Saturday, April 16, 2011

Knotty Gauge Theory IV

If we chop the bracketing information off the (right hand) Jacobi rule, we are left with the three $1$-circulant permutations:

If braided with an over under pair (which must be the case for a braided Jacobi law) these diagrams represent a particle triplet, such as $(\gamma, e^{-}, e^{+})$. The left handed rule then corresponds to the mirror $(e^{-}, e^{+})$ pair, leaving the photon identity invariant. Thus our discussion of mirror gravity makes sense in the context of Zvi Bern's double Jacobi QCD. It is difficult to understand why physicists are still loathe to consider the Bilson-Thompson diagrams, given their obvious connection to categorical physics.


  1. Of course no one is really paying attention, but this is an important observation: the pair production diagram comes directly from the braided Jacobi rule, where the left (resp. right) braided pair form an annihilating couplet under braid composition (ignoring the bracketing trees).

  2. Your knotty gauge theory seems very similar in spirit to some of my work 15 years ago e.g. look for "is string theory in knots". When I get back from hols I will take another look at these posts

  3. "It is difficult to understand why physicists are still loathe to consider the Bilson-Thompson diagrams, given their obvious connection to categorical physics."

    Do the diagrams have a meaning outside of loop quantum gravity? As I understand it, LQG can't even describe a particle moving through ordinary space.

    Braids show up in field theory in other ways, e.g. Witten's Wilson loops in 2+1 dimensions. I have no idea what happens if you try to impose Bilson-Thompson's interpretation (or yours) in that case.

  4. Because they fear the Octopi - muahahaha! No. On a more serious note I have considered the BT model and how one might be able to find a natural embedding of braids if we take into account discrete symmetries in LQG ( Perhaps this is the reason physicists are loath to give me a postdoc!

    It can be hard keeping the faith, and it is posts such as on your blog that give me hope. Very nice blog Marni. Cheers!

  5. Deepak, if you are doing good theoretical work, you will almost certainly not get a postdoc. I applied for over 1000 postdocs. Mitchell, WTF do you think I have been talking about for the last 5 years? Do you even read the blog?

  6. It's inspiring to see someone having a go in this way, and maybe you do have a valid criticism - of physics culture, of human nature - that so few people are taking such radical leaps; but most such attempts at finding a short-cut really do fail outright, whereas the cautious strategy advocated by Arkani-Hamed is guaranteed to yield something. And if you apply that strategy mathematically, to the notion of braids having a special role in fundamental physics, you don't get the B-T model, because it's based on taking the LQG leap of faith that you can build a realistic dynamics out of those loop states, and then making a "brittle" association between a simple braid algebra and some of the simple algebraic properties of the particles. A cautious approach would instead focus on something like arxiv:1101.1329, where complexified knots in twistor space provde an alternative derivation of the BCFW recursion relations. There, you're working with field theories which already provide a functioning context in which all the qualitative properties of real particles can be described. There, you get a theorem and not just a guess, so you *know* it's real mathematics, and may even be part of real physics.

    What you're doing has not yet graduated either to the status of mathematical proof or to the status of physical prediction. With respect to the latter, it might be helpful if you made a post enumerating what you regard as successful predictions or retrodictions made so far (e.g. Graham D's calculation). Even if you disagree with my other judgments, I hope you'll agree that the physical logic behind the calculations you *have* made so far needs to become a lot clearer, and that will only happen if people can see how they work at present. As things stand, to the casual reader this blog is a disconnected blizzard of category-theory diagrams, peculiar claims about mirror particles, and reportage of theory news and experiment news. I've probably made more of an effort than most readers to crack the Kea code, and large sections of it are still opaque to me; I can't tell if there's logic at work, or just free association and the hope that everything will make sense in the end.

    In an attempt to conclude constructively, I'll repeat this suggestion: Let's see a post listing the positive results of your program so far; the qualitative or quantitative features of the world that your ideas explain. MINOS results, leptophobic Z', whatever, let's see them all in one place, along with the logic behind them.


  7. No contact with motion? And what do you call particle masses then? Don't tell a blogger what to post ... it displays mind numbingly bad manners. And your apparent assumption that you understand everything I have said so far is incredibly arrogant. Don't lecture me on what I have and have not achieved. You are in no position to do so. It is quite clear that I have never taken LQG seriously, putting far more emphasis on the ability of category theory to illuminate true background independence.

    There are no mathematical proofs in physics, and I never aspired or claimed to be a mathematician. (In fact, when I was younger everyone thought I should be a mathematician, but I have always gone to a great deal of effort to avoid it.)

  8. Until now, I didn't know for sure your attitude towards LQG. It was always possible you thought of it as an intermediate level of description. I also just took a look at your withdrawn preprint from 2004. Is "space/matter duality" the bridge between quantum gravity and the usual space-time description?

    Regarding particle masses, apart from wanting to know when and how they end up in their usual context of things moving through space, I wonder whether the framework that you and Carl have devised can accommodate the running of particle masses. Could you have an RG equation for the Koide matrices in which the matrices have a functional dependence on energy scale? Does that make sense in your system? I think you'll need something like it.

  9. Mitchell, since modern renormalization theory is expressed in terms of Hopf algebras of trees (and I even mentioned this yet again recently) it already has a categorical context, and Carl has been busy working on getting quantum numbers from Hopf algebras. Yes, eventually the motivic description should be powerful enough to do everything. Nima seems to have realised this now, although he hasn't thought much about the actual physics. But you should be careful about thinking of QFT in traditional terms: the non local version of QFT is different, and there is no reason we should ever return to the usual point of view, so long as we can reproduce all experimental results.

  10. Mitchell, the only way I was able to truly appreciate what Kea is doing is by sitting around talking with her about it. This is much easier for the same reasons that learning a subject is easier when it's taught in a class (or better, when it's taught one on one).

    As a grad student I was unable to learn subjects by myself that I later found quite easy to learn in class. It's a matter of feedback.

    Remembering those conversations with Kea, what I'd really like to see is around a 12 page paper describing how she came to be working on the subject. From the outside it might look like she arbitrarily picked a subject but instead the subject forced itself on her for natural and logical reasons; I think that these can best be understood in terms of a description of the evolution of her ideas rather than the current state. This may be because I'm rather primitive in mathematics.

    Which reminds me, I'm currently reading Ivancevic & Ivancevic's paper 0810.2070 "Applied Categories and Functors for Undergraduates". It seems that my biggest issue is just the straight memorization of the zillions of words they define (which is what I've always hated about algebra).

  11. Yes, modern mathematics has an issue with jargon overkill. There are countless variations on the few really important definitions, all with their own terminology. I guess that's what happens after an age of isolated specialization. Thank goodness that age is coming to an end.


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