Re: Quantum Entanglement Explained by Jacobson Radical + PI

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 07/09/04 

Date: Fri, 9 Jul 2004 04:40:29 +0000 (UTC)

On 8 Jul 04 20:04:33 -0400 (EDT), Osher Doctorow wrote:
>LEMMA. P(A U B) < = P(A U B)_I < = P(A U B)_Q
>
>if A, B are positively quadrant dependent, that is to say P(AB) > =
>P(A)P(B) in the sense of Lehmann (1967) - at least in the case of
>P(A U B) on the far left hand side of the inequality. (Here Q
>replaces M since we refer to quantum probability.)

Positive quadrant dependence says in terms of probability-statistics
that A and B, or for random variables X and Y, that the pair in-
creases together rather than being either unrelated or one in-
creasing when the other decreases - again, emphasizing their
probabilistic and statistical behavior. So positive quadrant
dependence is quite appropriate in this inequality. To give the
corresponding expressions for X and Y (random variables of continuous
type, although discrete ones can be handled by extension), one uses
the usual A = {w: X(w) < = x}, B = {w: Y(w) < = y), or perhaps
more appropriately labelling A as A_x or Ax, B as B_y or By where
Ax means A with a subscript x in these particular contexts.

Notice that the Jacobson radical provides a deep connection between
probability-statistics and the rest of algebra once the x o y and
1 + y - x relationship is understood. Since fuzzy multivalued
logicians and other mathematical logicians usually claim to trace
themselves back to algebra rather than to probability-statistics,
the connection via the Jacobson radical resembles more the two-way
connection of geometry and topology via the Gauss-Bonnet theorem and
its corollaries than a one-way connection in which algebra allegedly
gives rise to everything else.

If algebra is really interpretable in two very different ways, one
by logic and one by probability-statistics, and similarly for logic,
then the "exceptional" position of quantum logic also would seem to
be called into question. Quantum logic is currently thought to
represent a "world of its own" separate from other logics and with
very deep algebraic underpinnings, and researchers into it generally
accept mathematical physics' divorce from most of probability-stat-
istics and don't question anomalies and paradoxes like the Heisenberg
Uncertainty Principle (HUP) that come from mathematical or theor-
etical physics. If it is not really exceptional, and relates to
probability-statistics fundamentally as much if not more than to
algebra and physics, then the results for Quantum Entanglement may
be expected to characterize much of the foundations of Quantum
theory.

Osher Doctorow

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