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From: OsherD on 27 May 2010 21:00
From Osher Doctorow

To simplify the discussion, we will simply use the symbol [T1, T2] =
T1T2 - T2T1 for operators T1, T2, without distinguishing here between
Poisson brackets, Dirac brackets, Moyal brackets, Lie brackets, and so
on.

Roughly speaking, the following regimes hold for linear operators:

1) Classical: [T1, T2] = 0 (but see below)
2) Quantum: [T1, T2] = ih, h Planck's constant divided by 2pi.

However, for nonlinear operators there is a remarkable example where
(1) fails, the simplest case being the "Translation" operator:

3) T(f) = f + k, k > 0

Notice that this is nonlinear, because:

4) T(f + g) = (f + g) + k which does not equal (f + k) + (g + k) = (f
+ g) + 2k.

Next, let us convert Probable Causation/Influence (PI), which has form
1 + y - x, to an operator of type (3) for the situation where y =
f(t), x = g(t) for t = time, and write h(t) = y - x = f(t) - g(t), so:

5) T(h) = h + 1 (which is y - x + 1 for y = f(t), x = g(t) )

This is so "absurdly" simple that it is easy to overlook a curious
property of T in (5) which "interpolates" in a certain sense between
(1) and (2), namely if Dt is the derivative operator with respect to
t, then:

6) Dt T(h) = Dt(h) = Dt [h + 1] = Dt[y - x + 1] = DtP--> where P-->
is the operator:

7) P--> = (definition) the operator such that P-->(x, y) = 1 + y - x

Now look at Dt(h) in (6):

8) Dt(h) = Dt(y - x) = Dt(y) - Dt(x)

Now add (1) to both sides of Dt(h) = Dt(y) - Dt(x), recalling that
Dt(h) = Dt T(h) and T(h) = y + x - 1 from (5):

9) 1 + Dt T(h) = 1 + Dt(y) - Dt(x)

For everything normalized, this says:

10) 1 + Dt T(h) = P(Dt(x) --> Dt(y)) = P-->(Dt(x), Dt(y)) = P--> Dt
(x, y)

Therefore from (10):

11) Dt T(h) = Dt P-->(x, y) = P--> Dt(x, y) - 1

and therefore finally:

12) Dt P-->(x, y) - P-->Dt(x, y) = -1

which is roughly speaking:

13) [Dt, P-->] = -1 using the bracket as in the first sentence of this
post.

Here -1 = i^2, so in a sense (13) interpolates between i and 0 and
introduces a 3rd scale separate from the scale of i (imaginary
numbers) and [0, 1] of probabilities, namely negative reals which
include i^2 = -1.

I will leave matrices and tensors for another time.

Osher Doctorow
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