Date: Wed, 19 Dec 2001
16:46:54 0600
Dear Chris,
Quaternions are a much
more advanced algebra than vectors or tensors, and so electrodynamics
expressed in quaternions allows a great many things to be done and seen
by the modeler, than exist in the tensor and electrodynamics models.
For example, you cannot even see what Tesla was doing in his circuits,
if you use the standard vector and tensor electrodynamics. That was
rigorously shown by one of the fine electrodynamicists, T. Barrett, in
his paper
"Tesla's
Nonlinear OscillatorShuttleCircuit (OSC) Theory," Annales de la
Fondation Louis de Broglie, 16(1), 1991, p. 2341.
I
recommend you read Graham P. Collins, "Fractional Success," Scientific
American, 286(1), Jan. 2002, p. 21  and particularly the side panel
discussing mathematics and the universe. A dimension, e.g., is a
"degree of freedom". Simply put, if you "look in higher dimensions",
even a familiar object is surprisingly different. And the article is
speaking of viewing the universe in a "fourdimensional flatland"
intersection of a five dimensional universe. It appears we may be able
to get all the various theory together into the "theory of everything",
although conventional scientists are skeptical, as the article reports.
But the
interesting thing is that this very much more advanced "look" at the
universe turns out to be directly related to quaternion algebra!
Quoting, "Only quaternions, complex numbers and real numbers 
corresponding to four, two and one dimensions, respectively  have the
right properties for making the required exotic quantum state."
Interestingly, Maxwell started  or tried to start  his
electrodynamics in quaternion algebra and quaternionlike algebra. But
this was so far ahead of the times (1865) that he himself participated
in starting the reduction of his theory to vectors (and later tensors).
Just because the vectors were simpler, and the "electricians" (as
electrical engineers were then called) would "never be able to learn
that exotic a mathematical exposition".
He must
have been correct; they haven't adopted it yet, but still hang in there
with vectors and, when they wish to be "advanced", use tensors.
You can't
see past the "old science" to a new science if you will only look at
things the way the old science does, and the old model does.
Anyway,
that's what all the fuss is about. There are many better algebras
already available in which electrodynamics can be embedded and has been
to some extent. Quaternions is one such; Pauli algebra is another.
Clifford algebra is even "higher", etc.
So some
problems of the world can be handled with arithmetic alone, e.g. Some
require ordinary high school algebra. Some require calculus and
vectors, or tensors. And some require much higher "modeling and
looking".
Quaternions is a very good way to start, because they also are
intimately involved or associated with many attempts to produce a
"theory of everything". I particularly like Mendel Sachs's unified
field theory, which again is surprisingly associated with quaternions.
In simple
language, it's sorta like this: If you wish to be able to see over the
intervening mountains to the other side and what's over there, you
cannot use straight beams of light from the desert floor far below. If
you can get curved beams, fine! Or if you can get up higher than the
mountains, you can see beyond them with "straight beams".
Quaternions give us a great deal more flexibility in the "search beyond
those present mountain barriers" and enable us to see what's out there
beyond the present limits.
Best
wishes,
Tom
Bearden
