= Half-integer open-string oscillator
{title2=$\alpha_r^i,\quad r\in\mathbb Z+\tfrac12$}
For a <Neumann-Dirichlet open-string boundary condition>, the transverse modes have frequencies $r\in\mathbb Z+1/2$. The <canonical commutation relations> give $[\alpha_r^i,\alpha_s^j]=r\delta^{ij}\delta_{r+s,0}$ and $\alpha_r^\dagger=\alpha_{-r}$. One derivation expands $X-y=\sum_{r>0}q_r\cos(r\sigma)$: spatial <orthogonality> reduces the <Polyakov action> to $(8\alpha')^{-1}\int\sum_r(\dot q_r^2-r^2q_r^2)d\tau$. These are <harmonic oscillators> of <mass> $1/(4\alpha')$, whose normalized <creation operators> satisfy $\alpha_{-r}=i\sqrt r\,a_r^\dagger$ in the conventional phased expansion.
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