Quantum Gravity 183.3: Heisenberg Uncertainty Principle (HUP) Misinterprets Opposites

From Osher Doctorow

Let U( ) be "uncertainty in" or "uncertainty of" (however it is
defined), let x be position, p be momentum, then HUP claims that:

1) U(x)U(p) > = (small) positive constant

in the quantum scenario.

This does not even mean anything in general, that is to say the
product of real operators or functions or values being greater than a
real positive constant has so many interpretations that it is by
itself basically mathematically meaningless without further context
and qualifications.

To see this, consider the equation which I derived earlier:

2) P(A ' --> B) = P(A) + P(B) - P(AB)

where the right hand side increases as Positive Quadrant Statistical
Dependence or its generalization decreases, where the latter is:

3) P(AB) - P(A)P(B)

but where A, B are (generalized) Positive Quadrant Statistically
Dependent iff:

4) P(AB) > P(A)P(B) or equivalent P(AB) - P(A)P(B) > 0

which for P(A) not 0 is equivalent to:

5) P(AB)/P(A) > P(B) or equivalently P(B|A) > P(B)

Here P(B|A) is the conditional probability of B "given" A or "with A
fixed".

Because of (4), we might hypothesize (falsely) that the Probable
Causation/Influence (PI) analog of (3) is:

6) P(AB) - (P(A) + P(B)

but we know from elementary probability that:

7) P(A) > = P(AB), P(B) > = P(B)

by monotonicity of probability. So (6) is negative.

However, we might do the "next simplest thing" and examine:

8) kP(AB) - (P(A) + P(B)), k large real constant number

and try to define the PI analog of "Positive Dependence" via:

9) kP(AB) > P(A) + P(B), some positive constant real k

We could for example try to take the minimum or maximum such k.

It turns out, as readers can try to verify, that this only works if we
put lower bounds on P(A) and P(B), for example:

10) P(A) > 1/k, P(B) > 1/k

We could even set 1/k = h/(2pi) or an even smaller positive number.

The difficulty is that with (8) through (10) we are obtaining
"effective" Dependence down to but not below some positive tiny real
number 1/k, and there are infinitely many such scenarios. This might
be sufficient for approximations, but one searches in vain for a
general theory of which these are approximations.

The above procedure is analogous to what the Heisenberg Uncertainty
Principle says in terms of multiplication as in (1).

It turns out that the "real" answer to the above problem with
Dependence, that is to say the more general solution, is not to use 1/
k at all, but to look at a different definition of what we intuitively
think is Dependence (and in the HUP case, what we intuitively think is
"Non-Dependence" or something like it).

Recall that Statistical Independence of A and B is equivalent to:

11) P(AB) = P(A)P(B)

or equivalently if P(A) is not 0:

12) P(B|A) = P(B) provided that P(A) is not 0

But what (12) means both intuitively and actually is that A "passes
through" B, that is to say nothing happens to P(B) or to B by having A
"given" or "fixed".

The opposite of this should be:

13) P(B|A) = P(A)

which says that A "totally influences" or even "becomes" B in the
sense that the result of having B with A "given" or "fixed" is P(A)
rather than P(B).

Now, (13) is equivalent to (for P(A) not 0):

14) P(AB) = P(A)^2

when expanding P(B|A) = P(AB)/P(A). The PI analog is:

15) P(A-->B) = P(A)

which when expanded is:

16) 1 + P(AB) - P(A) = P(A)

or equivalently:

17) P(AB) = 2P(A) - 1 for PI Dependence

When dealing with PI, we don't have to restrict P(A) to not be 0, so
let's consider the cases when P(AB) = 0 and P(AB) = 1.

18) P(AB) = 1 in (17) iff P(A) = 1 for PI Dependence
19) P(AB) = 0 in (17) iff P(A) = 1/2 for PI Dependence

In fact, since (17) involves a probability P(AB) which is by
definition nonnegative, P(AB) > = 0 applied to (17) yields:

20) 2P(A) - 1 > = 0, or equivalently P(A) > = 1/2 for PI Dependence.

Surprisingly, conditional probability P(B|A) in its own definition of
Dependence as P(B|A) = P(A) has some similarities. For example, its
equation is:

21) P(AB) = P(A)^2

and comparing with (17), we see that if P(A) is 1, we get:

22) If P(AB) = 1 in (21), then P(A) = 1 (compare (18)
23) If P(AB) = 0 in (21), then P(B|A) is undefined.

Of course, P(A)^2 and 2P(A) - 1 increase together as P(A) increases.

Notice that although "independence" means the same thing in
conditional probability and PI, namely P(AB) = P(A)P(B), "(positive)
Dependence" at least for the totally Dependent case means something
quite different, namely P(AB) = P(A)^2 for conditional probability and
P(AB) = 2P(A) - 1 for PI.

It should be rather understandable that HUP does not make any such
fine distinctions, and that as it is formulated it would not survive
distinctions between Independence, Non-Dependence, Total (Positive)
Dependence, etc.

Osher Doctorow

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