Closed Strings and Virasoro algebra

In summary, The discussion is about the closed string following Green, Schwarz, Witten. The question is about the point of constructing the Virasoro algebra when the corresponding generators are zero due to constraints. The response is that after quantization, only L_n with positive n are required to be zero, while L_n with negative n are "creation operators" that do not vanish. This is similar to the Gupta-Bleuler condition in the Lorentz-gauge quantization of QED. There is a reference for this in David Tong's QFT notes.
  • #1
earth2
86
0
Hey guys,

i just started taking a course on ST and so far we discuss the closed string following Green, Schwarz, Witten.

I don't see the point of constructing the Virasoro algebra (formula 2.1.85 in GSW) if the corresponding generators are zero due to the constraints. Or to put it differently, they are defined by
[tex]L_m\propto \int e^{...m} T_{--}d\sigma[/tex] but by the constraints on the system we have T_{--}=0 (similarly for the barred generator). In other words: i don't understand why we deal with the L's if they are zero anyways?

Hope someone can enlighten me on that...

Cheers,
earth2
 
Last edited:
Physics news on Phys.org
  • #2
They are classically zero, but after quantization you only require the operators L_n with positive n to be zero. L_0 acquires an anomaly and becomes 1. L_n with negative n are "creation operators" which don't vanish. I'm sure your course will discuss it soon.
 
  • #3
Hey, thanks for the answer. We had exactly this reasoning today in our lecture but we only DEFINED that L_n with n>0 vanishes. Is there a physical argument for that that makes this definition somehow intuitively accesible? I mean, why do we only require that n>0 vanishes and not all n?

Cheers
 
  • #4
earth2 said:
Hey, thanks for the answer. We had exactly this reasoning today in our lecture but we only DEFINED that L_n with n>0 vanishes. Is there a physical argument for that that makes this definition somehow intuitively accesible? I mean, why do we only require that n>0 vanishes and not all n?

Cheers

This goes way back before string theory. It's essentially the same as the Gupta-Bleuler condition is the Lorentz-gauge quantization of QED. One possible reference for this is David Tong's QFT notes:
http://www.damtp.cam.ac.uk/user/tong/qft.html
Open Chapter 6, and the relevant parts start at page 131.
 

Related to Closed Strings and Virasoro algebra

What are closed strings?

Closed strings are one of the two fundamental types of strings in string theory. They are one-dimensional objects that have no endpoints and are "closed" in the sense that they form a loop. This is in contrast to open strings, which have two distinct endpoints.

What is the Virasoro algebra?

The Virasoro algebra is a mathematical structure that describes the symmetries of string theory. It is named after Italian physicist and mathematician Michele Virasoro, who first introduced the algebra in the 1970s. The Virasoro algebra is a central part of the mathematical framework of string theory and plays a crucial role in the study of closed strings.

What are the implications of closed strings in string theory?

Closed strings are important in string theory because they allow for the possibility of interactions between different strings. This is in contrast to open strings, which can only interact with other open strings. Closed strings also have the ability to vibrate in different modes, which gives them a rich and complex structure that allows for the description of various physical phenomena.

How is the Virasoro algebra related to closed strings?

The Virasoro algebra is closely related to closed strings as it describes the symmetry of the closed string worldsheet. This means that the algebra governs the transformations that leave the closed string equations of motion invariant. Additionally, the Virasoro algebra plays a crucial role in the quantization of closed strings, which is necessary for their inclusion in string theory.

What are the practical applications of understanding closed strings and the Virasoro algebra?

While string theory is still a developing area of physics, understanding closed strings and the Virasoro algebra has potential practical applications in the fields of quantum gravity and cosmology. By studying these concepts, scientists hope to gain a better understanding of the fundamental nature of the universe and potentially develop new technologies based on the principles of string theory.

Similar threads

A "Monstrous Higgs and Leech Spacetime"
    • Beyond the Standard Models
Replies
0
Views
1K
Clarifying Open String Hamiltonian for Witten's Book
    • Beyond the Standard Models
Replies
7
Views
4K
Quantum Looking for a good String Theory text
    • Science and Math Textbooks
Replies
4
Views
1K
I Trying To Conceptualize Two Dimensions Of Time
    • Beyond the Standard Models
Replies
2
Views
2K
Understanding GSW 2.2.3: Questions on Vertexoperators and Symmetric Traces
    • Beyond the Standard Models
Replies
2
Views
3K
A Constructing massive N=1 SUSY Multiplets
    • Beyond the Standard Models
Replies
5
Views
2K
Classical Strings - constructing explicit solutions
    • Beyond the Standard Models
Replies
1
Views
2K
Linearized graviational field of a cosmic string
    • Advanced Physics Homework Help
Replies
5
Views
2K
Thomas Larsson's post on LQG-String
    • Beyond the Standard Models
2
Replies
44
Views
8K
How Do Waves Propagate on a String?
    • Other Physics Topics
Replies
0
Views
4K

Hot Threads

Recent Insights

Back
Top