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Line defects and String Theory

  1. Apr 1, 2012 #1
    The appearance here of threads such as Causes of loss of interest in the String program and Is string theory really science? are for me a symptom of the frustration of modern physicists struggling to formulate in a predictive way the concept of elementary objects that are geometrically linear, rather than old-style points.

    A century ago, when Einstein was engaged in his struggle to formulate his new concept: gravity as a distortion of simple Euclidian spacetime, his frustration became relieved by learning about tensors, a recent geometrical invention of the mathematicians Tullio Levi-Civita and Gregorio Ricci-Curbastro. Modern physics owes a great deal to these 19th Century Italians. Here I’d like to suggest that we might yet come to owe even more to like folk.

    Einstein singled out one kind of distortion of Euclidian geometry; that described in the 19th century by Bernhard Riemann. I’m guessing here that this step opened the door, as it were, to other kinds geometrical distortions being used to help describe yet more fundamental stuff, as string theory seems to be struggling to do.

    It’s not clear to me whether today’s string theorists are aware of another Italian
    mathematician’s work and its extensive later 20th century ramifications, which have to do with the physics of crystals. Like Luigi Bianchi, Vito Volterra was a student of the 19th Century differential geometer and topologist Enrico Betti. Volterra described stringy topological defects in continuous elastic media –called distorsioni; later dislocations and disclinations. In the 20th Century these stringy line defects were applied to explain much of practical importance in real crystalline materials. There’s a huge literature about this; see for example F.R.N. Nabarro, Theory of Crystal Dislocations, OUP, 1967.

    From my level of (great) ignorance I’d be interested to know whether Volterra’s work has had, or could have, any utility in string theory. Einstein is surely a good act to follow!
  2. jcsd
  3. Apr 21, 2012 #2
    A friend who publishes esoteric stuff in GR has remarked to me that the sad reality is that everyone is so busy with their own work that there’s not much time for kind of cross-fertilisation of ideas I suggested above; fleshing out Volterra’s ideas might take a lot of work. I agree. But:

    If there is no obvious reason why the only geometries useful for physics are exclusively either Euclidean or Riemannian, I’d like to add that, despite the press of work and the dread prospect of lots more, it might be worth taking the risk of revisiting Volterra’s excursion into the defect geometry of a continuum.

    I remark further that quantitative geometry can be more useful than qualitative geometrical concepts. In Euclidean geometry, for example, Pythagoras’s theorem seems more practical than the qualitative concept of the congruence of shapes. And in Riemannian geometry the qualitative concept of space(-time) curvature is rendered practical for predictive purposes when it’s quantified using coordinates, in terms of the curvature tensor Perhaps predictive physics Beyond the Standard Model could use a successful struggle to quantify Volterra - style defect geometry in terms of coordinate geometry. Or maybe one is already available that I could be pointed at?
  4. Apr 23, 2012 #3
    I think I’ve discovered why this thread is barking up the wrong tree, as it were. The
    idea that line defects may have any implication for physics ‘Beyond the Standard
    Model’ seems to have been explored pretty thoroughly, but impotently, by one
    Hagen Kleinert, starting as long ago as 1987 (more recently, see arXiv:gr-
    qc/0307033v1). Like other current non-Baconian proposals, his scheme seems
    quite remote from any verification/disproof by observation and experiment.

    Kleinert’s scheme may also be tainted with his specious (for me, maybe for others
    too) philosophy that

    Discussions of the present plethora of theoretical schemes designed to unite GR and QM seem now to me as futile as those of sports commentators about horse-racing prospects --- in a world with only imaginary horses. So I’ll close now.
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