Nehn Home Frequently Asked Questions. In discrete mathematics, bwez octonions provide an elementary derivation of the Leech latticeand thus they are closely related to the sporadic simple groups. The present Your calendar The alternative law gives the spinor identity that makes supersymmetry work for super-Yang-Mills theory and classical superstrings in dimensions 3, 4, 6 and Then multiplication is given by. But then the show stops: The problem with what has come out of string theory is not that it is based on too much abstract mathematics, but that it is based on a physical idea that turned out to be wrong. May 3, at I read the article and enjoyed it. John Baez on Octonion????
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There are exactly four normed division algebras: the real numbers , complex numbers , quaternions , and octonions. The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.
Most mathematicians have heard the story of how Hamilton invented the quaternions. In , at the age of 30, he had discovered how to treat complex numbers as pairs of real numbers.
Fascinated by the relation between and 2-dimensional geometry, he tried for many years to invent a bigger algebra that would play a similar role in 3-dimensional geometry.
In modern language, it seems he was looking for a 3-dimensional normed division algebra. His quest built to its climax in October He really needed a 4-dimensional algebra.
Finally, on the 16th of October, , while walking with his wife along the Royal Canal to a meeting of the Royal Irish Academy in Dublin, he made his momentous discovery. And for a while, quaternions were fashionable. They were made a mandatory examination topic in Dublin, and in some American universities they were the only advanced mathematics taught. Much of what we now do with scalars and vectors in was then done using real and imaginary quaternions.
Tait wrote 8 books on the quaternions, emphasizing their applications to physics. When Gibbs invented the modern notation for the dot product and cross product, Tait condemned it as a "hermaphrodite monstrosity". A war of polemics ensued, with luminaries such as Heaviside weighing in on the side of vectors. Ultimately the quaternions lost, and acquired a slight taint of disgrace from which they have never fully recovered [ 24 ].
The very day after his fateful walk, Hamilton sent an 8-page letter describing the quaternions to Graves. Graves replied on October 26th, complimenting Hamilton on the boldness of the idea, but adding "There is still something in the system which gravels me.
I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties. In January , Graves sent three letters to Hamilton expanding on his discovery. In July he wrote to Graves pointing out that the octonions were nonassociative: " , if be quaternions, but not so, generally, with your octaves. Meanwhile the young Arthur Cayley, fresh out of Cambridge, had been thinking about the quaternions ever since Hamilton announced their existence.
He seemed to be seeking relationships between the quaternions and hyperelliptic functions. Apparently as an afterthought, he tacked on a brief description of the octonions.
In fact, this paper was so full of errors that it was omitted from his collected works — except for the part about octonions [ 16 ]. Upset at being beaten to publication, Graves attached a postscript to a paper of his own which was to appear in the following issue of the same journal, saying that he had known of the octonions ever since Christmas, Still worse, Graves later found that his eight squares theorem had already been discovered by C.
Degen in [ 25 , 27 ]. Why have the octonions languished in such obscurity compared to the quaternions? Besides their rather inglorious birth, one reason is that they lacked a tireless defender such as Hamilton. But surely the reason for this is that they lacked any clear application to geometry and physics.
The unit quaternions form the group , which is the double cover of the rotation group. This makes them nicely suited to the study of rotations and angular momentum, particularly in the context of quantum mechanics.
These days we regard this phenomenon as a special case of the theory of Clifford algebras. Most of us no longer attribute to the quaternions the cosmic significance that Hamilton claimed for them, but they fit nicely into our understanding of the scheme of things. The octonions, on the other hand, do not. Their potential relevance to physics was noticed in a paper by Jordan, von Neumann and Wigner on the foundations of quantum mechanics [ 55 ].
However, attempts by Jordan and others to apply octonionic quantum mechanics to nuclear and particle physics met with little success. Work along these lines continued quite slowly until the s, when it was realized that the octonions explain some curious features of string theory [ 60 ].
The Lagrangian for the classical superstring involves a relationship between vectors and spinors in Minkowski spacetime which holds only in 3, 4, 6, and 10 dimensions. Note that these numbers are 2 more than the dimensions of and. As we shall see, this is no coincidence: briefly, the isomorphisms allow us to treat a spinor in one of these dimensions as a pair of elements of the corresponding division algebra.
It is fascinating that of these superstring Lagrangians, it is the dimensional octonionic one that gives the most promising candidate for a realistic theory of fundamental physics! However, there is still no proof that the octonions are useful for understanding the real world. We can only hope that eventually this question will be settled one way or another. Besides their possible role in physics, the octonions are important because they tie together some algebraic structures that otherwise appear as isolated and inexplicable exceptions.
As we shall explain, the concept of an octonionic projective space only makes sense for , due to the nonassociativity of. This means that various structures associated to real, complex and quaternionic projective spaces have octonionic analogues only for.
Simple Lie algebras are a nice example of this phenomenon. These were discovered by Killing and Cartan in the late s. At the time, the significance of these exceptions was shrouded in mystery: they did not arise as symmetry groups of known structures. Only later did their connection to the octonions become clear. It turns out that 4 of them come from the isometry groups of the projective planes over , and.
The remaining one is the automorphism group of the octonions! Another good example is the classification of simple formally real Jordan algebras. Minimal projections in this Jordan algebra correspond to points of , and the automorphism group of this algebra is the same as the isometry group of. The octonions also have fascinating connections to topology. In , Raoul Bott computed the homotopy groups of the topological group , which is the inductive limit of the orthogonal groups as.
He also computed the first 8: Note that the nonvanishing homotopy groups here occur in dimensions one less than the dimensions of , and.
This is no coincidence! In a normed division algebra, left multiplication by an element of norm one defines an orthogonal transformation of the algebra, and thus an element of. This gives us maps from the spheres and , and these maps generate the homotopy groups in those dimensions.
As we shall see, this is true. Conversely, Bott periodicity plays a crucial role in the proof that every division algebra over the reals must be of dimension 1, 2, 4, or 8. In what follows we shall try to explain the octonions and their role in algebra, geometry, and topology.
Each approach has its own merits. In Section 3 we discuss the projective lines and planes over the normed division algebras — especially — and describe their relation to Bott periodicity, the exceptional Jordan algebra, and the Lie algebra isomorphisms listed above.
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As I will argue, they are really abusing the stellar brand of string theory to promote their idiosyncratic bullshit. Also, the "other remarkable structures" that directly come from the R,C,H,O sequence are not that interesting. In the slow comments under the blog entry, Robert Helling argued that there is a lot of interesting fog about the closure of the supersymmetry algebra etc. I find this whole approach to these issues irrational. But the topics that Baez, Huerta, Helling, and others are talking about are pretty much exactly those where the mystery has already been fully eliminated. The mysterious links to pure mathematics continue to exist but you must get much deeper to uncover them.
BAEZ OCTONIONS PDF
Submitted by plusadmin on January 1, January John Baez is a mathematical physicist at the University of California, Riverside. He specialises in quantum gravity and n-categories, but describes himself as "interested in many other things too. In a two-part interview in the previous issue of Plus and this one, Helen Joyce, editor of Plus, talks to Baez about complex numbers and their younger cousins, the quaternions and octonions. Photo courtesy of Tevian Dray We resume our story with the invention of quaternions - lists of 4 numbers that come with a rule for multiplying and dividing. Describing the moment when inspiration struck, their inventor, William Rowan Hamilton said: "And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples