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Mode Expansion of Closed String with Compact Dimensions

  1. Feb 8, 2010 #1
    Im working through derivations of string equations of motion from the Nambu-Goto Action and I'm stuck on something that I think must be trivial, just a math step that I cant really see how to work through.
    At this point Ive derived the equation of motion for the closed string from the wave equation in a general [tex] \tau [/tex] gauge, and now im trying to do the same but for a string in space with compactified dimensions. So im trying to implement

    [tex] X^i (\tau, \sigma) = X^i (\tau , \sigma +2\pi) +2\pi R_i W^i [/tex]

    Where W is wrapping number of ther string around the dimension.

    Now, without the compactified dimension it was simple to see the periodicity of the left moving wave and the right moving wave separately, and to expand using fourier expansion and combine the 2 to get the equation of motion. But now im stuck on how to use the constraint to again rederive the equation of motion from the general solution to the wave equation. how do you do the mode expansion now that you have this extra term of

    [tex] 2\pi R_i W^i[/tex] ?

    Thanks for any help.

    B
     
  2. jcsd
  3. Feb 9, 2010 #2
  4. Feb 9, 2010 #3
    Thanks, but Im afraid I wasn't able to find what I'm looking for there. I'm sure that because this is such a trivial step, it is not explicitly referred to in the literature. Ive also looked in Zwiebach. Its a trivial step that eludes me however.

    It seems from what I have read, that what arises from the implementation of the closed string in compact space constraint, is a constraint on the values of the constants [tex] a_0 ^i[/tex],
    where [tex] a_n ^\mu[/tex] are the coefficients of the oscillatory modes in the equation of motion. ( \mu is regualr space coordinates and i denotes compact space coordinates).
    I cant see how this comes about mathematically though.

    Is what I'm asking clear?

    Thanks again.
     
  5. Feb 9, 2010 #4
    You need a zero-mode contribution that takes care of this. If I recall correctly from conformal field theory, which has a similar calculation, you need to implement something like

    [tex]X(\tau,\sigma) = \ldots + R_iW^i \sigma +\ldots[/tex]

    The terms in front are [itex]\sigma[/itex] independent. The latter terms are exponentials in [itex]\sigma[/itex] and do not change.

    However, there is also a term [itex]\tau[/itex] times some coefficient. This coefficients also changes, but I dont know how.

    Hope this helps a bit.
     
  6. Feb 9, 2010 #5
    Since a circular dimension can be represented as \mathbb{R} where one identifies all points separated by 2*pi*R. So you can use the mode expansion for the non-compactified case, and just impose the restriction that comes from the space-identifications. Are you not doing this?

    The periodicity condition only affect the components of X in the i-directions. In addition, the higher order terms [tex] a_n^{\mu}[/tex] for n>1 won't be accected by the circular dimension, because their accompanying sin() function in the mode expansion will automatically have the same value both at [tex]\sigma=0[/tex] and at [tex]\sigma=2\pi[/tex] (beware, some only use [tex]\sigma\in[0,\pi][/tex]).

    So in the constraint equation in the i-direction reduces from

    X(sigma=0) = X(sigma=2pi) + 2*pi*R*w

    to something only involving the [tex]a^i_0[/tex] because all the other terms disappear automatically. I.e. to the usual formulas for the a0's in terms of the winding number w and the momentum number n.

    Torquil
     
    Last edited: Feb 9, 2010
  7. Feb 9, 2010 #6
    Thank you both for your replies.
    Torquil, spelling it out for me helped thanks! I just wasn't looking at it correctly. its amazing how much time one can spend on these small math steps sometimes.

    Cheers!
     
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