From Osher Doctorow
Susumu Okubo, U. Rochester New York USA, in "Representations of
Clifford algebras and its applications," 30 pages, arXiv: hep-th/
9408165 v1 29 Aug 1994, was one of the earliest Clifford algebra
researchers to get into arXiv, and defines N-dimensional real Clifford
algebra C(p, q) for N = p + q in terms of Dirac matrices g_u or gu, u
= 1 to N, such that:
1) gu gv + gv gu = 2 n_uv E
where n_uv is 0 if u is unequal to v, 1 if u = v = 1, 2, ..., p, and
n_uv = -1 if u = v = p+1, p+2, ..., N, E = unit matrix.
While some people may think that the 2 is coincidental (one researcher
claims that another representation differs from it in some
circumstances by an "irrelevant factor 2"), the viewpoint of Probable
Causation/influence shows that 2 and 1/2 have deep Causal properties
from 1 + P(A<-->B) = 2 at optimum (maximum) Probable Correlation P(A<--
B) = 1. As mentioned in previous Sections, 1/2 comes from
(normative) scaling by 1 + P(A<-->B) (division).
G. Bergdolt, IRES France, http://clifford-algebras.org/v9/v91/bergdo91.pdf,
accepted April 27, 1999, 7 pages, "Isomorphism groups in Clifford
Algebras," defines a real Clifford algebra as generated by a set of
basis vectors {ei, i = 1 to 3} and defining relations:
2) eiej = -ejei (i not equal to j)
3) ei^2 = +/- 1
yielding a similar equation to (1).
Going back to Okubo, a remarkable relationship (p. 7) is:
4) if either p ' - q ' = p - q (mod 8) or p ' - q ' + p - q = 2 (mod
8), then real Clifford algebra C(p ', q ' ) is isomorphic to real
Clifford algebra C(p, q) where we're given p ' + q ' = p + q ( = N by
definition).
So here we get not only a 2 but the importance differences p - q which
are fundamental to Probable Causation/Influence (PI) which is based on
addition/subtraction fundamentally.
Osher Doctorow
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