THE OCTONIONS. JOHN C. BAEZ. Abstract. The octonions are the largest of the four normed division algebras. While somewhat neglected. The expository article about octonions by John (Baez) that appeared in the AMS Bulletin (copy here, a web-site here) is one of the best pieces. After the octonions comes a dimensional number system called the ” sedenions”. They have some nice features, which I describe in this.
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I love octonions but I realize that they don’t play much role in most of physics – and not even in most of string theory. So why is there so much ado about nothing? Hossenfelder’s pathetic attack against CERN’s future collider. B Not being associative, the nonzero elements of O do not form a group.
While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of ocgonions. But so far, all I naez is that classical superstrings favor dimensions 3, 4, 6 and 10, thanks to the four normed division algebras.
So, we get supersymmetry when the total number of dimensions is 3, 4, 6, or Octonions have also arisen in the study of black hole entropy and quantum information science.
Graves called his discovery octaves, and mentioned them in a letter to Hamilton dated 16 December Algebra Sedenionsnot alternative but power associative.
The ictonions integral octonions are exactly those of prime norm, and every integral octonion can be written as a product of irreducible octonions. So when John Huerta finally made it really clear, it seemed worth explaining — in detail in some math papers, and in a popularized way in Scientific American. April 29, at 8: The existence of a norm on O implies the existence of inverses for every nonzero element of O.
They’re just algebras in four dimensionalities D such that D-2 – the number of physical polarizations of a gauge boson – is a power of two. 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.
John, Thanks a lot for the detailed explanations. Since, the ORDER of the octonions determines the value, has anyone tried to make a consistent theory of spacetime using octonions?
I never understood the big deal with octonions. But this is octomions real excuse because Scientific American shouldn’t be publishing stuff written by people who don’t know what they’re talking about. The article says quite clearly that superstring theory makes no testable predictions:. HTML code with the link pointing here.
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We also touch upon their applications in quantum logic, special relativity and supersymmetry. Zathras, String theory is not based on rigorous mathematics.
At this point we should emphasize that string theory and M-theory have as of yet made no experimentally testable predictions. Freeman and Company, San Francisco, Octonions are related to exceptional structures in mathematics, among them the ooctonions Lie groups.
There’s lots of fascinating, still poorly understood mysteries about string theory’s internal mathematical consistency. Elon Musk has just unveiled the Hyperloop, well with the air in the tunnels included.
April 30, at 5: From Wikipedia, the free encyclopedia. You can even go ahead and define the octonionic projective plane and PSL 3,Owhich turns out to be a certain real form of E6.
Every haez is a real linear combination of the unit octonions:. By way, I just won a case of scotch from Dave Ring: There is only one String theory.
Octonion – Wikipedia
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Moyses has pointed out that I haven’t discussed the July paper by Maldacena-Milekhin-Popov Princeton about a new wormhole soluti The lines are directional. Joe Polchinski boasts that volume I of his book derives the critical dimension of the bosonic string theory in 7 different ways.
These four algebras also form the only alternative, finite-dimensional division algebras over the reals up to isomorphism.