Quantum Gravity 183.5: HUP's Mis-Diagonalization
- From: OsherD <mdoctorow@xxxxxxxxx>
- Date: Sat, 06 Oct 2007 23:52:13 -0700
From Osher Doctorow
I pointed out that many Physicists and even Non-Probability-Statistics
Mathematicians appear to believe that population means (expectations)
and population variances (or standard deviations, which are positive
square roots) "characterize" probability distributions and so are
allegedly unique for each probability distribution. This is false,
as I pointed out. Instead of the above, probability density functions
(for continuous random variables) or probability mass functions (for
discrete random variables), pdfs for short, and cumulative
distribution functions (cdfs) are unique for each probability
distribution and so totally characterize probability distributions,
and characteristic functions do the same thing. You can learn some of
this from Kai Lai Chung's (Stanford U.) A Course in Probability
Theory, Harcourt, Brace, & World: N.Y. 1968 or its later Editions.
It is a graduate level text.
The Heisenberg Uncertainty Principle (HUP) "weaves a thread" using its
inequality:
1) U(x) U(p) > = k (k very small constant, U( ) uncertainty of)
allegedly through the quantum (microscopic) part of the Universe,
somewhat analogously to Cantor's diagonalization proofs such as the
uncountability of the reals, but in the case of HUP we could construct
whole sequences (infinitely many of them, in fact) of probability
distributions for which U(x) U(p) > = k, U(x) U(p) > = k + 1, ...
etc., or we could increase k by some very small constant amount for
each distribution.
The allegation of the HUP people in the above process would be that
regardless of what probability distribution we actually choose, as
long as we know U(x) and U(p), then we "know what we're doing" and
have real relationships cutting across all these possible probability
distributions.
However, there are infinitely many possible probability
distributions. If you tell me that you have a probability
distribution with mean 5 and standard deviation .01, then you might be
talking about a Gaussian/normal pdf, a Gamma pdf, an F pdf, a
(Student's) t pdf, a power function pdf, a Beta pdf, a Logistic pdf, a
Pareto pdf, a Weibull pdf, etc. They don't look anything like each
other usually, and often their shapes relate to totally different
dynamics.
But it gets even worse for HUP. In order to really select pdfs for
which the means and/or variances go in "opposite directions" as in HUP
more or less, or even more dangerously such that the pdfs themselves
go in "opposite directions", you have to violate cardinal rules of
Probability Theory according to which pdfs approach 0 at their
extremes (for example, at -infinity, infinity, or for finite interval
distributions on [a, b] outside a and b, or for positive real line
distributions at 0 and infinity, etc.). The pdfs have to DECAY
asymptotically! Two pdfs cannot be "opposites" if they both decay
asymptotically. In fact, they cannot even be opposites in between
because there are not any negative pdfs to "counter-balance" positive
pdfs.
Why do continuous pdfs approach 0 at their extremes? It is because
for continuous random variables, probability is an AREA or volume
under a curve, and if a pdf is nonzero on the whole real line or on
the nonnegative real line for example, then if the pdf does not
approach 0 at infinity, the area under its graph is infinite! For
readers who aren't very familiar with probability, consider an
analogous example in calculus, where the integral of x dx from 0 to t
0 approaches infinity as t --> infinity, and x of course or y = x >0 is unbounded on [0, infinity) and does not approach 0 as x
increases.
For an example of what happens EVEN IN THE SAME pdf, with different
values of expectations and variances and time scales, see "Toward a
physics of evolution: existence of gales of creative deconstruction in
evolving technological networks," by Rudolf Hanel, Stuart A. Kauffman,
Stefan Thurner (respectively 13 papers in arXiv, 13 papers in arXiv,
and 50 papers in arXiv) of Medical U. Vienna, U. Calgary, and Medical
U. Vienna respectively, arXiv: 0703103 v.1. physics.soc-ph, 7 pages.
Just by selecting possibly different time scales, they are able to
model 3 phase transitions including rapidly increasing diversity,
stationary diversity, and rapidly decaying diversity (diversity being
the number of existing elements over time). They use a famous
equation which generalizes the Lotka-Volterra replicator, hypercycles,
and Turing gas, and is applicable not only to technology but to
various biological, social, and physical systems. I have discussed
the close relationship between the Lotka-Volterra equations and the
Riccati Differential Equation previously.
Osher Doctorow
.
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