M Theory Lesson 43
Terence Tao asks the question, what is a quantum honeycomb? Recall that the usual honeycombs for 3x3 complex matrices involve a single hexagon in the plane. Let us resolve the vertices of this hexagon into hexagons. 
M theory requires octonion Hermitean matrices. Instead of a combination of three matrices resulting in another complex matrix, with 9 external edges, we consider 27 external edges in the triple strand case. Note the scale invariance of the diagram in zooming either inwards or outwards. Altering the local geometry does not affect the rigidity of the pattern. Apologies for the omission of the top Y from the diagram and for the wonky edges.
Drawing a triangle around the triple strand diagram with a vertex at the top of the diagram, we observe that outside the central hexagon there will be 6 pentagons and 3 squares. Thus the cylinder between the triangle and the hexagon happens to have a tiling by the faces of a Stasheff polytope. We could glue two such cylinders together to make a pretty torus geometry.
M theory requires octonion Hermitean matrices. Instead of a combination of three matrices resulting in another complex matrix, with 9 external edges, we consider 27 external edges in the triple strand case. Note the scale invariance of the diagram in zooming either inwards or outwards. Altering the local geometry does not affect the rigidity of the pattern. Apologies for the omission of the top Y from the diagram and for the wonky edges.Drawing a triangle around the triple strand diagram with a vertex at the top of the diagram, we observe that outside the central hexagon there will be 6 pentagons and 3 squares. Thus the cylinder between the triangle and the hexagon happens to have a tiling by the faces of a Stasheff polytope. We could glue two such cylinders together to make a pretty torus geometry.
posted by Kea | 9:35 AM
8 Comments:
Those dang hexagons only managed to have 24 outputs in the diagram. But this is very interesting. How do you get numbers or other results from it?
Also, did you take a good look at the latest from Matti? Observational evidence in favor of tripled Pauli statistics.
Must rush... it's 27 if you put the missing Y on the top.
Carl, I'm thinking tripled Pauli statistics is applicable the entangled three qutrit description of supergravity black holes. To get a feel for handling three qutrits, read Higuchi's On the one-particle reduced density matrices of a pure three-qutrit quantum state.
I am somewhat amused by Tao's desision to imrove his own work and to go ahead with "quantum" honeycombs as if other honeycombs have something to do with non quantum (that is classical) reality. In Heisenberg's honecomb paper its is shown how, if needed, the Schrodinger picture can be restored. By inventing "quantum" honeycombs one can play games indefinitely (as long as situation permits) but one should not be mislead by the title "quantum" when one wants to think about applications to reality (this is usually what most mathematicians try to avoid at all cost)
OK, Carl, I saw your comments over on Matti's blog. Yes, it sounds pretty cool - quantitative evidence! Somehow, 'though, I don't think anybody is going to notice.
Dear Kea,
you are right. No one will notice! I have collected during these 28 years enormous pile of material, anomalies after anomalies finding a nice explanation in TGD framework and constructed refined theoretical framework but only very few colleagues notice it. It is not time for physics now.
My luck is that working with such a radical idea as TGD is so a rewarding an adventure that there remains no time wasted to bitter feelings. It is however sad to see that people stubbornly blowing their head on the wall again and again.
Something has gone badly wrong in the culture of theoretical physics when communication is virtually impossible although possibilities to communicate are better than ever and you can learn almost anything you want by just going to Wikipedia. Despite this too many theoretical physicists know nothing about what happens outside their specialization. The lack of these stimuli means an enormous loss of inspiration and leads to a loss of sense of proportionality.
Matti
Michael,
I've seen those papers, but they're a combination of over my head and under my intuition.
I like the tripled Pauli statistics and the dark matter / dark energy stuff because it is very simple and easy to interpret.
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