From Osher Doctorow
COPYRIGHT NOTICE
Dirac Matrices Derived From "Generalized PI"
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
In several of my earlier postings here and in sci.stat.math and
geoemtry.research and math-history-list (of Math Forum) and elsewhere,
I pointed out that Probable Influence (PI) generalizes more or less as
follows.
A. To all alternating series, finite or infinite.
B. To all measures or indicators of proximity, partial or "complete",
"one-sided" or "two-sided".
Proximity when used without qualification refers to the opposite in
some consistent sense of some distance-function or metric or distance,
while one-sided proximity refers to the imposition of an inequality
constraint between the two objects whose distance is considered (e.g.,
y < = x), and partial proximity means that the resulting measure or
indicator that's the "opposite" of distance is not itself a metric or
distance-function but has some of the properties of a
distance-function.
The Dirac matrices listed in Wolfram's Mathworld "Dirac Matrices,"
http://mathworld.wolfram.com/DiracMatrices.html, can all be derived
from the following principle 1 (although other forms of these matrices
exist which "permute" the matrices listed).
Principle 1. A Dirac matrix is a 4 x 4 matrix which is either (a) a
main diagonal matrix of alternating sign 1's or (b) which minimizes the
proximity and maximizes the distance (see below) between subdiagonals
occupied by 1's or i's provided that if i occurs then -i occurs in the
same subdiagonal, or (c) is the identity matrix (1 on main diagonal,
0's elsewhere) or the matrix of (a) above with the alternating sign 1's
grouped (permuted) into the main diagonal 1 1 -1 -1.
Subdiagonals are parts of a matrix parallel to either the main or
second diagonal (upper left to lower right, or lower left to upper
right diagonals), and here the term is used in the "proper" sense (so
that a diagonal is not a subdiagonal), and maximum distance between
subdiagonals a, b means that the number of parallel subdiagonals
between a and b is maximum for 4 x 4 matrices.
Of course, the Dirac matrix is composed of Pauli 2 x 2 submatrices.
Osher Doctorow
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