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From: Osher Doctorow on 22 Jun 2010 17:55
From Osher Doctorow

I have shown that optimum Probable Causation/Influence (PI) occurs
when P(A) = P(B) for A the Cause, B the Effect, which generalizes to
y(y - x) = 0 or for matrices A, B, diag(A)^2 = diag(A)diag*(B) where
diag*(B) reverses the elements on the diagonal of the diagonal matrix
B for a 2 x 2 matrix (which appears to be generalizable to other
dimensions). Reversing, in turn, is related to Palindromes (see
"Palindrome" online, especially the Wikipedia article) and to Quantum
Erasure, and the latter is related to Wheeler's Delayed Choice
GedankenExperiment and can be combined with the latter. The result
at least locally looks like a time reversal or backward Causation,
which would be interesting for changing from Attraction to Repulsion
of interactions if feasible, although the usual interpretation has not
been extended that far.

Look at:

1) "Random delayed-choice qunatum eraser via two-photon imaging,"
Giuliano Scarcelli, Yu Zhou, Yanhua Shih, U. Maryland USA and also for
the first author U. di Bari Italiy, arXiv: quant-ph/0512207 v2 14 Jun
2007, 10 pages, 2 columns per page.

The paper has some interesting history of the Quantum Eraser
discovered by Scully and Druhl in 1982, which has become one of the
most interesting topics in studying the foundations of QM.

One interesting result of Scarcelli et al is that 2-photon phenomena
as in the double slit appear to be completely different from 2
INDEPENDENT photons or from 1-photon phenomena, almost like a
different dimension (in my words). Only the former can explain
entanglement and eraser physics, not the latter. The concept of a
"biphoton" wavepacket is introduced for non-local spatio-temporal
correlations of states, and relats to geometric optics. The classical
electric field appears to be replaced by 2 non-independent photon
probability amplitudes.

"Delayed Choice" keywords bring up 27 papers in arXiv, all related to
Wheeler or to the Quantum Eraser.

Probable Causation/Influence 1 + P(AB) - P(A) or 1 + P(B) - P(A) for
P(B) < = P(A) and Conditional Probability P(AB)/P(A) for P(A) not 0
both study Independence and Dependence. The Conditional Probability
notion of Independence is:

2) P(AB)/P(A) = P(B) for A, B independent, if P(A) is not 0.

The Probable Causation/Influence notion of Independence has several
different interpretations:

3) 1 + P(AB) - P(A) = 1 + P(A)P(B) - P(A) (equivalent to (2))
4) P(A-->B) = 1 + P(AB) - P(A) = P(B) (independence of effect from
cause)

Dependence holds whenever none of (2), (3), (4) hold, or the
appropriate subsets when restricted to only PI or to only Conditional
Probability.

Osher Doctorow
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