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The Euler-Lagrange equation of each component of the string embedding map is the wave equation $({\partial }_{\tau }^2-{\partial }_{\sigma }^2)X^{\mu }=0$. Its left- and right-moving solutions are identified by the endpoint Neumann boundary conditions, so the resulting open-string mode expansion isIndeed, the Fourier cosine series makes ${\partial }_{\sigma }{X}^{\mu }$ vanish at $\sigma =0,\pi$. The canonical momentum density is ${\Pi }_{\mu }=(2\pi {\alpha }^{\prime }{)}^{-1}{\dot{X}}_{\mu }$. Imposing the equal-time canonical commutation relations $[X^{\mu }(\sigma ),{\Pi }_{\nu }({\sigma }^{\prime })]=i{\delta }_{\nu }^{\mu }\delta (\sigma -{\sigma }^{\prime })$ gives the covariant quantization of the bosonic stringwith ${\alpha }_0^{\mu }=\sqrt{2{\alpha }^{\prime }}{p}^{\mu }$.

An infinitesimal Lorentz transformation is $\delta X^{\mu }={\omega }^{\mu }{}_{\nu }{X}^{\nu }$ with ${\omega }_{\mu \nu }=-{\omega }_{\nu \mu }$. Applying Noether theorem to this continuous symmetry gives the conserved Lorentz currentIts Noether charge is $M^{\mu \nu }={\int }_0^{\pi }d\sigma J_{\tau }^{\mu \nu }$. Substituting the open-string mode expansion and using orthogonality of the cosine modes yieldsThe first term is orbital angular momentum; the sum is the contribution of the string oscillators.