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Analytical methods in theoretical physics

  1. Jul 12, 2015 #1
    Hi everybody!
    I'm very interested in the methods of complex analysis in theoretical physics. By these methods I mean such things (tools) as (for example) analytical properties of scattering amplitudes (or S-matrix) in QFT, regge calculus in QCD, analytical continuations of Einstein's equations in GR etc.
    I'm working through the books: 1) Eden, Landshoff, Olive, Polkinghorne "The analytic S-matrix";
    2) Chew "The analytic S-matrix";
    3) Hepp-Epstein's book on Wightmann&LSZ and scattering amplitudes in QFT;
    4) Collinz "An introduction to regge theory and high-energy physics";
    5) Frederick Pham's books on Landau singularities.
    6) ... and some literature on dispersion relations and current algebra;
    They are interesting textbooks, but sometimes it seems to me that they (may be) quite well-fashioned and I just study too old things.
    So my question is: are there any newer reviews, advanced textbooks or interesting papers on such methods or relates topics in QFT? What's about applications in classical and quantum gravity? I'm interested in all modern developments of these methods (including with quite sophisticated maths as differential topology etc) and probably will defend Bachelor's degree in this field.
    Thanks in advance and sorry for my halting English!
     
    Last edited: Jul 12, 2015
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  3. Jul 12, 2015 #2

    fzero

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    You might find studying conformal field theory in 2d interesting, since it uses complex analysis heavily and is very relevant to modern topics in many areas of physics. The lectures by Ginsparg are an excellent place to start. If you find it interesting, a more complete text is di Francesco et al, Conformal Field Theory, Springer 1999 (aka "The Big Yellow Book").
     
  4. Jul 13, 2015 #3
    Thank you very much! The first books is really nice it seems to me.
    I've also found several interesting papers which connect Regge field theory with CFT.
    How do you think, mathematical background from Nicahara's books will be enough to study CFT?
     
  5. Jul 13, 2015 #4

    fzero

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    If you mean Nakahara's Geometry, Topology, and Physics, yes, that's a good background.
     
  6. Jul 13, 2015 #5

    radium

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    I was just about to mention CFTs since the conformal group in 1+1d or two spatial dimensions are just the analytic functions in the complex plane. In 1+1D radial quantization has very interesting consequences. You can map the plane to the cylinder with time as the length.
     
  7. Jul 13, 2015 #6
    Thanks everybody! I Think I need a little more complex analysis to search is these topics.
    I need recommendations in some advanced (but not too large and not too strict, as I m not a mathematician) books in complex analysis. Including analytical continuation of functions of several complex variables, general theory of Riemann surfaces with geometrical aspects and other contemporary topics
     
    Last edited: Jul 13, 2015
  8. Jul 14, 2015 #7

    radium

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    Other books to look at would be stone and goldbart (there is a preprint on stone's webpage) as well as Hassani, Arfken and Weber (although the material in there is not at as high of a level), dennery, and of course nakahara. A first course in differential equations by Weinberger is also very good.

    Of course Di Francesco is where you should go eventually as it is the bible of CFT. It also has a very useful part 1 which reviews QFT and statmech very well, gets to the core ideas in each.
     
  9. Jul 14, 2015 #8
    I know all of that. I need some advanced complex analysis books to read simultaneously with CFT
     
  10. Jul 14, 2015 #9

    radium

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    Unless you want to do really rigorous stuff you really don't to look at advanced math books. Most physicists learn math as they need it, math books are not as helpful since they are usually too rigorous.
     
  11. Jul 14, 2015 #10
    It is a good advice, thank you. I don't know, but I think not all math books are rigorous. Nicahara for example is not rigorous and I enjoyed reading it.
     
  12. Jul 14, 2015 #11

    radium

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    I don't really consider Nakahara as a math book though. It is more a math book written for physicists who use a lot of advanced math but who have not yet crossed the line to mathematical physics. As of late I would characterize myself that way as a grad student. It's definitely more rigorous than Arfken and weber though.

    There is a complex analysis book that might be good for your purposes by Elias stein. It is published by some Princeton lectures thing. I haven't looked at it carefully but it seemed ok. Stone and goldbart is very useful though since it introduces advanced stuff like differential geometry, topology, tensor analysis, and has two chapters on complex analysis all with examples from physics. It's best for someone who has prior knowledge of these topics.

    So I would look for a whole book on complex analysis which presents physical examples. I know a few in differential geometry but can't think of one for complex analysis off the top of my head.
     
  13. Jul 17, 2015 #12
    Thank you for your recommendation. I agree that Nicahara really is not a math book. It's something like bridge between physics and advanced mathematics.
     
  14. Jul 17, 2015 #13

    radium

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    I wouldn't even call it that, I would say it is a reference for physicists who work in fields that involve more advanced math. The topics Nakahara covers would be enough for 4-5 math books.
     
  15. Jul 17, 2015 #14
    But also (as i think) it is a good pedagogical book for physicists
     
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