I'll certainly give it a thorough read and see what conclusions I come to.
A few words on quaternions. I approach quaternions from a very basic algebraic viewpoint and operational stance. I went back to the old books of Tait and Hamilton and learned how they approached it. My/their method is not difficult at all. First just the definition of a quaternion. A quaternion is defined as a scalar or vector or a combination of the two. This is straight forward and nothing spooky. And with this definition you can see it is applicable to any and every physical expression. The rules of manipulating certain types of quaternions are rather simple but it can get enormously tricky trying to keep track of signs. There are however certain types of quaternions that are members of a Caley Dickson construction such as quaternions, octonions, septenions, etc., that lose certain mathematical operational abilities. For example a quaternion does not commute. An octonion is a quaternion that does not commute and is non associative.
I pretty much try to stay with quaternions because they are i, j, k, and a scalar, or x,y, z, and a scalar like zero the origin. The rules of quaternion manipulation can be simple. One of the rules is that the reciprocal of a quaternion i.e., 1/j, is called the conjugate of j and can be designated as 1/j=j. Also ij does not equal ji. ij=ji. And ii=jj=kk=ijk=1. Lastly ij=k, jk=i, and ki=j. Those are the basic rules.
Note: i does not equal k, does not equal j, does not equal i. None of the three equal any of the other two. This is one reason why it's conceptually difficult to wrap your brain around just what quaternions are. I call them units of precession, allow them to act algebraically, and call it day. I leave the heavy lifting to the number theorists.
We can always designate the xyz axes as ijk. This the basis of my current induction equations. Nothing complicated. Just basic algebra. If you take the Lorentz force, E is along one axis, B is along another, and qv along the third. Thus they must behave as quaternions and obey the rules of quaternions. This is why via GFT principles we may write both E/B=qv and ExB=qr/t. This is basically why Maxwell and Steinmetz found quaternions to be so useful because electrons are quaternions.
Lastly, to my knowledge I"m the only one who has done this but I unify all of trigonometry and Hamilton's quaternions with ii=jj=kk=ijk=1=e^(ipi). Again I may be mistaken but no one except me seems to see Euler's formula as being the mathematical foundation of the whole of quaternion mathematics.
Again, I have to read much more in depth Wheeler's work but Steinmetz and Dollard talk of and explain this concept of innerspace. I too have proposed something akin to this concept of inner space via conjugates. (this was prior to reading about Steinmetz and Dollard) I've also proposed that the empty space inside an atom is not just empty space but is actually a quaternion space. A space that is dynamically active and quantifiable. This is why I find Wheelers book interesting. He touches upon several subjects I too have looked into.
« Last Edit: 20140719, 18:22:30 by GFT »
