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The object ( k ) is a tensor which can be decomposed into symmetric ( ) ( k ) and antisymmetric ( ) ( k ) parts (see Relativity Demysti ed if you aren t sure about
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CHAPTER 4 String Quantization
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this). The symmetric part corresponds to a massless spin-2 particle which is the graviton. The linearized metric g = + ( x ) satis es the linearized Einstein equations ( x ) = 0, and ( x ) = 0. By taking the Fourier transform of ( ) ( k ), it can be shown that these equations are satis ed. The trace ( k ), which de nes a scalar, is also important. This corresponds to a massless scalar particle called the dilaton.
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Light-Cone Quantization
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We now turn our attention to a different method of quantization. We began the chapter with a discussion of covariant quantization. This method is a straightforward application of the imposition of commutation relations. We took this approach rst because it may seem familiar from your studies of ordinary quantum mechanics. In addition, it preserves the Lorentz invariance of the theory. Physicists say that this approach is manifestly Lorentz invariant, colloquially meaning that the Lorentz invariance is obvious. The technique has the disadvantage in that negative norm states appear. Although this is a problem, it is instructive to go through the process of eliminating the negative norm states. Another approach is possible which avoids the negative norm states at the cost of losing manifest Lorentz invariance. This is called light-cone quantization. We brie y discuss it here, considering the open string case. We begin by using light-cone coordinates, which were introduced in Chap. 2 in Eq. (2.34): X = X 0 X D 1 2 (4.30)
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The remaining coordinates X i are transverse coordinates. The center of mass coordinate x and momentum p are also written as light-cone coordinates. In the light-cone gauge, we choose X + = x+ +
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(4.31)
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+ which leads to n = 0 for n 0, that is, the modes are zero for X +. The Virasoro constraints will lead us to a description based on transverse oscillators. We have the freedom to set 2 p + = 1, giving the center of mass position as s
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X = x +
p p+
The Virasoro constraint becomes
String Theory Demysti ed
1 X X = ( X i X i )2 2
(4.32)
So we have a relation between light-cone coordinates and the transverse coordinates. The mode expansion of the worldsheet coordinates for an open string is given by X = x + In particular X = x +
2 s 2 s
p + i
n in e cos n s n 0 n
p + i
n in e cos n s n 0 n
We can then solve for the nonzero modes of X in terms of the transverse oscillators. These are
n =
1 1 D 2 i i + 2 : n m m : a n , 0 p s i=1 m=
(4.33)
In the case of the zeroth mode, we can derive an expression for the hamiltonian H = p = 1 i i 1 i i p p + n n + 2p n 0 (4.34)
De ne a conjugate momentum P . The system is quantized by imposing commutation relations on the transverse components of position and momentum: [ x i , p j ] = i ij , [ X i ( ), P j ( )] = i ij ( ) The mass-shell condition becomes M 2 = 2 p+ p pi2 =
i =1 D 2
(4.35)
( N a)
(4.36)
The number operator is given by
i i N = n n i =1 n =1 D 2
(4.37)
CHAPTER 4 String Quantization
Normal ordering leads to 1 D 2 i i 1 D 2 i i D 2 n n n = 2 n : n n : + 2 n 2 i=1 = n =1 i =1 =
The regularization trick can be applied to make the second term nite. We nd D 2 D 2 n = 24 2 n=1 Again taking a = 1, one nds D = 26.
Summary
In the previous two chapters, we constructed a relativistic theory of the string, the classical theory. In this chapter we have introduced the simplest possible quantum extension of the classical theory. This is a theory that consists only of bosons. While the theory is not realistic since it does not include fermionic states, it is easier to deal with and introduces important concepts and methods that will play a role in the full quantum theory. The classical theory was quantized using two different methods. The rst method, called covariant quantization, is a straightforward approach that imposes commutation relations on the X and their conjugate momenta. This leads to negative norm states. The Virasoro constraints are imposed to rid the theory of these states. When this is done, we nd that the theory must have 26 space-time dimensions. We concluded the chapter with a different approach, known as light-cone quantization.
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