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From: OsherD on 22 Apr 2010 23:29
From Osher Doctorow

Chris Isham of the U.K. (Oxford U. or Cambridge U., I forget which) in
a paper today in arXiv continues the usual U.K. Loop Quantum Gravity/
Penrose attempts to destroy Probability, this time via Topos theory,
to such a degree that I am reconsidering whether to consider the U.K.
as an over-bureaucracized nation and to "demote" it from front leading
research inventiveness. One reason for the attempt to "outlaw"
Probability in quantum theory and quantum gravity is somewhat
surprising: it shows the "coincidental" nature of the Heisenberg
Uncertainty Principle (HUP).

Simply stated, HUP claims that:

1) VAR(x) Var(p) > = k where k is a simple function of the very tiny
h (Planck's constant), x position, p momentum.

However, variances (VAR) are expectations of a particular type, and so
are "means", which are much less rigorous and specific than
Probabilities. In fact, it is rather easy to prove the following
with Probability:

2) If (1) above were "non-coincidental", or in a sense "non-trivial",
then P(AB) > 0 would be equivalent to it where A is a set
representation of x and B a set representation of p. However, it can
be shown that P(AB) = 0 must hold for some x and/or some p.

There are various ways of going about establishing (2) and using it.
Notice that:

3) P(AB) < P(A)P(B) is "negative quadrant statistical dependence" (E.
Lehmann, 1960s), but if one or both of P(A), P(B) are 0, then so is
P(AB). Also, negative quadrant statistical dependence is what the
"non-coincidental" version of (1) would require unless it is involves
statistical independence.

4) P(AB) = P(A)P(B) is statistical independence of sets A, B. But if
P(A) = 0 and/or P(B) = 0, then P(AB) = 0.

Moreover:

5) P(x U p) = P(x) + P(p) - P(xp) = 1 iff P(xp) = P(x) + P(p) - 1 so
that P(xp) can easily be 0 by proper choice of P(x) and P(p), and
moreover the first equation (where P(x) means P(A) for A representing
x, etc.) must hold since Momentum and/or Position hold everywhere in
physics. The expression P(xp) is P(AB) for A representing x, B
representing p here.

It is still possible for (1) to hold, but since it has no set theory
analog, and since topos theory is a claimed generalization of set
theory, and in view of (2)-(5), it has no relevance to actual position
or momentum as (random or non-random) variables.

Osher Doctorow

From: OsherD on 23 Apr 2010 00:36
From Osher Doctorow

Some readers may miss the curiosity that P(AB) < P(A)P(B) cannot hold
if P(A) and/or P(B) = 0 since no probability (that is, in this case,
P(AB)) can be < 0!

Osher Doctorow
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