Neumaier's Modification of Heisenberg 4: The "Generalized HUP"

>>From Osher Doctorow mdoctorow@xxxxxxxxxxx

In section 5, Neumaier (2003) generalizes the Cauchy-Schwarz
inequality:

1) /<f*g>/^2 < = <f*f><g*g>

for all f, g in E where E is "quantities" that include complex numbers,
to the (generalized) "ucertainty relation":

2) Var(f)Var(g) > = /cov(f, g)/^2 + /(1/2)<f*g - g*f>/^2

for all f, g in E, where notice that the second term on the right hand
side would vanish for real f, g since then f*g = g*f, while for complex
non-real f, g we would get for "canonically conjugate" f, g, the second
term as h^2.

If we go back to the definition of <f>, Neumaier gives on page 16 of
his paper two cases: (a) for random variables only assuming a finite
number of discrete values, we have:

3) <f> = sum pk fk

for probabilities pk of assuming the value fk, and the pk add up to 1
and are nonnegative. (b) In the quantum algebra E of bounded linear
operators on a Hilbert space H now, a "pure ensemble" or "pure state"
has:

4) <f> = w*fw

But this has all the defects discussed earlier in my threads of w not
being able to depend on more than one variable and not even changing
its notation with f. It doesn't do Neumaier any good to tell us that
in quantum thermodynamics an equilibrium ensemble is described by:

5) <f> = tr(exp(-S/k)f), k > 0 being Bolzmann constant, S a Hermitian
quantity, tr(exp(-S/k)) = 1 called the entropy.

Neumaier has not generalized (3) to continuous random variables, not to
mention complex expressions of continuous type which could substitute
for random variables.

In short, the "brilliant" generalization (2), which admittedly is a
nice piece of mathematical work, just lacks one thing: meaning, since
the quantities have nothing to do with probabilities or probability
densities in general. They might work out with some uniform
distributions, and again the interpretations with only uniform
distributions would be rather bizarre since "conjugate uniform
distributions" if they existed would influence each other from my
earlier postings in recent threads.

Is it possible for something without meaning probabilistically to
reduce to the Cauchy-Schwarz inequality? Yes! That inequality is not
purely probabilistic. It has plenty of non-probabilistic scenarios.

So what does (2) prove? It proves that some non-probabilistic
expression is bounded below by h^2.

Next question :>)

Osher Doctorow

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