Closed Strings and Virasoro algebra

[earth2]

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
$$L_m\propto \int e^{...m} T_{--}d\sigma$$ 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

 

[petergreat]

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.

 

[earth2]

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

 

[petergreat]

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.