Clarifying Open String Hamiltonian for Witten's Book

[cuerdero]

trying to get the open string hamiltonian I use
<br /> H=\int\,d\sigma(\dot{X}.P_{\tau}-L)=\frac{T}{2}\int(\dot{X}^{2}+X&#039;^{2})d\sigma<br />
as in Witten´s book, but we are integrating the Virasoro constraint equal to zero.
So, Is not the Hamiltonian zero?
Please, clarifyme this equation.

 

[cuerdero]

Please, I can´t get the answer..
Thank you

 

[petergreat]

Yes, it's zero! In fact the Hamiltonian is zero (on shell) for any system with diffeomorphism invariance. In the Hamiltonian formulation of general relativity, the Hamiltonian is also zero. (Therefore it doesn't make much sense to talk about energy in GR, though people can get agitated about this issue...)

 

[negru]

Yeah any parametrization invariance leads to this. You can consider just a simple point particle, not even a string, for the hamiltonian to vanish.

 

[mitchell porter]

See the end of this page

http://www.physics.thetangentbundle.net/wiki/String_theory/relativistic_point_particle/action

 

[cuerdero]

Thanks to everyone, it was really helpful...but my next question is:
what about the Virasoro operators and the mass of the string?

 

[mitchell porter]

http://www.scribd.com/doc/17025413/A-First-Course-in-String-Theory-2nd-Edition-Cambridge-2009" derives the mass of an open string, first classically (equation 9.83) and then quantum mechanically (equation 12.108). We can discuss the logic of this derivation if you like.

 

[cuerdero]

thanks a lot..now I can see it better