Strap yourselves in; here comes the juice

Float to the top or sink to the bottom. Everything in the middle is the Churn.

I wrote a paper, and, when requested, I published it (https://link.springer.com/article/10.1007/s00006-018-0820-8). It’s a very pretty paper, extending the notion of 8-d Bott periodicity in Clifford algebras to a 24-d periodicity. This mirrors the exceptionality of the dimensions 2, 8, and 24 in lattice theory. Hyper-cool.

“There was a button,” Holden said. “I pushed it.” 
“Jesus Christ. That really is how you go through life, isn’t it?”

As is true of almost everything I’ve written and often published in theoretical and mathematical physics, the paper linked to above (call it Bobbie) utilizes the 3 division algebras, C, H, O, which are linked to even more fundamental topological objects, the parallelizable spheres of dimensions, 1, 3, and 7.

I frequently encounter papers exploiting the split octonions in their theory building. This requires replacing 4 of the 8 basis elements of O with i times those elements, where customarily i is taken to be the imaginary unit of C. One can also make split versions of H by similarly modifying 2 of its basis elements. But this method does not extend to C, as i is an element of C already. This is rather annoying.

In Bobbie this problem was remedied by replacing i by a new imaginary unit ι, and supplementing C,H,O with a distinct complex number field, C. Via the imaginary unit ι, all 3 of the division algebras have split versions. It’s very cool.

I am that guy.

Then it occurred to me …

C is generally assumed to be the number system from which arises all of complex analysis. C is recognized to be part of a unique and exceptional finite series of division algebras, and inspired by this fact over the past several decades many have tried to turn H and O into analytical doohickeys in the manner of C. These efforts have met with varying degrees of failure. And why?

Because C is not the foundation of complex analysis, C is. C,H, and O, are a set of mathematical objects relating to topology, geometry, and algebra. They are architectural. They can be viewed as real algebras, but they are really C algebras, where C is the mathematical field from which all our nice analysis – Green’s functions, Cauchy-Riemann equations, QFT, … – arise. So, because something is true of C does not mean it can be generalized to H and O. The field C is separate; it is not part of a series of mathematical objects.

Got it? All the division algebras have split versions. H and O are not the foundations of higher dimensional analytical theories. And C isn’t either.

Mother Nature suspects you will doubt this obvious Truth. Punishment is being prepared to meet your obstreperous petulance.