Poisson Brackets

We use the following shorthand notation:

${\dot{X}}^{\mu }={\partial }_{\tau }{X}^{\mu }{X}^{\prime \mu }={\partial }_{\sigma }{X}^{\mu }$

such that the gauge-fixed Polyakov action is

$S=\int d\tau \mathcal{L}=\frac{T}{2}{\int }_0^{\pi }d\sigma \left[{\dot{X}}^2+X^{\prime 2}\right]$.

The world-sheet canonical momenta are:

${\Pi }_{\mu }=\frac{\delta \mathcal{L}}{\delta {\dot{X}}^{\mu }}=T{\dot{X}}^{\mu }$.

Then, the gauge-fixed action becomes (canonical form of the action)

$S=\int d\tau \left[{\int }_0^{\pi }d\sigma {\dot{X}}^{\mu }{\Pi }_{\mu }-H\right]$

with Hamiltonian

$H=\frac{1}{2}{\int }_0^{\pi }d\sigma \left[\frac{{\Pi }^2}{T}+T{X}^{\prime 2}\right]=\frac{1}{2}{\int }_0^{\pi }d\sigma ({\dot{X}}^2+X^{\prime 2})$.

Thus,

$\{X^{\mu }(\sigma ),{\Pi }_{\nu }({\sigma }^{\prime })\}=\delta (\sigma -{\sigma }^{\prime }){\delta }_{\nu }^{\mu }$ with all other Poisson brackets vanishing.

The Poisson brackets for the coefficients in the mode expansion are

$\{x^{\mu },p_{\nu }\}={\delta }_{\nu }^{\mu }$

$\{{\alpha }_n^{\mu },{\alpha }_n^{\nu }\}=-im{\delta }_{m+n}{\eta }^{\mu \nu }$

$\{{\tilde{\alpha }}_n^{\mu },{\tilde{\alpha }}_n^{\nu }\}=-im{\delta }_{m+n}{\eta }^{\mu \nu }$

with all other Poisson brackets vanishing.