= Oscillator symplectic form of an open string
{title2=$\{\alpha_j^m,\alpha_k^n\}=-ij\eta^{mn}\delta_{j+k,0}$}
The Fourier kinetic term $\sum_{k>0}i\alpha_{-k}\cdot\dot\alpha_k/k$ supplies conjugate pairs, with $\alpha_{-k}=\alpha_k^*$. Inverting its <symplectic form> yields the displayed <Poisson brackets>. Nonzero modes are independent of the center-of-mass pair, while $\alpha_0=\sqrt{2\alpha'}p$ is the momentum zero mode. The imaginary factors are compatible with a real action up to a total derivative.
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