On 2/22/2019 11:39 AM, Lawrence Crowell wrote:
This sounds almost tautological. I have not read Masanes' paper, but
he seems to be saying the Born rule is a matter of pure logic. In some
ways that is what Born said.
The Born rule is not hard to understand. If you have a state space
with vectors |u_i> then a quantum state can be written as
sum_ic_i|u_i>. For an observable O with eigenvectors o_i the
expectation values for that observable is
sum_{ij}<u_j|O|u_i> = sum_{ij}<u_j|o_i|u_i> = sum_ip_io_i.
So the expectations of each eigenvalue is multiple of the probability
for the system to be found in that state. It is not hard to
understand, but the problem is there is no general theorem and proof
that the eigenvalues of an operator or observable are diagonal in the
probabilities. In fact this has some subtle issues with degeneracies.
Doesn't Gleason's theorem show that there is no other consistent way to
assign probabilities to subspaces of a Hilbert space?
Brent
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