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Now we work on the second term, in two steps. First we lower an index, and then we swap the labels used for the dummy indices and again exploit the
CHAPTER 3 Symmetries and Worldsheet Currents
symmetry of the metric, but this time it s the worldsheet metric we are talking about:
LP =
T 2 T = 2 T = 2 T = 2 T = 2
h h X b h h h h h h h h
( ( (
= T h h
( (
T 2 T X b 2 T X b 2 T X b 2 T X b 2
h h b X h h b X h h b X h h b X h h X b
) ) ) )
X b
Now notice we have a term multiplied by b , which is the small parameter we used to vary X . The rest of this expression is the conserved current we re looking for: P = T h h ( X ) (3.20)
If we use reparameterization and Weyl invariance to take h then we have P = T X P = T X The conservation equation for the current is P = 0 (3.21) P = T X
Dropping the string tension T and ignoring the minus sign we see that the conservation equation for the current becomes the equation of motion for the string worldsheet: P + P = 0 (3.22)
String Theory Demysti ed
P has an immediate physical interpretation. It is the momentum density of the string. We integrate along the length of the string xing to get the total momentum carried by the string, which we label p : p = d P
(3.23)
The other conserved current associated with the global symmetries of the action comes from invariance under Lorentz transformations. In this case
X = X
We can show that the lagrangian in Eq. (3.19) is invariant under a Lorentz transformation in the following way:
X X = X X + X X
= =
) ( ) ( X ) X + X ( X ) ( X ) X + X ( X )
= X X + X X = X X + X X = X X + X X = X X + X X (lower indices with ) w (relabel dummy indices ) (relabel dummy indices ) (antisymmetry = )
= X X + X X = 0
So the lagrangian is invariant under a Lorentz transformation, but what are the currents This is easy to nd, since T h h X X 2 LP T T = h h X = h h P 2 2 ( X ) LP =
Using J = [ LP / ( X )] X where X = X together with the antisymmetry of , we conclude that the Lorentz current is J = T X P X P
(3.24)
CHAPTER 3 Symmetries and Worldsheet Currents
The Hamiltonian
We have introduced the energy-momentum tensor and looked at some conserved currents that arise due to symmetries of the lagrangian. The next major piece of the dynamics puzzle is the hamiltonian which governs the time evolution of the worldsheet. It can be written down simply using formulas from classical mechanics: H = d X P LP =
d ( X 2 + X 2 )
(3.25)
Summary
In this chapter we have extended our classical analysis of the string. We did this by introducing the energy-momentum tensor and by describing the symmetries of the Polyakov action. Then we derived conserved currents of the worldsheet, and wrote down the hamiltonian. In the next chapter, we will conclude our classical description of the string by writing down mode expansions of the hamiltonian and energymomentum tensor, and describing the Virasoro algebra. After writing down a mass formula for the string, we will proceed to quantize the theory.
Quiz
1. Let + ( ) be an in nitesimal reparameterization. Considering only terms that are rst order in , nd the variation of the worldsheet metric h . 2. Assuming that you can move the derivative d/d inside the integral in Eq. (3.23), explain conservation of momentum for the cases of the open and closed string. 3. The energy-momentum tensor has zero trace. Show that this is a consequence of Weyl invariance. 4. Consider light-cone coordinates and derive the Virasoro constraints for the energy-momentum tensor. 5. Consider the Lorentz current J in Eq. (3.24). What is the equation that describes that the current is conserved What are the conserved charges and what do they describe
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