published it<\/a> (https:\/\/link.springer.com\/article\/10.1007\/s00006-018-0820-8). It\u2019s 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.<\/p>\n\n\n\n<\/p>\n\n\n\n
\u201cThere was a button,” Holden said. “I pushed it.” As is true of almost everything I\u2019ve written and often published in theoretical and mathematical physics, the paper linked to above (call it Bobbie<\/em>) utilizes the 3 division algebras, C<\/strong>, H<\/strong>, O<\/strong>, which are linked to even more fundamental topological objects, the parallelizable spheres of dimensions, 1, 3, and 7. <\/p>\n\n\n\nI frequently encounter papers exploiting the split octonions in their theory building. This requires replacing 4 of the 8 basis elements of O<\/strong> with i times those elements, where customarily i is taken to be the imaginary unit of C<\/strong>. One can also make split versions of H<\/strong> by similarly modifying 2 of its basis elements. But this method does not extend to C<\/strong>, as i is an element of C<\/strong> already. This is rather annoying.<\/p>\n\n\n\nIn Bobbie<\/em> this problem was remedied by replacing i by a new imaginary unit \u03b9, and supplementing C<\/strong>,H<\/strong>,O<\/strong> with a distinct complex number field, C.<\/em> Via the imaginary unit \u03b9, all 3 of the division algebras have split versions. It\u2019s very cool. <\/p>\n\n\n\n<\/p>\n\n\n\n
I am that guy.<\/strong><\/p>\n\n\n\nThen it occurred to me \u2026<\/p>\n\n\n\n
C<\/strong> is generally assumed to be the number system from which arises all of complex analysis. C<\/strong> 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<\/strong> and O<\/strong> into analytical doohickeys in the manner of C<\/strong>. These efforts have met with varying degrees of failure. And why?<\/p>\n\n\n\nBecause C<\/strong> is not<\/em> the foundation of complex analysis, C<\/em> is. C<\/strong>,H<\/strong>, and O<\/strong>, 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<\/em> algebras, where C<\/em> is the mathematical field from which all our nice analysis – Green\u2019s functions, Cauchy-Riemann equations, QFT, \u2026 – arise. So, because something is true of C<\/em> does not mean it can be generalized to H<\/strong> and O<\/strong>. The field C<\/em> is separate; it is not part of a series of mathematical objects.<\/p>\n\n\n\nGot it? All the division algebras have split versions. H<\/strong> and O<\/strong> are not the foundations of higher dimensional analytical theories. And C<\/strong> isn\u2019t either. <\/p>\n\n\n\nMother Nature suspects you will doubt this obvious Truth. Punishment is being prepared to meet your obstreperous petulance. In a private conversation with Mother Nature, she told me, rather firmly: \u201cThis is not a suggestion.”<\/p>\n\n\n\nA brief word on futility. There is a high probability that everyone who might have found this entry of interest is someone I\u2019ve met, or connected to someone I\u2019ve met. The degrees of separation are small. And none of these fellow hapless mavericks is in a position to influence the currents of scientific history. Anyway, I don\u2019t know about the rest of you, but I\u2019d not have done anything differently even knowing that recognition of such ideas as these will not even occur posthumously. <\/p>\n\n\n\nWell, that was fun.<\/figcaption><\/figure>\n\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"
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\u2019s 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, … <\/p>\n
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