My Response to Professor Shahn Majid's Reply to Me about 2 + 1 quantum gravity noncommutative phase logic
Kauffman:
"All of this points out how the complex numbers, as we have previously
examined them, live naturally in the context of the non-commutative
algebras of iterants and matrices.... A natural non-commutative algebra
arises directly from articulation of discrete process and can be
regarded as essential information in a Fermion. It is natural to compare
this algebra structure with algebra of creation and annihilation
operators that occur in quantum field theory. ..."In the notion of time
there is an inherent clock and an inherent shift of phase that enables a
synchrony, a precise dynamic beneath the apparent dynamic of the
observed process....By starting with a discrete time series of
positions, one has immediately a non-commutativity of observations,
since the measurement of velocity involves the tick of the clock and the
measurement of position does not demand the tick of the clock....In
this sense, i [square root of negative one] is identical in concept to a
primordial time."
On Mon, Jul 13, 2020 at 10:57 AM Shahn Majid <s.majid@qmul.ac.uk> wrote:
Sorry too much philosophy for me. I would not take Connes noncommutative music theory too seriously its a lot to draw from one example of a dirac operator which happens to be noncommutative and where the eigenvalues are arranged in geometric progression rather than linear, he was having fun and not a serious basis for anything as far as I am concerned but refer you to Connes. S
Thank you again for the kind response Professor Majid. By geometric
progression you mean as a matrix? So what about Kauffman's work?
Math
professor Louis Kauffman:
"A first mathematical direction is to see
how i, the square root of negative unity, is related to the simplest
time series: ..., -1,+1,-1,+1,... and making the above analysis of time
series more algebraic leads to the following....If η is the order two
permutation of two elements, then [a, b]η= [b, a]. We can define i=
[1,−1]η and then i2 [squared]= [1,−1]η[1,−1]η= [1,−1][1,−1]ηη2=
[1,−1][−1,1] = [−1,−1] = −1. In this way the complex numbers arise
naturally from iterants. One can interpret [1,−1] as an oscillation
between +1 and −1and η as denoting a temporal shift operator. The i=
[1,−1]η is a time sensitive element and its self-interaction has square
minus one. In this way iterants can be interpreted as a formalization of
elementary discrete processes. A more general approach to discrete
processes [18] includes this interpretation of iterants and the square
root of negative unity. The more general approach is worth reprising in
this context. Given a sequence of discrete algebraic elements Xt(t=
0,1,···) (we take them to be associative but not necessarily commutative
for this discussion), we define an invertible shift operator J"...We see
that, with temporal shifts, the algebra of observations is non-commutative....The
square root of minus one is not really living in a commutative world.
It's living in a non-commutative world....It just happens to be
commutative for you when you're doing complex analysis because you only
looked at the combination....
"The
"Quaternion Handshake" illustrates the fundamental
orientation-entanglement relation that interlocks the structure of the
quaternions with the geometry and topology of an object connected to a
background in three dimensional space. In this case the objects are
human hands, the background is the body and the connection is the arm
that links hand to body. ... "
Starring: Martial Arts By: Louis Kauffman John Hart Eddie Oshins ...
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