Notes on string theory: Hamiltonian formalism and Poisson brackets

So far the picture of the relativistic string has been built in terms of the Lagrangian formalism. We can now also spend a few minutes engaging in a short review of the Hamiltonian description in the flat gauge (before formally moving to study the spectrum of the string).

Recall the S_{P} action in the flat gauge, restoring explicit notation for the string tension,

    \[S_{P} = -\frac{T}{2} \int d^{2}\sigma \partial_{a}X \partial_{b}X\eta^{ab} \]

Where \mathcal{L} = -\frac{T}{2} \int d^{2}\sigma \partial_{a}X \partial_{b}X\eta^{ab}.

For the Hamiltonian we may write what is below, recalling the conjugate momenta previously defined:

    \[ \mathcal{H} = \int_{0}^{\sigma_1} d\sigma(\dot{X}_{\mu}P_{\mu}^{\tau} - \mathcal{L}_{P}) = \frac{T}{2} \int_{0}^{\sigma_1}d\sigma (\dot{X}^2 + X^{\prime 2}) \]

Where we have canonical fields X^{\mu}(\tau, \sigma) and conjugate momenta,

    \[ \prod^{\mu}(\tau, \sigma) = \frac{\partial \mathcal{L}}{\partial \dot{X}_{\mu}(\tau,\sigma)} =T\dot{X}^{\mu}(\tau, \sigma) \]

Finally, in anticipating what is to come in the next few posts, a few brief comments on the Poisson brackets is necessary. Firstly, in string theory, our concern is with Poisson brackets for all fields at time \tau (Weigand, p. 21). This will be our starting point when we finally begin discussing the quantisation of the modes of the string. Moreover, as stated in (Weigand, p.21), we can define the Poisson brackets in terms of two arbitrary fields F(\tau, \sigma) and G(\tau, \sigma^{\prime}).

    \[ \left\{F, G\right \} = \int d\tilde{\sigma} (\frac{\partial F(\tau,\sigma)}{\partial X^{\mu}(\tau, \tilde{\sigma})} \frac{\partial F(\tau, \sigma^{\prime})}{\partial\prod_{\mu}(\tau, \tilde{\sigma})} - \frac{\partial G(\tau, \sigma^{\prime})}{\partial X^{\mu}(\tau, \tilde{\sigma})} \frac{\partial F(\tau, \sigma)}{\partial\prod_{\mu}(\tau, \tilde{\sigma})}) \]

From this we arrive at canonical equal time Poisson relations,

    \[ \{X^{\mu}(\sigma), \prod^{\nu}(\sigma^{\prime})\} = \eta^{\mu \nu}\delta(\sigma - \sigma^{\prime}) \]

    \[ \{X, X\} = 0 = \{ \prod, \prod \} \]

It also follows that one can use the Fourier expansion to derive Poisson brackets for the modes along the classical string,

    \[ \{\alpha_{m}^{\mu}, \alpha_{n}^{\nu} \} = -im\delta_{m+n, 0} \eta^{\mu \nu} \]

An identical result is also found for the case of \tilde{a}, while the mixed brackets equal zero. It can also be stated that the commutation relations for \{x^{\mu}, p^{\mu} \} = \eta^{\mu \nu}.

References

Joseph Polchinski. (2005). “String Theory: An Introduction to the Bosonic String“, Vol. 1.

David Tong. (2009). “String Theory” [lecture notes].

Timo Weigand. (2015/16). “Introduction to String Theory” [lecture notes].