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About twistor theory

  1. Sep 18, 2013 #1
    some years ago when I was studying Nuclear engineering in Turin ( Italy ), I attended
    some courses with prof. Tullio Regge about groups theory, spinors and twistors.

    I was too young to fully understand what he was telling us, but now, after some years
    of private study about Einstein's gravity and a differential geometry, I gifted myself for
    my birthday with a copy of both volumes of Spinors and Space-Time.
    This is, of course, really different to the normal "tensorial" approach I was used to and I'm really
    fascinated by this topic. Attending some extra courses in particle physics, I saw how spinors born
    to life naturally in Dirac's theory, but I wasn't aware they could be used successfully in Einstein's gravity as well.
    I've read the spinors approach is handier for describing some massless phenomena, and for other the tensor approach is more appropriate.
    The real interest to me is how complex numbers theory comes in help in physics here and there.
    Usually, when you are able to switch some theory rearranging it under the complex line of sight, you add to it some more physics content, and I'm curious about that.

    So the question is: from your experience, does spinors/twistors Penrose's approach worth the pain to break my head on it? Or it just remain actually, after 40 years, a sterile theory, which destiny is to remain in the maths tool games?

    Indeed, the first volume would worth the pain even only for its interpretation of Lorentz transformation as complex Mobius transformation. This connection is really, really interesting to me.

    I know recently ( where recently I meant 10 years ago ) Ed Witten has given new life to it, encompassing it somehow in string theory, but the math is too advanced to me for now( I'm studying algebraic topology and knots theory for it, but it takes a long time to master it, as you may know ).

    Thanks, regards


    EDIT: I'd like to stress my point here: since I'm a physics enthusiast any theory is actually fascinating me a lot,
    twistors included, and I'm constantly wondering every day about its beauty. But my time is limited and I'm putting my efforts from since last year to have a better grasp on Einstein's gravity and differential geometry, mainly doing exercises or so. This is of course really time consuming and I'd avoid to jump into "exotic" theories before having a better grasp on the main ones. Nevertheless, from my experience about complex number theory, I'd bet this should open a new insight to me ( like Riemann sphere, which impressed me so much at the very first sight), just, for instance, about the "new" induced (2,2) metric signature from it.
    Last edited: Sep 18, 2013
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  3. Sep 18, 2013 #2
    In the ROAD TO REALITY,2004, Penrose offers a chapter [#33] which may be of interest:

    Under 33.14, about three and a half pages, 'The future of twistor theory' Penrose says:

    I have no personal opinion except that you would probably find the entire chapter of interest.
  4. Sep 18, 2013 #3


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    Spinors aren't as common in classical GR as they are in QFT; even today what you'll mostly see in classical GR is tensor calculus as opposed to 2-spinor calculus. If I had to dig deep, the most famous application of spinors in classical GR would have to be Witten's proof of the positive energy theorem. Needless to say, Penrose/Rindler's two texts are probably worth reading for a variety of reasons; for one, the first volume gives a very detailed and systematic exposition of the abstract index formalism.
  5. Sep 18, 2013 #4
    Hi WannabeNewton,
    I'm pleased to ear from you, since I've read a lot of your posts, they are always interesting to me.
    thanks for your answer.
    I know you have a big background on tensor calculus ( or at least it seems so from your posts ), so what
    do you think about my point about complex treatment of GR...do you think you can have a sort of dual description
    using 2-spinors instead of pure tensor calculus?
  6. Sep 18, 2013 #5


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  7. Sep 19, 2013 #6
    There may be a renewed interest in twistor theory with the 17-SEP-13 amplituhedron paper.
    One of the PF form threads about it is here.
  8. Sep 20, 2013 #7
    Thanks for this!!!

    I will look further to it deeply for sure.

    I was puzzling myself about why spinor theory has not become so popular in GR as well. I've not entered too much in it yet, but from Penrose's POV it seems giving more "natural" insights for GR as well ( of course he did the theory as first, so he was trying probably to push it a bit ). From my limited experience, through my studies, when I met complex numbers I've always seen a lot of new ideas from theories involving them, so I cannot a priori think Penrose's was too optimistic about that, i.e. usually complex structures have a richer content than "simple" theories on real fields. What's your opinion about that? the maths is too difficult compared to the insights it gives back?
    I will study it for sure despite it's not popular, but I was just wondering about that.

    Last edited: Sep 20, 2013
  9. Sep 20, 2013 #8


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    I think the simplest answer is that most things in classical GR, to put it lightly, can be described beautifully and perfectly using tensor calculus so there's no need for a formal development of spinor calculus in texts or elsewhere; spinor calculus (and the theory of spinors in general) requires more extensive background in mathematics and more care in development so if I had to guess, I would say that the tradeoffs in switching to a spinorial treatment (such as offering different geometric insights) aren't enough to completely replace tensor calculus, which has both ease of development and ease of use.
  10. Sep 20, 2013 #9
    Thanks very much WannabeNewton,
    I have not yet enough knowledge about both of them to open any further discussion about it.
    I hope to have a deeper lead on them in future, carefully pushing them in parallel in my studies.


  11. Sep 20, 2013 #10
    it does seem that way.

    some examples to consider........

    more in this discussion:
    What is a particle
  12. Sep 20, 2013 #11


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    But that's QFT; spinors are extremely natural in QFT. The OP was asking about classical GR.

    One thing at least is that once you get past the mathematical formalities of spinor theory and get to the actual computational aspects of spinor algebra/calculus, the mechanics is very similar to that of tensor algebra/calculus (with some caveats e.g. you can't just freely raise and lower indices with complete disregard of the ordering of indices like you can when dealing with just tensors).
    Last edited: Sep 20, 2013
  13. Sep 25, 2013 #12
    I was wondering someone could address me eventually to some textbooks on spinor/twistor theory, which could help me in understanding better this really important ( to me ) aspect of them.

    I landed to these Penrose's books from "Visual Complex Analysis" Needham book ( absolutely one of my favorites, covering almost all fields of complex analysis ) in which there's a very deep and important chapter about Mobius transformations and their analysis on Riemann sphere. There we can see how Penrose was suggesting in its first chapter how Lorentz transformations are really connected to Mobius transformations ( that's why, I guess, Riemann sphere is so important for Penrose, appearing in almost its slides on twistor theory conferences I found online ).

    I'm already going to buy the original Cartan "Theory of Spinors", but this is quite more for collection purposes, since I consider it a "classic" for the argument.
    May be you ( WBN, for instance ) could suggest a more modern approach to it.

    Thanks, regards

    EDIT: I actually this PF thread about twistor textbooks. I was more oriented towards spinors, first.
    Last edited: Sep 25, 2013
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