M Theory Lesson 20

Kholodenko’s paper ‘Heisenberg Honeycombs Solve Veneziano Puzzle’ shows how important the honeycomb concept is for physics. Take a look at Tao’s cool honeycomb applet. Your left and right mouse buttons will shrink/expand the hexagons in the diagram.

For a $3 \times 3$ Hermitean matrix a typical honeycomb looks like: Er, no. That’s a bit messy. The Ys should look alike. Anyway, note that the infinite lines go off to the north east, north west and south.

Now why would such hexagon diagrams be so important for physical ampitudes? Remember we tiled the real moduli $M(0,4)(\mathbb{R})$ with a line segment corresponding to the 1-dimensional Stasheff polytope for the associator, labelled by a 1-level 3-leaved tree. The 2-dimensional analogue is one of two kinds of hexagon, each labelled by a 2-level tree. Could this be what this is really about?

9 Responses so far »

  1. 1

    Anonymous said,

    It should be noted that the honeycomb’s idea is present already in the Nobel winning paper by Heisenberg. Neither Heisenberg nor anybody else till Kholodenko’s paper took advantage of such formulation of quantum mechanics. It is quite remarkable that develpment of this line of thought brings to life quite effortlessly Gromov-Witten invariants for small quantum cohomology ring. The whole machinery could be discovered much sooner should physicists and mathematicians listen to each other at those times. At the same time, a typical physicist of that generation would most likely say that such invariants are irrelevant for physics.

  2. 2

    Kea said,

    Yes, this is remarkable, both physically and from an historical point of view. I’m still working through Kholodenko’s papers, but I will certainly be continuing with this train of thought – although I am supposed to be doing many other things. Thanks, anonymous.

  3. 3

    Doug said,

    Hi Kea, I was unaware of the honeycomb idea in QM.

    Bees are associated with the honeycomb.

    Benzene rings are honeycomb-like when drawn in 2D.

    Why does nature continue to use similar structures at different scales?

    Still the most ubiquitous structure in nature appears to be the helix.
    It is theorized in QM by Hestenes; found in nucleic acids; used as trajectory space in game theory, ballistics and mechanics; associated with electromagnetic reconnections and has been actually been imaged in large scale near the galactic core.

    Mark Morris, [PDF] ‘The Galactic Center Magnetosphere’, figure 2.

  4. 4

    CarlBrannen said,

    Whoa! This is a remarkable insight, and I think I will be able to find uses for it very quickly.

    My interest, of course, is in the 3×3 matrices of primitive idempotents. We already know that these can be converted into 3×3 circulant Hermitian matrices and that these can represent the leptons.

    So the idea is that you take linear combinations of these to produce quarks. This sounds suspiciously like adding three 3×3 circulant Hermitian matrices. For the u, it would be 2x the positron plus 1x the anti neutrino, etc.

  5. 5

    Kea said,

    I think I will be able to find uses for it very quickly.

    That’s the idea, Carl!

  6. 6

    Kea said,

    Er, Carl … with a baryon mass formula, it won’t be so easy for people to dismiss these coincidences, will it?

  7. 7

    CarlBrannen said,

    Kea, I am quite certain that no one reads my Clifford algebra papers because I don’t write so well that I am that easily understood, and I have my share of typos, but no one points them out or asks questions.

    All they do is look at the very simplest of equations (Koide, or the new baryon equation) and say that they are just coincidences. It really doesn’t matter how many coincidences we write down, that will always be the answer. My Clifford algebra approach is even more lonely than your functors.

    But I am greatly heartened that this thing is solvable because the baryon “preon fine structure” is 1/729. It converges even faster than alpha, and it shows up in that damned number as well as the neutrino masses. Because of this, I am quite confident we will eventually have a completely unified solution with simple calculations.

  8. 8

    Kea said,

    Because of this, I am quite confident we will eventually have a completely unified solution with simple calculations.

    Oooh, that’s nice. Not that I doubted it. Sorry if I haven’t been too careful reading all your papers – I just feel my time can be better spent trying to understand things the M Theory way.

  9. 9

    Anonymous said,

    Addendum to Heisenberg’s honeycombs. I was waiting that somebody would notice that in this paper it is shown (quite explicitly) what is wrong with quantum mechanics as we know it from the the available texbooks. That, actually, it was discovered most likely by Kramers (not by Heisenberg as officially believed) and, that all these people who got their Nobel Prizes for its creation hid very skillfully who did what and when. By knowing this information, fortunately,situation can be repaired to a large extent, say, by using current mathematical results by Kerov, Vershik and Okounkov, for example, in view of their combinatorial and topological nature nicely compatible with available spectroscopic data. Sad but true that it looks like nobody is interested in the history of science. It is always good to know history before jumping with whatever conclusions regarding to what else can be done. Please, keep in mind that quantum mechanics first and foremost came out as an attempt to explain the available experimental data as good as possible. Such a task may look completely ugly for some mathematicians but in the end of the day, when it is used in physics, unless the theory describes experiment logically well enough, it is going to be dismissed irrespective to current mathematical fashions.

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