Extra, Extra
From Motl's latest post (which discusses this paper):
Conventional physics uses quadratic Lagrangians, two-dimensional worldsheets, second-rank tensors under Yang-Mills groups, commutators between two objects, and similar structures based on the number "2" all the time. We know them quite well.Hmmm. Sounds familiar.
Still, it looks likely that there exists a whole realm of wisdom that remains mostly hidden in a cloud of mystery ... There exist hints that these largely unknown structures might be based on the number "3" in a similar way as the known theories are based on the number "2". This comment looks extremely vague but there are many reasons to see this prophesy.
posted by Kea | 11:57 AM
6 Comments:
Sounds familiar indeed. Perhaps stringy colleagues are finally starting to learn and begin to discover TGD. I wonder however how this horrible theory monster could be transfored to become hiddenly TGD in all its elegance. But some kind of clumsy mimicry is certainly possible.
Chern Simons action, the only possible variational principle for light-like 3-surfaces, is certainly the basic realization of threeness and makes topological QFT almost topological. It brings in 3-dimensionality and by holography you end up with 4-D space-time.
Matti, some M theorists are extremely talented physicists, and by no stretch of the imagination is their work undeveloped or clumsy.
I went and read some of the papers but they're too abstract for me to understand.
I'm reminded of the MUB stuff. For a Clifford algebra, everything shows up in 2s and the commuting roots of unity are square roots. The projection operators are linear: (1+A)/2.
For the MUBs of the 3-d Hilbert space, the commuting roots of unity have to be cubed roots of unity. The projection operators are quadratic: (1+A+A^2)/3/.
The papers are not so much abstract as couched in the traditions of string theory. I went to an excellent series of lectures by Mukhi a year or two ago on matrix models, which were very concrete and not abstract at all.
Thanks for bringing this to light Kea. The nonassociative field theory papers appear to be quite important. Following references, I came across a paper by Berman: arXiv:0710.1707v2. I found the section on nonassociative membrane theory to be quite potent (pgs. 55-61). There, Berman discusses Bagger and Lambert's original idea of having membrane fields lie in a nonassociative algebra. The idea is that an interacting M2-brane model would contain such fields, transverse to the three-dimensional worldvolume.
The nonassociative algebra is assumed to satisfy certain properties (eqs. 164-168, Berman pg. 59) and a carefully crafted example is given. What I find interesting is most of the properties (eqs. 164-167) are also satisfied by the exceptional Jordan algebra. In this case, the use of the associator is natural because the nonassociative octonions prevent one from re-writing associators as double commutators.
Bagger and Lambert have found the simplest non-trivial model to describe three M2-branes (arXiv:0712.3738v2). This physical pictures seems sensible, and may allow one to treat the three branes on equal footing via triality transformations.
It is nicer than the things LM said about Bee and me.
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