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Homework Help: Open-string mode expansion

  1. Jan 21, 2006 #1
    Compute the mode expansion for an open string with Neumann boundary conditions for the coordinates X^0,..., X^24, while the remaining coordinate X^25 satisfies Dirichlet boundary conditions at both ends:
    [tex]
    \[
    X^{25}(0, \tau) = X^{25}_0 \quad \text{and} \quad X^{25}(\pi, \tau) = X^{25}_\pi \, .
    \]
    [/tex]

    -------

    I know that the mode expansion is a sum of left- and right-moving modes
    (in the light-cone coordinates [tex]\sigma - \tau[/tex] and [tex]\sigma+\tau[/tex].)

    The expression for the X^\mu in the open-string expansion is
    [tex]
    \[
    X^\mu(\tau, \sigma) = x^\mu + \ell^2_s p^\mu \tau + i\ell_s \sum_{m \neq 0} \frac{\alpha^\mu_m}{m} \, e^{-im\tau} \cos (m\sigma) \, ,
    \]
    [/tex]

    but I'm not sure how to fit the Dirichlet conditions in the twenty-fifth coordinate.

    Edit: Sorry, here are definitions for some of the symbols:

    tau and sigma are timelike and spacelike coordinates, respectively. They parameterize the world-sheet and appear in the embedding functions X^\mu.

    The [tex]\alpha^\mu_m[/tex] are actually operators that have harmonic-oscillator-like commutation relations:
    [tex]
    \[
    [\alpha^\mu_m, \alpha^\nu_n] = m \eta^{\mu\nu} \delta_{m+n, 0} \, .
    \]
    [/tex]
     
    Last edited: Jan 22, 2006
  2. jcsd
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