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A closed string is a little loop moving through space-time. In this case, the worldsheet is a cylinder or a tube. The boundary conditions are periodic, described by X ( , ) = X ( , + ) (2.23)
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Now let s write down the conjugate momenta for the string. First recall the lagrangian L = T ( X X ) 2 ( X ) 2 ( X ) 2 The conjugate momentum corresponding to the coordinate s is P = L = T ( X X ) 2 ( X ) 2 ( X ) 2 X X
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1/ 2 T ( X X ) 2 ( X ) 2 ( X ) 2 2 ( X X ) X 2 X 2 X 2
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= T
( X X ) X X 2 X ( X X )2 ( X )2 ( X )2
String Theory Demysti ed
and we also have a conjugate momentum corresponding to the coordinate : L = T ( X X ) 2 ( X ) 2 ( X ) 2 X X
P =
(2.25)
1/ 2 T ( X X ) 2 ( X ) 2 ( X ) 2 2 ( X X ) X 2 X 2 X 2
= T
( X X ) X X 2 X ( X X )2 ( X )2 ( X )2
Now, let s vary the action to obtain the equations of motion for the string. First, if it s been awhile since you ve had eld theory, convince yourself that X = ( X ) X = X X = = ( X ) Then, we can vary the action, and using the conjugate momenta we get
f L L S = T d d ( X ) + ( X ) i 0 X X f = T d d ( X ) + ( X ) 0 i
We can rewrite this expression so that we can get terms multiplied by X by using the product rule from calculus. For example, P X + X P X = P
P P X = P X X
CHAPTER 2 Equations of Motion
Similarly, P X = P P X X
This means that the variation of the action can be written as P S = T d d P X X i 0 P f T d d P X X i 0
Recall from classical mechanics that a variation is de ned such that variation at the endpoints is 0, that is, at the initial and nal times X = 0. In the case of the endpoints of the string, we can apply either Neumann or Dirichlet boundary conditions so we will have to handle each case differently (more on this as we go along). For now, let s take X = 0 for simplicity. This means that we can throw away the terms in the above expression which are integrals of total derivatives:
P X = 0
P X = 0
This leaves us with P P f f 0 = S = T d d X + T d d X i 0 i 0 P P f + = T d d X 0 i This gives us the equation of motion for the string, derived from the Nambu-Goto action: P + P =0 (2.26)
String Theory Demysti ed
The Polyakov Action
Quantization using the Nambu-Goto action is not convenient due to the presence of the square root in the lagrangian. It is possible to write down an equivalent action, equivalent in the sense that it leads to the same equations of motion that does not have the cumbersome square root. This action goes by the name of the Polyakov action or by the more modern term the string sigma model action. Look back to the start of the chapter when we considered the point particle. There too, we ran into a situation where the action had a square root and we dealt with it by introducing an auxiliary eld a( ). We can use the same procedure here, to rewrite the action for the string in a more convenient form. This is done by introducing an intrinsic metric h ( , ), which acts like the auxiliary eld. We use the notation h because the metric can be written as a matrix. We use the indices to denote rows and columns in this matrix. Then, using the notation h = det h , the Polyakov action can be written as SP = T d 2 h h X X 2 (2.27)
A historical aside: While Polyakov did important work with this action, it was actually proposed by Brink, Di Vecchia, and Howe and independently by Desser and Zumino. Polyakov got his name attached to it by using it in a path integral quantization of the string. It is also called the string sigma action.
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