|Date: Wed, 19 Dec 2001
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 Oscillator-Shuttle-Circuit (OSC) Theory," Annales de la Fondation Louis de Broglie, 16(1), 1991, p. 23-41.
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 "four-dimensional 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 quaternion-like 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.