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The string sigma model in a spacetime metric is the Polyakov actionTake conformal gauge $h_{ab}=\mathrm{diag}(-1,1)$, and choose spatial period $2\pi$ for the closed string. Substitution of the plane-wave metric givesThe stipulated light-cone gauge in string theory $X^{+}=\tau$ sets ${\dot{X}}^{+}=1$ and $X^{\prime +}=0$. Its longitudinal term $-T\int {\dot{X}}^{-}$ is a total time derivative. The transverse string sigma model in an isotropic plane wave is thereforeThe resulting Euler-Lagrange equations are the massive wave equationsThese conventions fix the mass term directly from $X^{+}=\tau$. More generally $X^{+}=\kappa \tau$ would replace $m$ by $|m\kappa |$, with $p^{+}=2\pi T\kappa$ for this spatial period.

Use the real orthonormal Fourier basisFor the plane-wave string mode frequencies, let ${\omega }_0=|m|$ and ${\omega }_{n,c}={\omega }_{n,s}=\sqrt{n^2+m^2}$. The complete real mode expansion for nonzero frequencies isEvery term obeys the wave equation and is periodic. Orthonormality and completeness let its coefficients match arbitrary appropriate initial position and velocity data. For $m=0$, replace the zero-mode bracket by $q_0^I(0)+\tau p_0^I/T$; the zero mode becomes a free particle rather than an oscillator.

The canonical momentum density is ${\Pi }_I=T{\dot{X}}^I$. The transverse Hamiltonian isWrite $X^I={\sum }_a{q}_a^I{f}_a$ and $p_a^I=T{\dot{q}}_a^I$. Since $\int f_a{f}_b={\delta }_{ab}$ and $\int f_a^{\prime }{f}_b^{\prime }=n_a^2{\delta }_{ab}$, substitution gives the plane-wave string mode HamiltonianThis is a collection of independent harmonic oscillators. The last equality follows by substituting the sine-and-cosine time dependence; each oscillator energy is constant.

For canonical quantization of each nonzero-frequency mode, putThen $H_{\perp }={\sum }_{I,a}{\omega }_a(a_a^{I†}{a}_a^I+1/2)$. Opposite travelling-wave oscillator combinations give the equivalent formFor $m\ne 0$, the formal zero-point energy is $E_0=\frac{24}{2}[|m|+2{\sum }_{n\ge 1}\sqrt{n^2+m^2}]$. At $m=0$, replace the zero-mode occupation term by ${\sum }_I(p_0^I{)}^2/(2T)$. The zero-point expression is divergent and needs a regulator or a specified normal ordering convention; no mass-independent intercept has been assumed.

The remaining Virasoro constraints determine the longitudinal coordinate:Periodicity of $X^{-}$ imposes ${\int }_0^{2\pi }{\Pi }_I{X}^{\prime I}d\sigma =0$, the closed-string level matching condition. Labelling the travelling oscillators by their opposite worldsheet momenta gives ${\sum }_{n\ge 1,I}n(N_n^I-{\tilde{N}}_n^I)=0$. The light-cone energy $p^{-}=-p_{+}$ equals $T\int ({\dot{X}}^{-}+m^2{X}^I{X}^I)d\sigma =H_{\perp }$, consistent with the positive mass term in the Hamiltonian.

This is the classical reduction in the specified background. As a full quantum background with constant dilaton and no Kalb–Ramond field, this metric has $R_{++}=24m^2$ and fails the lowest-order sigma-model beta function equation when $m\ne 0$; additional background fields would be required for quantum Weyl invariance.

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.