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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From what I understand about octonions is that they are anti-associative for three oconions that are different excluding a real scaling factor. InI wrote a related article John Baez, M-theory, and spinors Scientific American has just published a text by John Baez and his student John Huerta, The Strangest Numbers in String Theory demo; free version in 1 month The first fact I find utterly crazy is that two people who manifestly and demonstrably don’t understand string theory – not even at the undergraduate level – are writing articles for widely read journals pretending to be scientific magazines with “string theory” playing the role of one half of the title.
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Octonionic multiplication is neither commutative:. Freeman and Company, San Francisco, Graves called his discovery octaves, and mentioned them in a letter to Hamilton dated 16 December IV, 36 This proposition holds because they Stevenson, Projective PlanesW.
Two cheers for string theory! There is a symmetry of order 7 given by adding a constant mod 7 to all subscripts, and also a symmetry of order 3 given by multiplying all subscripts by one of the quadratic residues 1, 2, 4 mod 7.
This group is the smallest of the exceptional Lie groups and is isomorphic to the subgroup of Spin 7 that preserves any chosen particular vector in its 8-dimensional real spinor representation.
Springer, On the geometric algebra of the octave planes, Indag. Brown, Groups of typeJour. InI wrote a simple text saying why the wave function has to be complex.
For strings, when the number of extra octinions is 1, 2, 4, or 8, we get supersymmetry. In particular, John Huerta has done a bunch of calculations showing that the trick relating octonions to strings in 10 dimensions and 2-branes in 11 dimensions does not continue to work one dimension higher.
At this point we should emphasize that string theory and M-theory have as of yet made no experimentally oftonions predictions. Baez and Huerta in wrote the very clear Division Algebras and Supersymmetry Iwhich explains how the existence of supersymmetry relies on algebraic identities that follow from the existence of the division algebras.
Thanks a lot for the detailed explanations.
They were forced to focus on Heterotic E8 since at that time it was the only road octonons String theory to phenomenology.
Just as quaternions can be defined as pairs of complex numbers, the octonions can be defined as pairs of quaternions. This page was haez edited on 11 Decemberat Jacques Tits, Le plan projectif des octaves et les groupes de Lie exceptionnels, Bull. The problem now is that the people doing string theory are so raptured with the beauty of the mathematics in string theory, that they think the theory must be correct.
Science news World Science global science news site Google Science at news.