D of "Theory of Nothing", as it is
slightly more up to date than in "Why Occam's razor", and the
relationship of measure to the magnitude of the quantum state is more explicit.
The interesting thing to note about the P_\psi function is that it is
linear in its argument, and can be interpreted as the measure of the
observer moment \psi_A - see equation (D.7).
P_\psi( a{\cal P_A}\psi + b{\cal P_B}\psi) =
aP_\psi({\cal P_A}\psi) + bP_\psi({\cal P_B}\psi) (D.7)
Given some reference state \psi, and an observer moment \phi=m\psi
having measure m, then we do see that the measure of an observed state
{\cal P_a}\phi is given by P_\psi({\cal P_a}\phi) = m P_\psi(\psi_a),
and so the probability P_\psi(\psi_a) is equal to a simple ratio of
measures of observer moments.
But hang on, you say, isn't the linearity equation (D.7) inconsistent
with the first line of (D.10), which appears to be bilinear?
P_\psi(\psi_a)=<\psi_a,\psi_a> (D.10)
The answer is that \psi_a is not actually a general vector, but related back to
the reference vector \psi by means of a partition (D.8)
\psi = \sum_{a\in S} \psi_a
So it is consistent after all. It would be nice to relate this back
into the verbal terms somehow, ie is this result the
"non-contextuality" property that Mallah talks about? As I mentioned a
couple of days ago, I found Mallah's paper very hard to follow.
Anyway, I still stand by the proposition that the Born rule problem is
solved - the only thing now is to establish whether it is the same problem.
Cheers
-------------------------------------------------------------------------
A/Prof Russell Standish Phone 0425 253119 (mobile)
Mathematics
UNSW SYDNEY 2052 hpcoder.domain.name.hidden
Australia
http://www.hpcoders.com.au
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A/Prof Russell Standish Phone 0425 253119 (mobile)
Mathematics
UNSW SYDNEY 2052 hpcoder.domain.name.hidden
Australia http://www.hpcoders.com.au
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Received on Mon Sep 24 2007 - 23:52:11 PDT