Independent/Dependent Phases 21: 0 and 1 as Phase Boundaries

>>From Osher Doctorow mdoctorow@xxxxxxxxxxx

Readers who've looked at section 20 of this thread may be surprised to
find that it's moved way down on their monitor screens (at least, the
ones with Google Beta), so if you want to read my reply to hhc Harry
C.'s calling me: "Hey Kook," you may have to look a bit. It's worth
it.

By now, I've arguably established that if variables are restricted to
the interval [0, 1], then 1/2 is a "natural" phase boundary (phase not
in the angle sense but in the usual
gas/liquid/solid/superfluid/superconductor/plasma/condensate/arguably
even black hole and others senses). If somebody like hhc Harry C.
finds that impossible to understand from reading one post in one
section of a thread, then as an alleged mathematically literature
physicist (he claims to be a physicist) it must be the first time that
he's encountered arguments requiring reading previous arguments. If I
randomly selected a section of a thread or a post in a subsection, I
wouldn't call somebody a Kook if I didn't understand what they were
saying.

To be able to analyze 0 and 1 in this context of variables restricted
to [0, 1] with phase boundary 1/2, from one viewpoint it would be
hypothetically "symmetric" for 1 and 0 to also be phase boundaries if -
and that's a big if - there were some "hyper-universe" outside [0, 1].
Even if there were nothing outside [0, 1], 1 and 0 are equally far from
1/2, so might be regarded as somehow special in reference to phases.
In fact, as we know with conditional probability fY/X=x(y/x) =
f(x,y)/fX(x) and FY/X=x(y/x) = F(x,y)/FX(x) for f and fX bivariate and
univariate probability density functions (pdfs) and F and FX
respectively bivariate and univariate cumulative distribution functions
(cdfs), FX(x) = 0 and fX(x) = 0 are "forbidden" or "blow-up" cases, and
so the value 0 is indeed qualitatively and quantitatively very
different from the rest of the interval [0, 1].

One clue that [0, 1] can be generalized beyond this range of a random
or even deterministic variables is arguably for values that exceed one
end of the interval [0, 1] in value to be on a changed scale, as for
example imaginary numbers. Thus, for example, sqrt(1 - v^2/c^2)
becomes imaginary formally for v > c.

Richard Feynman in one of his papers toward the end of his life
similarly found a use for "negative probabilities," which would be
below the 0 end of [0, 1] in a probabilistic scenario. It is in one of
the volumes honoring David Bohm (with whom (David Bohm) I disagree in
almost everything).

I again remind readers that Sir Arthur Stanley Eddington in his
Mathematical Theory of Relativity (1922 and a 1948 or 1952 edition,
Cambridge University Press: U.K.) pointed out that a superluminal
regime was formally possible mathematically but that in his view we
couldn't communicate with it. We now know that mathematical models
sometimes "blow up" at phase boundaries as for example when it would
require dividing by 0 or taking log(0) or sqrt(below 0). So if SR were
valid except at or very near the "phase boundary" v = c or v/c = 1, it
wouldn't be too surprising.

Osher Doctorow

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