8 years ago

## Wednesday, August 10, 2011

### Theory Update 100

Kostant's $e_8$ algebra is clearly about qubits and qutrits. One considers the qubit cube $Z_{2}^{5}$, corresponding to the field $F_{32}$ with $32$ elements. This sits inside the $2 \times 2$ matrix group $SL(2,F_{32})$, conveniently in the form which is used for braids in emergent geometry. A $32$ dimensional information space could be a plane over the bioctonions. Triality also appears, because $248 = 192 + 56$.

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And recall that the oft appearing number $288 = 2^{5} 3^{2}$ satisfies $1/288 = 0.0035$. This is a parameter in the (triality respecting) form of the CKM quark mixing matrix. The other $2$ parameters are $24$ and $-0.231$.

ReplyDeleteAnd remember that there are three nice ways to place a $2 \times 2$ matrix inside a $3 \times 3$ one, as in the tripling of spin for the generation quantum number, resulting in the Koide relations for particle masses.

ReplyDeleteMarni, the link you provided to Konstant is also about John Baez talking about Garrett's e8. But his model has lots of higg's fairy fields, as you call them. Isn't that a bad thing?

ReplyDeleteDaniel, if you read two paragraphs of one of my papers, you would know that I have never, in any way or form, advocated a Lisi model. Lisi did not invent the Lie group $E_8$. Read the Kostant paper.

ReplyDeleteBy which I mean to say, mathematics is one thing and physics is another thing entirely. As kneemo has often pointed out, we need a $57$ dimensional picture for $E_8$ (in its Jordan algebra guise). This can be $57=56+1$, but observe that we also have $57-3 = 54$ with $54$ the $CO$ dimension, and the $54$ can be split into three lots of $18$ so that there is a nice $3 \times 3$ packaging for the $57$ dimensions.

ReplyDeleteOops, must correct the post then. Done.

Kea,

ReplyDeleteIn case you have not seen it my new graphs have reached many of the same number relations although I don't see how people do it without a picture.

In retrospect the idea of mass and fairy fields and SUSY seem a small part of the concepts- not to those bloggers with sensible imaginations. I like to count the empty spaces also.

The PeSla

Yes, ThePeSla, one should always aim for a viewpoint from which everything one was taught at school is but a small mound in the landscape below.

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