Clarifying Open String Hamiltonian for Witten's Book

In summary, the conversation is discussing the Hamiltonian for an open string, where H is equal to the integral of the dot product of X and P_tau minus L, and is also equal to half of T multiplied by the integral of X and X' squared. The Virasoro constraint is also being integrated, which leads to the Hamiltonian being zero. This is due to diffeomorphism invariance, which is also seen in the Hamiltonian formulation of general relativity. The conversation then shifts to discussing the mass of the string, which is derived both classically and quantum mechanically in a resource provided by one of the speakers.
  • #1
cuerdero
6
0
trying to get the open string hamiltonian I use
[tex]
H=\int\,d\sigma(\dot{X}.P_{\tau}-L)=\frac{T}{2}\int(\dot{X}^{2}+X'^{2})d\sigma
[/tex]
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.
 
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  • #2
Please, I can´t get the answer..
Thank you
 
  • #3
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...)
 
  • #4
Yeah any parametrization invariance leads to this. You can consider just a simple point particle, not even a string, for the hamiltonian to vanish.
 
  • #5
See the end of this page

http://www.physics.thetangentbundle.net/wiki/String_theory/relativistic_point_particle/action
 
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  • #6
Thanks to everyone, it was really helpful...but my next question is:
what about the Virasoro operators and the mass of the string?
 
  • #7
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.
 
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  • #8
thanks a lot..now I can see it better
 

Related to Clarifying Open String Hamiltonian for Witten's Book

1. What is the purpose of clarifying the open string Hamiltonian in Witten's book?

The purpose of clarifying the open string Hamiltonian in Witten's book is to provide a better understanding of the mathematical and physical concepts involved in open string theory. By breaking down and explaining the Hamiltonian in detail, readers can gain a deeper understanding of the behavior and properties of open strings.

2. What is the significance of Witten's book in the field of string theory?

Witten's book, "String Theory and Noncommutative Geometry", is considered a seminal work in the field of string theory. It introduced the concept of open strings and their connection to noncommutative geometry, paving the way for further research and developments in this area of physics.

3. Can you explain the key components of the open string Hamiltonian?

The open string Hamiltonian consists of three main components: the kinetic energy term, the potential energy term, and the boundary conditions. The kinetic energy term describes the motion of the open string as it propagates through spacetime. The potential energy term takes into account the interaction between the string and its environment. The boundary conditions specify how the string behaves at its endpoints.

4. How does the open string Hamiltonian differ from the closed string Hamiltonian?

The open string Hamiltonian differs from the closed string Hamiltonian in several ways. Firstly, the open string Hamiltonian has boundary conditions that are absent in the closed string case. Additionally, the potential energy term in the open string Hamiltonian takes into account the interactions with external objects, while the closed string Hamiltonian only considers self-interactions.

5. What implications does the open string Hamiltonian have for string theory?

The open string Hamiltonian has many implications for string theory. It allows for the study of open strings, which have different properties and behaviors compared to closed strings. This has led to a better understanding of the duality between open and closed strings, and has also opened up new avenues for research and applications in the field of string theory.

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