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BRST Quantization
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So far we have seen two methods that can be utilized to quantize the string: the covariant approach and light-cone quantization. Each offers its advantages. Covariant quantization makes Lorentz invariance manifest but allows for the existence of ghost states (states with negative norm) in the theory. In contrast, light-cone quantization is ghost free. However, Lorentz invariance is no longer obvious. Another trade-off is that the proof of the number of space-time dimensions (D = 26 for the bosonic theory) is rather dif cult in covariant quantization, but it s rather straightforward in light-cone quantization. Finally identifying the physical states is easier in the light-cone approach. Another method of quantization, that in some ways is a more advanced approach, is called BRST quantization. This approach takes a middle ground between the two methods outlined above. BRST quantization is manifestly Lorentz invariant, but includes ghost states in the theory. Despite this, BRST quantization makes it easier to identify the physical states of the theory and to extract the number of space-time dimensions relatively easily.
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String Theory Demysti ed
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BRST Operators and Introductory Remarks
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We begin by considering our old friend, Lie algebra. This is the algebra that is obeyed by the familiar spin angular momentum operators of ordinary quantum mechanics. In general, let some physical theory contain a gauge symmetry with operators K i. These operators satisfy the Lie algebra: [ K i , K j ] = fij k K k (6.1)
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where the fij k are called the structure constants of the theory (note we are using the Einstein summation convention, so repeated indices are summed over). The structure constants satisfy fij m fmk l + f jk m fmi l + fki m fmj l = 0 (6.2)
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The BRST quantization procedure begins with the following. We introduce two ghost elds that are denoted by bi and c j which satisfy an anticommutation relation given by {ci , b j } = ij (6.3)
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Furthermore {ci , c j } = {bi , b j } = 0. Here the ci are ghost elds and the b j are ghost momenta. Notice that since an anticommutation relation is satis ed by the ghost elds, these elds are fermionic. Now, recall that a eld ( z , z ) has conformal dimension (h, h ) provided that it transforms under some conformal transformation z w( z ) as follows: w w ( z , z ) = ( w, w) z z
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(6.4)
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The elds b and c are chosen such that they have conformal dimension 2 and 1, as we will see later. There are two operators that are constructed out of the ghost elds and the K i . The rst of these is the BRST operator which is given by Q = ci K i 1 k i j fij c c bk 2 (6.5)
CHAPTER 6 BRST Quantization
It is assumed that Q = Q . We say that the BRST operator is nilpotent of degree two. This means that if we square the operator we get zero: Q2 = 0 (6.6)
Notice that Eq. (6.6) can also be expressed as {Q, Q} = 0. We label the BRST operator with a Q to imply that this is a conserved charge of the system, we often call this the BRST charge. A second operator that is composed solely of ghost elds is called the ghost number operator U. This is given by U = ci bi (6.7)
(Again note the Einstein summation convention is being used.) This operator has integer eigenvalues. If the dimension of the Lie algebra is n, then the eigenvalues of U are the integers 0,...,n. A state has ghost number m if U = m . EXAMPLE 6.1 Show that Q raises the ghost number by 1. SOLUTION Using the anticommutation relations for the ghost elds [Eq. (6.3)], notice that Uci K i = cr br ci K i = cr ( ri ci br )K i
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