Conclusions

What's been done
What's new
What needs to be done

What's been done

It is an old dream (or perhaps more accurately, a recurring nightmare) to express laws of physics using quaternions.  In these web pages, quaternion operators were employed to express central laws of physics:  Newton's second law, the Maxwell equations, and the Klein-Gordon equation for relativistic quantum mechanics.  Applications of quaternions to special relativity were done in detail, with over 50 problems worked out explicitly.  Quaternions do not make problem solving easy.  Rather, they help unite the laws themselves.  Significantly, an analysis of the length of a quaternion interval if the origin is moved establishes a connection to the machinery of general relativity, the affine parameter.

What's new

One might suspect that the reason for the success claimed above is that nothing proposed is new.  After all, quaternions are a linear combination of tensors of rank zero and one, and while used in a new way here, does anything genuinely novel appear?

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.

What needs to be done

Nothing presented was proven rigorously.  There were few references to the literature.  This body of work is more like a skeleton of work, a reflection of the author's semi-formal training and isolation from the professional physics community.  There is a need to flesh these ideas out only if there is the potential for new insights.  Although I appreciate the standard approach, I feel like I have gained new insight into why Maxwell's equations are necessary, have a new way to view relativistic quantum mechanics, and cling to a novel toehold on general relativity.  And since each of these is a quaternion, it becomes possible to mix and match them to create new areas of study.  I hope this work generates interest in the physics community.

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...