How Does Dirichlet Boundary Condition Affect Open String Mode Expansion?

[Themetricsystem]

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

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

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

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 $$\alpha^\mu_m$$ are actually operators that have harmonic-oscillator-like commutation relations:
$$\[ [\alpha^\mu_m, \alpha^\nu_n] = m \eta^{\mu\nu} \delta_{m+n, 0} \, . \]$$

 

[Jhero]

Hi there,

To incorporate the Dirichlet boundary conditions for the 25th coordinate, we can start by defining a new coordinate \tilde{X}^{25} as

\[
\tilde{X}^{25}(\tau, \sigma) = X^{25}(\tau, \sigma) - \frac{X^{25}_0 + X^{25}_\pi}{2} \, .
\]

This shifts the origin of the coordinate to the midpoint between the two boundary conditions. We can then rewrite the mode expansion for X^{25} as

\[
X^{25}(\tau, \sigma) = \tilde{X}^{25}(\tau, \sigma) + \frac{X^{25}_0 + X^{25}_\pi}{2} \, ,
\]

and substitute this into the original expression for X^\mu. This will give us a new mode expansion for X^\mu that satisfies the Dirichlet boundary conditions for the 25th coordinate.

Hope this helps! Let me know if you have any further questions.