String sigma model in an isotropic plane wave ($d{s}^2=-2,d{X}^{+}d{X}^{-}-m^2|X{|}^2(d{X}^{+}{)}^2+d{X}^{I}d{X}^I$)

In conformal gauge and light-cone gauge in string theory $X^{+}=\kappa \tau$, the transverse part of the string sigma model in a spacetime metric reduces toThe omitted longitudinal term is a total time derivative. Each transverse coordinate obeys a massive wave equation. If the spatial period is $L$, the longitudinal momentum is $p^{+}=T\kappa L$ in this normalization. Keeping the gauge slope explicit prevents missing a factor of $p^{+}$ when changing conventions.

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 ${\omega }_a(a_{Ia}^{†}{a}_{Ia}+1/2)$ 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.

For a closed string in a string sigma model in an isotropic plane wave, insert $e^{-i{\omega }_n\tau +2\pi in\sigma /L}$ into the transverse wave equation. Periodicity requires integer $n$, giving the displayed dispersion relation. The zero spatial mode is an oscillator of frequency $|m\kappa |$; when this mass vanishes it becomes a free particle mode and needs the position-plus-linear-time expansion.