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Question on world-sheet-metric

  1. Oct 9, 2004 #1
    What I understand mathematically:
    The Polyakov action has three symmetries of the world sheet(reparametrization-inv.,Weyl inv.). These can be used to take the worldsheet metric into the form diag(1,-1), i. e. it is a flat space.

    But I don't understand that when I try to visualize the worldsheet of a string oszillating in some (spacelike) direction because the corresponding worldsheet is not flat then anymore. There are these bulgs in the oszillating direction which make the shape of the worldsheet not a flat one.

    Please help me out.

    If you can't (or don't want to) my questions, maybe you know some forum or text where it can be infered but please don't tell me the standard answer to browse the arXiv or Green, Schwarz, bla bla bla.
    Just tell me references where you know that I can find an answer.
    Last edited: Oct 9, 2004
  2. jcsd
  3. Oct 9, 2004 #2


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    First of all, when the STRING vibrates, the worldsheet doesn't move, it is the history of the string, just as the worldline of a particle is the history of the particle. Secondly the vibration and other movements of the string appear on the world sheet as potentials. The spatial coordinates that the string passes through are the potentials X^mu which appear in the equations. From point to point on the worldsheet, that is at different points in the string's history, these potentials vary, because a) you are at a different part of the string, and b) at a different moment of that point's history.

    Locally the worldsheet is flat, as the worldline of a point is locally flat.
  4. Oct 9, 2004 #3
    I never claimed that the worldsheet is moving when the string moves. It's clear to me that is the history of the string.

    I know that the X^mu are scalar fields when viewed from the worldsheet and that they are the embedding functions of the string when viewed from the spacetime. Of course they vary scince they describe the worldsheet, i.e. more than one point and therefore the functions X^mu are not constant (as you can see from the solutions of the eqns of motion).
    But, that does not say that the worldsheet is flat. That it is locally flat is clear scince every diffeomorphism invariant manifold is locally isomorphic to an R^n.
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