Science > Physics > Nonrenormalization vs Renormalization 11: Generalized Binomial Theorem
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Science > Physics |
| User: |
"OsherD" |
| Date: |
11 Apr 2006 12:58:31 AM |
| Object: |
Nonrenormalization vs Renormalization 11: Generalized Binomial Theorem |
From Osher Doctorow
Edwin Hewitt and Karl Stromberg of respectively U. Washington and U.
Oregon in their classic Real and Abstract Analysis, Springer-Verlag:
New York. 1965, generalize the binomial coefficient that I've written
as C(n, r) with r, n integers to what I'll type as C(a, n) with a any
real number:
1) C(a, n) = a(a - 1)(a - 2)...(a - n + 1)/n!, n = 1, 2, 3, ....
The well-known generalized binomial theorem in their notation except
for my use of C instead of vertical parentheses is (Sum is convergent):
2) (1 + x)^a = Sum C(a, n)x^n, Sum over n = 0 to infinity, x in (-1,
1)
When a is a positive integer greater than n, C(a, n) of (1) becomes the
usual a!/(n!(a - n)! of the previous postings which equals the right
hand side of (1) for such positive integer a.
The remarkable thing about (2) is that the right hand side is a sum
(usually infinite) with nth term a product of two factors, C(a, n) and
x^n, so that the coefficient of x^n is always constructed by
(generalized) gamma functions or generalized factorials. Thus, gamma
functions or generalized factorials (factorials are particular gamma
functions) "construct" expressions of form (1 + x)^a for real a and for
x in (-1, 1). If a is positive, (2) also works for x = -1 and/or +1.
In a sense, we're running out of things not constructed by gamma
functions.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Nonrenormalization vs Renormalization 11: Generalized Binomial Theorem |
11 Apr 2006 01:06:59 AM |
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From Osher Doctorow
In the third to last paragraph of the previous posting, I should have
typed a!/[n!(a - n)!] instead of a!/(n!(a - n)! because the latter
leaves out one right part of a pair of parentheses.
Here's some interesting homework: what happens if x = - v^2/c^2 and a =
1/2 or -1/2?
Osher
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| User: "OsherD" |
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| Title: Re: Nonrenormalization vs Renormalization 11: Generalized Binomial Theorem |
11 Apr 2006 01:28:10 AM |
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From Osher Doctorow
I don't know whether Hewitt and Stromberg may have been thinking about
x = -v^2/c^2, but it does look very much as though Special Relativity
(SR) and hence Quantum Field Theory is constructed from the
gamma/factorial function generalized. Since Probable
Influence/Causation (PI) yields gamma/factorial functions and PI has no
finite speed of light restriction, the idea that imaginary sqrt(1 -
v^2/c^2) (for v^2 > c^2) represents a different phase from our own
gains considerable plausibility. Likewise, PI has no discrete
spacetime/matter constraints, so the interpretations of the Loop
Quantum Gravity (LQG) school about discrete Universe appear to be
implausible. Of course, Renormalization as a "universal principle" is
also implausible from this and prior sections of the thread, though I
would never (I think) discard finite physics under certain scenarios
since for example the finite interval probability distributions are
maximum entropy in PI among continuous random variable distributions
with two unknown parameters, and they are also Shannon maximum entropy
for the particular case of the uniform distribution, e.g., on (0, 1).
To put it another way, PI suggests "live and let live". Both infinite
and finite physics plausibly have much to contribute, and they should
stop trying to destroy each other so to speak as well as attempting to
become "the one and only theory". Both anti-Renormalization and
pro-Renormalization physics should be heavily researched.
Osher Doctorow
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