Working with the bosonic string in a background space-time with one compact dimension, i.e.: $$ R^{1,24}\times S^1 $$ I have been able to calculate the mass-squared: $$ M^2 = \frac{n^2}{R^2} + \frac{m^2R^2}{\alpha^{'2}} + \frac{2}{\alpha'}\left( N + \bar{N} - 2\right) $$ Here n and m are integers related to the quantisation of the string momentum and winding respectively.

I would now like to calculate the Hamiltonian of the closed string in question. My first thought was to sum this with the momentum-squared but I can't seem to get it in the proper form.

I also thought that perhaps I could start from the Lagrangian density in the Polyakov action: $$ S = \frac{-1}{4\pi\alpha^{'}}\int d\tau d\sigma \sqrt{-h}~h^{\alpha\beta}\partial_\alpha X^\mu\partial_\beta X^\nu\eta_{\mu\nu} $$

Could someone please give me a nudge in the correct direction, I feel like I'm overcomplicating this.

String Theory Newbie

I've figured it out now, I'll type my workings up as an answer tomorrow morning. Hopefully they will aid any weary travellers that reach this final bastion of hope.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.