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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Supersymmetric world from a conservative viewpoint. John Baez has always been obsessed by octonions. He sees octoions everywhere. 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.

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.

As I will argue, they are really abusing the stellar brand of string theory to promote their idiosyncratic bullshit. They’re not even the most occtonions or most symmetric ones. 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.

It has to be a power of two in a minimal supersymmetric gauge theory because the number of bosonic polarizations has to match the number of fermionic polarizations and the latter come from a spinor; I don’t need division algebras to prove that. Also, I don’t need division algebras to prove bwez minimal super Yang-Mills theories can only exist in these four dimensions.

Also, the “other remarkable structures” that directly come from the R,C,H,O sequence are not that interesting.

AMS :: Bulletin of the American Mathematical Society

He also talks about membranes in 4,5,7,11 dimensions not surprising that they have the same counting of physical degrees of freedom – it’s just double dimensional reduction. Except for the last one, they’re not terribly bwez theories or vacua. So the detailed data refute Baez’s hypothesis; he clearly doesn’t care. You simply can’t understand the secrets of string theory by learning the sequence 1,2,4,8.

In the slow comments under the blog entry, Octonkons 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. There’s lots of fascinating, still poorly understood mysteries about string theory’s internal mathematical consistency. 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. However, counting of the dimensions is one of the most elementary facts about string theory.

This Week’s Hype

All the cancellations that Baez et al. Joe Polchinski boasts that volume I of his book derives the critical dimension of the bosonic string theory in 7 different ways. So why is there so much ado about nothing? Our ability to derive results in 7 different ways shows that we kind of understand it. Moreover, as hinted in the previous sentence, what I am really irritated by is Baez’s obsessive tendency to reduce all the mathematical cleverness of string theory to the division algebras in general and octonions in particular.

His way of looking at all these things shows that he is nothing else than an irrational numerologist who can never distinguish real insights from superficial distractions – and who apparently doesn’t want to distinguish them.

The octonions as an algebra are just one possible corollary or intellectual projection of the structure behind it. And even all these structures combined are just a totally minuscule portion of the string theory’s wisdom, much like John Baez’s knowledge of string octoninos is an infinitesimal fraction of the knowledge of a good octinions student. So the article hyping a set of a few simple mathematical observations is just pathetic.

It’s not really demonstrably wrong – unlike Garrett Lisi’s pseudoscientific “theories of everything” that can’t agree with the most elementary facts of particle physics such as parity violation.

But it’s still morally wrong because it totally distorts what is understood and what remains mysterious about the remarkable mathematical structure we still call string theory. This distortion bae be unexpected from authors who don’t have a clue about string theory. But this is no real excuse because Scientific American shouldn’t be publishing stuff written by people who don’t know what they’re talking about.

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[math/] The Octonions

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