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String Theory Demysti ed
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Using the Minkowski metric, we have seen that in nitesimal distances in spacetime can be de ned according to ds 2 = dx dx = ( dx 0 )2 ( dx1 )2 ( dx 2 )2 ( dx 3 )2 Since, dx 0 = dx + + dx 2 dx 1 = dx + dx 2 (2.35)
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we can rewrite ds 2 in terms of light-cone coordinates as ds 2 = 2dx + dx (dx 2 )2 (dx 3 )2 So, we can de ne distances in terms of a light-cone Minkowski metric 0 1 1 0 = 0 0 0 0 0 0 1 0 0 0 0 1 (2.36)
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(2.37)
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Using Eq. (2.37), distances can be written compactly as ds 2 = dx dx (2.38)
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Working with vectors is a simple extension of what we ve written for coordinates. That is, de ne light-cone components of a vector v as v+ = v 0 + v1 2 v = v 0 v1 2 (2.39)
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Using the metric from Eq. (2.37) the inner product between two vectors can be calculated as v w = v w = v w = v + w v w + + vi w i
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(2.40)
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CHAPTER 2 Equations of Motion
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where generally, i = 1, , d 1. We can apply index raising and lowering to the light-cone components of vectors using a sign change v + = v v = v+
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The other components of the vector are left unchanged, that is, vi = vi . Now that we ve seen how to de ne light-cone coordinates for space-time, let s see how to de ne them for the worldsheet and hence for the string. In this case, we de ne
+ = +
=
(2.41)
Now since d + = d + d and d = d d , it should be clear that ds 2 = d + d (2.42)
This tells us that we can write the induced metric in Eq. (2.30) indexing a matrix as ( + +, + , +, ) giving 0 1/ 2 h = 1/ 2 0 (2.43)
You can quickly verify that the determinant is h = det h = 1/ 4 and the inverse of Eq. (2.43) is 0 2 h = 2 0 A relationship can also be written down between the derivatives with respect to the coordinates , and the light-cone coordinates. For notational convenience, we use the relativistic shorthand notation for derivatives i = and write 1 + = ( + ) 2 1 = ( ) 2 (2.44) x i
String Theory Demysti ed
Let s see how the action for the string is written using light-cone coordinates. The Polyakov action, which we reproduce here for your convenience, is SP = Using Eq. (2.43), we nd that h h X X = 1/ 4 h + + X X 1/ 4 h + X + X = 2 + X X Hence, using light-cone coordinates we nd the Polyakov action can be written as SP = T d 2 + X X We can nd the equations of motion by varying SP. We have (2.45) T d 2 h h X X 2
SP = T d 2 + X X
= T d 2 ( + X X ) = T d 2 ( + X ) X + T d 2 + X ( X ) The following fact helps us proceed:
Therefore,
X ( X ) =
SP = T d 2 + ( X ) X + T d 2 + X ( X )
Now integrate by parts to move the derivative away from the X term. Remember that
udv = uv vdu
CHAPTER 2 Equations of Motion
In our case, we get
SP = T d 2 ( X ) + X T d 2 + X ( X )
We ve dropped the boundary terms, which must vanish for Neumann boundary conditions in the case of open strings or for the requirement of periodicity for closed strings. Since X is arbitrary and SP = 0, it must be the case that + X = 0 (2.46)