Mottle calls it a twistor minirevolution. To stringers, this is the idea that one can completely reformulate the Standard Model without in any way affecting the physical basis for ordinary string theory. (I have personally heard some twistor mathematicians laugh at such nonsense, but what would they know?) Anyway, it seems that most stringers are rather unkeen to investigate ternary extensions of basic twistor geometry.
Thanks to kneemo's excellent advice I have lately been reading two wonderful papers (by Corinne Manogue and collaborators) on an octonion analogue of the complex group with which one begins to study twistors, namely $SL(2,C)$. The 2009 paper defines a group $SL(3,O)$ over the octonions. This behaves like the group $E_6$, which contains many subgroups of interest to stringers, such as three copies of $SO(9,1)$, acting on a $10$ dimensional Lorentzian space. This group is like the octonion group $SL(2,O)$, built from complex octonion matrices. We could also make $3 \times 3$ matrices with $2 \times 2$ blocks from $SL(2,O)$.
14 years ago
Do you think it would help if I stood on a hill here in New Zealand and screamed "PAY ATTENTION!!!!!".
ReplyDeleteI'm paying attention ! Thanks for reminding about the Manogue & Dray papers. Even with the smaller complex octonions it is interesting to think of what Pauli would have done with this - perhaps he would have seen the muon and tau as well - not likely in the twenties, but one might daydream.
ReplyDeleteHi Joel. Theory never gets that far ahead of experiment. The stringers should have remembered that.
ReplyDeleteIf I may hazard a guess about Complex Octonions and Physics - it seems that octonions want to put Bosons and Fermions on the same footing, as if bosons are just another generation, except that the signature is +++ rather than +-- in the even subalgebra. It might also like to assume the opposite of the usual assumption that particles need to be massless - if all have 3 polarizations. a rough first pass might make a Z look like a kind of "anti-photon", which sounds totally absurd, like maybe there is a different kind of symmetry breaking.
ReplyDeleteYou are on the right track, Joel. Indeed, in the real M theory, supersymmetry does magical things with bosons and fermions, without needing to add arbitrary extra particles.
ReplyDeleteI am a bit surprised that physicists seem to have jumped over plain old complex octonions on their way to supersymmetry (eg Baez-Huerta) as usually there is a step by step. After all, complex octonions should inherit any physics that exists in complex quaternion algebra (same as pauli algebra). That is why I question Geoff Dixon's approach - you end up with more dimensions. Of course OxHxC looks like SU3xSU2xU1. I see some of the rationale for Susy - but there is an aesthetic appeal when complex octonions give you +--- and -+++ for free (and no other signatures). What other little goodies are hiding in there - for free ? Perhaps Susy is a leap too far. As to cancellation, it is very cool that octonions can cancel antiassociativity and anticommutativity to recover the ordinary Peano axioms of ordinary arithmetic. But I tend to focus on the Generation Problem just because it is driving Veltman crazy - along with N other physicsts. Generation structure looks easy in complex octonions - you just consider +---, -+--, --+-, ---+ and that gives you four classes of things that look like oscillators - basically for free. No Susy required. No boson-fermion pairing. In fact: no Lagrangians, no conservation laws, no PDEs, hardly any group theory. Of course, you don't get much physics except generations of thingies that sort of have something to do with particles because duality oscillators are a generalization of photons - they all have a complex phase, which makes it easy to connect to Feynman's approach in QED. One does need both the real 16 component frames, as well as the 8 component complex frames, for the unitary reps. Instead of "beyond" the standard model, it looks like it is more fundamental than the standard model - underneath it, and simpler in its own way, as well as being an alternative algebra, whereas Dixon and Baez-Huerta are not.
ReplyDeleteWhat other little goodies are hiding in there?
ReplyDeleteAh, your mind will be blown when you see it. And it IS more fundamental than the SM! That is why it is called M Theory. Lagrangians belong to the old local classical continuum physics, which is why it is important to study twistors if you want to understand all the steps from M Theory back to SM and GR.