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Insights Why Higher Category Theory in Physics? - Comments

  1. Jan 4, 2017 #1

    Urs Schreiber

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  2. jcsd
  3. Jan 4, 2017 #2
    Just a few days ago, for the first time, a paper by Patricia Ritter gave me an explanation that I could understand, for the relevance of n-categories to objects like branes - the categorical identities express the equivalence of different ways of doing certain integrals over a volume, e.g. where in effect you might first integrate in the x direction, then along the xy plane, then throughout the xyz volume; but you might have done all that for a different order of x,y,z... the result needs to be the same for all orderings, and that leads to the categorical formulation of higher gauge theory.

    I want to emphasize, that's not exactly what she says, that's me trying to dumb it down to the simplest way of saying it. But am I even approximately correct in this interpretation?
  4. Jan 5, 2017 #3

    Urs Schreiber

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    Yes, that's one good way of thinking about it. This is the motivation from "higher parallel transport".

    Like so: the structure of a group (an ordinary group) is exactly what one needs in order to label the edges in a lattice gauge theory: the group product and associativity give that edge labels may be composed, and inverses gives that going back and forth along the same edge picks up no curvature. Of course this is not restricted to the lattice. In general, group-valued gauge fields are exactly the right data to have consistent Wilson line observables

    Now a 2-group (categorical group) is, similarly, exactly the data needed to consistencly label edges AND plaquettes in a consistent way (with possibly different labels for each). For instance associativity now includes a 2-dimensional codition which says that with four plaquettes arranged in a square, then first composing horizontally and then vertically is the same (in fact: is gauge equivalent to) first composing vertically and then horizontally.

    Again this is not restricted to the lattice. Generally, 2-group valued gauge fields are exactly what one needs for consistent Wilson surfaces.
    Last edited: Jan 5, 2017
  5. Jan 5, 2017 #4
    Fascinating topic Urs!
  6. Jan 29, 2017 #5
    This is not even unconvincing, to coin a phrase. If you cannot connect this mathematics to a physical problem other than gravity, it's almost certainly useless, since you are unable to make tangible statements about experimental reality.

    There's lots of physics out there in need of new mathematical models. Non-equilibrium quantum/classical physics of few or many bodies, soft matter, fluids, strongly correlated materials etc. Why, a great example would be in 2D materials; we already know that some exhibit Dirac excitations, and speculation has been ongoing for years on condensed phase simulations of gravity. Hawking radiation in dumb holes has already been verified or at least strongly suggested in Bose gases if I am not mistaken.

    Without that, there is absolutely no reason for any physicist to take this seriously.
  7. Jan 30, 2017 #6
    It's maths, @Crass_Oscillator ,if one day some physicists will find applications for it, then why not?

    BTW there's the book by Lawvere on Categories in continuum mechanics, so it seems mathematicians are working hard of finding applications of this work in the physical sciences.

    Will it work?

    Who knows, but for the maths sake I still will like to learn this stuff, eventually something beneficial will come from it, even if not in physics.

    BTW, @Urs Schreiber do you work at the maths or physics department? :-D

    Anyway, how did Feynman once said about sex and physics:" sure, it's practical, but that's not why we do it".
  8. Jan 30, 2017 #7
    Well, Feynman is also reputed to have said that "Math is to physics as masturbation is to sex." ;)

    However that's not really the point of my gripe. Urs is trying to sell higher category theory to physicists judging by the title alone, but he doesn't really seem to grasp how to make a compelling case to physicists. I could have been polite, but instead I decided to give him a hard prod. I'm very mathematically conservative and am no fan of math driven physics research, but I do recognize the value of more sophisticated mathematics. He needs to make a much better case.

    Mathematicians often get lost in how, for instance, the machinery they've constructed shortens previously elaborate proofs, or paves the way to proving that certain mathematical structures have exciting, exotic properties, but I use these mathematical structures to construct models. I don't care about proofs, save quick and dirty ones, and much of physics still doesn't get far past super-charged 19th century calculus, with topology and advanced algebra sprinkled in here and there.

    Perhaps starting with Lawvere's book would be the place to begin. And it had better simplify calculations, not make them more complicated. Or, allow me to calculate something that previously was difficult. If Lawvere's book suggests algorithms for solving continuum mechanics problems that nobody had thought of which are competitive with simpler 20th or 19th century algorithms or allows you to build models that obtain experimental observables that are hard to describe otherwise, it's interesting to a physicist. Otherwise, it's a waste of time.
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