Quantum Gravity 319.2: U.K., USA, Canada, Japan Break Through on Non-Gaussianity Via 3-Point Correlation Functions and Comparison with 3-Set Probable Causation/Influence (PI) Correlation

From Osher Doctorow

An important breakthrough in the analysis of Non-Gaussianity from
researchers in the U.K., USA, Japan, and Canada and Italy and a few
other places is "Non-Gaussianity as a probe of the physics of the
primordial universe and the astrophysics of the low redshift
universe," E. Komatsu, N. Afshordi, N. Bartolo, et. al., 8 pages,
arXiv: 0902.4759 v3 [astro-ph.CO] 14 Mar 2009, where the authors
include J. Khouri of U. Pennsylvania USA, J. Lehners of Princeton U.
USA, Paul Steinhardt of Princeton, S. Weinberg of U. Texas Austin USA,
X. Chen and M. Tegmark of MIT USA, J. E. Lidsey of U. London U.K., M.
Sasaki of Kyoto U. Japan, E. I. Buchbinder of Perimeter Institute
Canada, A. Linde and E. Silverstein of Stanford University USA, J.
Maldacena of Princeton Institute for Advanced Study USA, L. LeBlond of
Texas A&M USA, F. Takahashi and E. Komitsu of U. Tokyo Japan.

The 3-point (correlation) function or its Fourier transform is key to
this paper, and I will compare it with the Probable Causation/
Influence (PI) Probable Correlation here, except to mention that
Khouri was mentioned in a previous post here as being an Ekpyrotic
Universe theorist, and that Non-Gaussianity is optimal (at least for
particular types of Non-Gaussian distribution) in PI.

Roughly speaking, n-point (Correlation) Functions in standard
mathematical probability-statistics and physics including quantum
physics have the form:

1) E(X1X2...Xn), where E is a multiple integral in general with
argument X1X2...Xn or the product of their values x1x2...xn with
respect to dx1dx2...dxn except that an extra factor is typically
inserted such as a probability density function or even a time-
ordering operator (in the latter case, a non-commutative "factor"), n
integer > = 2.

The "analog", roughly speaking, of the n-point (Correlation) Function
(1) in Probable Causation/Influence (PI) is the Probable Correlation
(Function):

2) P(A1<-->A2<-->... <--> An) = P(A1A2...An) + P(A1 ' A2 ' ... An
' ), n > = 2 integer

where adjacent sets as in A1A2...An or A1 ' A2 ' ... An ' are
intersected, and A1 ' is the complement (part of the Universe outside)
of A1, etc.

There is nothing corresponding to the Probable Correlation Function
(2) in mainstream mathematical probability-statistics or non-PI
physics, since the operation <--> has not been explored there. Here
we have by definition:

3) P(A1<-->A2) = P{(A1-->A2)(A2-->A1)} = P{(A1 ' U A2)(A2 ' U A1)} = P
(A1A2 U A1 ' A2 ' ) = P(A1A2) + P(A1 ' A2 ' ), adjacent parentheses
are intersected.

The nth case, (2), simply replaces two parentheses products as in (3)
by n parentheses products, which are of course evaluated in pairs (two
at a time).

Note that the decomposition in (2) is not simply a definition - it can
be proven from the definition of the argument of P( ) on the left
hand side of (2) and probabiltiy theory and mathematical induction.

How does (3) relate to (1)? Well, (3) is defined on a set or sets,
while (1) is defined as a type of average on operators or their
values, for example real-valued. It turns out that with proper
choice of the sets, (3) is a POINTWISE correlation, in contrast to (1)
which is AVERAGED. This is very advantageous, because pointwise
quantities provide more information (at each point) than averaged
quantities and they can in turn be used for averages if desired.

For example, define:

4) Ai = {w: Xi(w) < = xi} where Xi is a continuous random variable
for i = 1 to n and xi is its value, here taken to be a real number,
and w is an element of the Probability Space.

Then:

5) P(Ai) = P(Xi < = xi)

and also:

6) P(AiAj) = P(Xi < = xi, Xj < = xj), where the "comma , " indicates
intersection ("and").

Similarly this in generalized to P(A1A2...An) and so on. Moreover:

7) P(Ai ' ) = P(Xi > xi)

Some Readers will recognize (6) as the Joint Cumulative Probability
Distribution Function (cdf) which is usually symbolized F(xi, xi).
Similarly, for n randm variables the corresponding cdf is F(x1,
x2, ..., xn). The cdf is a fundamental concept in mainstream
mathematical probability-statistics.

Osher Doctorow
.

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