I believe that a new very powerful idea drives this work, namely, that events represented as quaternion are a topological algebraic field. This implies that any collection of events can be generated by an appropriate quaternion function. Scalars and vectors mix under multiplication, so quaternions are a mixed representation.
A new view of relativistic quantum mechanics was outlined. The Klein-Gordon equation is a scalar equation. When quaternion operators are employed, the Klein-Gordon equation is part of a larger set, including a scalar and vector identity analogous to the Maxwell equations. These additional identities are also valid by the conventional analysis, but they do not naturally arise, so the parallel to the Maxwell equations is less clear.
A new link to general relativity has been proposed which is slightly different. The invariant interval in special relativity was the first term of the difference between two events quaternions squared. If the origin changes, then the first term of the difference between the two event quaternions and the origin quaternions squared is similar to the square of the affine parameter of general relativity. The only difference lies in the cross term.
Every event, every function, every operator used was a member of the field of quaternions. This might strike some as a comic reliance on a solitary tool. I prefer to think of it as a great democratic principle. Physics is impressively democratic, with each photon or electron obeying the same collection of laws interchangeably. The mathematics underlying the laws of physics should reflect this interchangeability.
More problems need to be solved. My upcoming focus will be the Dirac function and Fourier analysis using quaternions. If I can build these functions, it should be possible to approach problems in electromagnetism and quantum mechanics. I won't be easy, it never is, but it might be elegant. And maybe I'll dabble a little with curves...
Copyright © 1997, doug <sweetser@alum.mit.edu> All rights reserved worldwide.