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:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\[

X^{25}(0, \tau) = X^{25}_0 \quad \text{and} \quad X^{25}(\pi, \tau) = X^{25}_\pi \, .

\]

[/tex]

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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 actuallyoperatorsthat have harmonic-oscillator-like commutation relations:

[tex]

\[

[\alpha^\mu_m, \alpha^\nu_n] = m \eta^{\mu\nu} \delta_{m+n, 0} \, .

\]

[/tex]

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# Homework Help: Open-string mode expansion

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