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Quantum Gravity: Lie Groups vs. Banach Algebras & Spectral Theory

  1. Jul 19, 2013 #1
    Quantum Gravity: "Lie Groups" vs. "Banach Algebras & Spectral Theory"

    I'm interested in researching quantum gravity & non-commutative geometry. I am planning to take one math course outside of my physics classes this Fall to help, but can't decide between two: "Lie groups" or "Banach algebras & Spectral Theory". Your input is much appreciated.
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  3. Jul 19, 2013 #2


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    Lie groups, like SU(2) and SL(2,C)
    and representations of Lie groups are used all thru LQG from the start onwards.

    Did you already have a course in Lie Groups or that covered them in your physics classes? If not you probably should go with the Lie Groups. Is the teacher good?

    You can get more advice in the Academic Guidance section of the forum.

    IMHO Banach algebras, the maximal ideal space, Gelfand stuff, spectral theory, is very beautiful but it comes into QG at a more advanced level. Also, if you go that way, find out about the Gelfand Naimark Segal construction. C* formulation of quantum mechanics, Tomita flow. Lot of beautiful stuff. But more advanced level than Lie Groups and representations. Walk before running.

    If you are in the LA area you could contact Matilde Marcolli at Caltech and ask her. She is the person in the US who knows the most about combining QG+NCG. (what you say you want to investigate)

    She has a paper that combines LQG+NCG. She could advise you in a minute what would be best to do.
    Here is her LQG+NCG paper (came out in January this year)
    Last edited: Jul 19, 2013
  4. Jul 19, 2013 #3
    Lie groups also show up fairly early in String theory, and are used extensively in quantum field theory (that is, in the standard model and beyond), so it's not just a LQG thing.
  5. Jul 19, 2013 #4
    On the other hand, Banach and C* algebras are the language of noncommutative geometry. They show up immediately when you try to do noncommutative things.
  6. Jul 19, 2013 #5
    re: marcus

    Thanks for your feedback. I have studied Lie groups at the level of "geometrical methods of mathematical physics" by Bernard F. Schutz and "Tensors and Group Theory for Physicists" by Nadir Jeevanjee. I haven't begun studying them more formally. I'm just starting graduate level coursework. So far I have taken GR and will take QFT this Fall at Berkeley.

    Marc A. Rieffel is teaching "Banach Algebras & Spectral Theory". His research is in non-commutative geometry with an eye to physics. In the spring he will then offer "von Neumann Algebras". I have contacted him and he has given me a great deal of prerequisite reading on functional analysis. He also suggested I might just take the "Lie Groups" course, as it is more approachable.

    Vera Serganova is teaching "Lie Groups". The second semester will cover quantum groups and their representations.

    Or in replace of a math course I could take "String Theory" taught for the first time by Mina Aganagic.

    Does this change your recommendation at all?

    I have contacted Matilde Marcolli by e-mail. Thank you for the tip.
  7. Jul 19, 2013 #6
    If I had an opportunity to take a course with the great Rieffel, I would do so in a heartbeat...
  8. Jul 19, 2013 #7


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    It doesn't matter now,the other stuff. I recall dropping in on a weekly seminar at Evans that Rieffel grad students were holding on NCG. Just do whatever you can to get up to speed for that Rieffel course. (IMHO)

    Given your situation, if Marcolli takes the time to answer the chances are very good that she will advise Rieffel course on Banach algebras and spectrum. Indeed it is the basis of NCG and she has been a frequent collaborator of Connes. She has to recommend what is an ideal entrance to NCG.

    As long as he doesn't go traveling and leave somebody else giving the lectures.
    Last edited: Jul 19, 2013
  9. Jul 20, 2013 #8


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    Not trying to hijack the thread, but I have a related question. I'm aspiring to go into QFT/Quantum Gravity/String Theory...or something of that nature myself. I've still got a ways to go, but that's where my interests keep directing me.

    As an undergrad, would Lie Groups be more important than Topology? I've still got time to figure it out, but I've looked ahead a little bit, and this is something I've been wondering. I know they're both important, but which would be more important to learn sooner?
  10. Jul 20, 2013 #9


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    Lie Algebras and Representation Theory have applications to just about all branches of physics. Lie Groups are obviously closely related, but their study would involve some some differential geometry as a prerequisite, as well as topology, depending on the treatment. So it really depends on how you're planning to study the subjects. If it is by taking courses, pay close attention to the prereqs. If you are going to self-study from books, a book like Cahn or Georgi on Lie algebras for physicists would be fine without knowing topology. Differential geometry would be used a bit more than topology, but there are plenty of books for physicists (like Nakahara) that cover the basics of both.
  11. Jul 20, 2013 #10


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    The two are not mutually exclusive. If you want to learn about Lie groups properly from a math text (I don't know about you personally but I absolutely hate math for physicists books with all my heart) then you will need the latter (as well as differential topology). Good luck.
  12. Jul 20, 2013 #11
    Not saying I disagree. But I recommend against taking a first course in Lie groups that deals with differential topology. It is far too much abstraction for a first course. Taking a course on Lie groups that just deals with matrix groups really is good enough for a first course. It will teach the main ideas and the main methods without too much technicalities. Of course, once you're already a bit acquainted with Lie groups, then studying it in a more general point of a view of differential topology is the way to go. Just not as a first introduction.
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