Why quaternions?

greenspun.com : LUSENET : quaternions : One Thread

Why do you want to emphasize quaternions? There is a lot of redundancy in mathematics. Why add to it? There are more powerful tools available that will accomplish the same thing and with less redundancy. Geometric Calculus (based on clifford algebra is one). Clifford ALgebra was developed by combining Hamilton's quaternions and grassman's algebra. The fundamental question is what are the geometric relations you want to model and is there not already a unified way to express those geometric relations? That you will find links to QM and GTR is well known. Mendel Sachs even developed "Quantum Mechanics from General Relativity" using a quaternion metric. Due to the relationship between complex numbers,spinors and quaternions QM is sure to appear together with Maxwell's EM

-- Paul Gowan (pcgowan@yahoo.com), April 19, 2002


Hello Paul:

Physics has some deep but subtle problems. I know there are a bunch of different ways to work with physics, and different people have different STRONG opinions about which one is the most elegant. There is no way to decide such an issue.

Of all the possible Clifford algebras, is there an entirely technical reason to choose one over another? Birkhoff and von Neumann looked at the logical foundations of quantum mechanics in the 1930s. They came to the conclusion that one must work over the field of the real numbers, the complex numbers, or the quaternions. The real numbers are dull, there being no interference effects. Everyone does quantum mechanics over the complex numbers. Adler has a book on "Quaternionic Quantum Mechanics", but in my opinion it falls short because quaternion analysis is not used (that field being very poorly developed at this time).

People who work with Clifford algebras often do not mind if the algebra is not a division ring. It makes a big difference when thinking about quantum mechanics, the relationship between field equations and propagators (which are in effect the inverse of the field equations). If an equation cannot be inverted, then the propagator of the field equation cannot be determined. This happens with all gauge theories: they cannot be inverted so the propagator cannot be found. If one chooses a gauge, then it is possible to find the propagator. This is what happens with the Maxwell equations for example. The classical radiation field cannot be quantized until the gauge is chosen. I consider this a deep but subtle problem.

If one uses quaternions for the operators and potentials, then a field equation is necessarily invertible and a propagator can be found. That is new. Granted, most of the equations on my site are not novel, and I hope I was not projected that the majority were. However, operators as part of a division algebra are new and that is important in my opinion.

Does Mendel Sachs use quaternions or quaternions with an extra factor of i in it? The later is not a division algebra. It is very common to see in the literature because tossing in the i makes the link to Dirac's algebra simple. People prefer the simpler road :-) I will stay true to the more difficult work of hanging only with quaternions the division algebra due to the message from Birkhoff and von Neumann (Adler was the one who actually presented the technical arguments the best thought).

Good luck playing with geomentric algebra. You will at least have plenty of company! There are yearly conferences of GA, whereas the conference on quaternions in math and physics only happens once every five years.


-- Douglas Sweetser (Sweetser@theworld.com), April 24, 2002.

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