= Plane-wave string mode Hamiltonian
{title2=$H_\perp=\sum_{I,a}\left(\frac{p_{Ia}^2}{2T}+\frac T2\omega_a^2q_{Ia}^2\right)$}
Expand every transverse <closed string> coordinate in a real orthonormal <Fourier basis> of the spatial circle. The <canonical momenta> are $p_{Ia}=T\dot q_{Ia}$, and spatial orthogonality reduces the field <Hamiltonian> to the displayed independent <harmonic oscillators>. Canonical quantization gives $\omega_a(a_{Ia}^\dagger a_{Ia}+1/2)$ for each nonzero-frequency mode; a zero-frequency mode retains the free particle term $p_{Ia}^2/(2T)$. The formal <zero-point energy> requires a regulator; it is not automatically the massless flat-background intercept. Opposite travelling-wave oscillator combinations diagonalize the translation generator and satisfy <closed-string level matching>.
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