Can you find a physical meaning for this?greenspun.com : LUSENET : quaternions : One Thread
You several times give the following equation:
(d/dt, del) (c, D) = (dc/dt - del D, dD/dt + del c + del x D)
and typically you have the note "Dense notation :-)".
Then you have some more-or-less complicated operators you can apply to cancel out all but the terms you are interested in. And much of the complexity of your work involves dealing with the consequences of those operators whose only purpose is to remove the terms you don't want.
My question is, can you find any use for the result that includes all the terms? Is it good for anything as it stands?
If you can find a use for it, you can probably deal with it easier than the individual pieces.
-- Jet Thomas (email@example.com), August 17, 2002
A deep understanding of physics involves opposite ideas under tension. Take special relativity. In the popular culture, it is presented that "everything is relative." Just as important as the Lorentz covariant quantities like time, space, energy and momentum, are the Lorentz invariants such as the interval and mass. Some measurements depend on the inertial reference frame, others are completely independent of the frame. Both are at work, side-by-side.
One pair of opposites that has made it to the mainstream is the wave/particle duality of light. Light does come in clumps. Those clumps do interfere with each other. It might be hard to picture, but it is true.
A similar thing happens with quaternion multiplication, but I don't know that I have discussed it before. In math, we are taught to keep vectors separate due to how they transform. Don't add a polar vector to an axial vector, it makes no sense! Still, what if one object is moving linearly and spinning? A complete description would involve both a polar vector for the linear motion and an axial vector for the spinning.
Quaternion multiplication creates objects that transform differently under time and space inversion, along with mirror reflections:
time reversal space reversal mirror reflection dc/dt - del D + + + dD/dt + del c - - + del X D + + -
Because these three transform differently, they will describe separate things in Nature. All physical laws are written this way. Quaternion algebra gives a reason to group these objects with different transformation properties.
The one major use I may have found is a proposal to unify gravity and electromagnetism. It does involve these tricky grouping of different things issue. Will have to see how well I can explain that proposal to professionals. It is my opinion that if (c, D) is a potential, the first of the three listed is the gravitational field, the second the electric field, and the third the magnetic field. The last two are part of standard electromagnetic theory, but the first has nothing to do with gravity by current analysis.
-- Douglas Sweetser (firstname.lastname@example.org), August 21, 2002.