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CHAPTER 4 String Quantization
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A normal ordered product is denoted using two colons, that is : a a :. In the case of the Virasoro operator, we write Lm = 1 : m n n : 2 n
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Lm will lower the eigenvalue of the number operator by m. Looking at the commutation relation Eq. (4.10), you can see that m n and n commute when m 0. This means that when m 0 we can simply move raising and lowering operators where we want in the expression for the Virasoro operator because no extra terms will be added from the commutator. Normal ordering L0 gives 1 2 L0 = 0 + n n 2 n =1 Now, to get this result, note that 1 1 1 1 n n = 2 0 0 + 2 n n + 2 n n 2 n= n =1 n =1
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1 1 D 1 1 D = 0 0 + n n + n = 0 0 + n n + n 2 2 =0 2 2 n=1 n =1 n =1 n =1
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(4.19)
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To get from the rst to the second line, the commutator for the modes was used together with the fact that n n represents the dot product in D space-time dimensions. To get the normally ordered result we have thrown out the sum D /2 n. n =1 At rst glance, you might think that this sum is in nite, but regularization can be used to compute its nite value. To see how this works, recall the geometric series:
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a 1 r
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Taking the derivative, we have d d a a ar n = a n r n = dr 1 r = (1 r )2 dr n n Now let a = 2 / and multiply n by e a, then
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n =1
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d d e a 1 1 1 (e a )n = n (e a )n = da 1 e a = (1 e a )2 = a 2 12 da n n
String Theory Demysti ed
It follows that D/ 2 n = D/ 2( 1/12) = D/ 24. Another undetermined constant n =1 piece is missing from the difference between the general expression of L0 and the normal ordered expression of L0. This is a normal ordering constant which is denoted by a. Therefore in any calculation L0 is replaced by L0 a, where a is a constant. The point of all this is to write down the commutation relations for the Virasoro operators. Using Eq. (4.10), one nds that [ L m , Ln ] = ( m n ) L m + n + D 3 ( m m ) m +n , 0 12 (4.20)
We call this commutation relation the Virasoro algebra with central extension. The central charge is the space-time dimension D which has shown up in the second term on the right-hand side. This is also the number of free scalar elds on the worldsheet. It is clear that if m = 0, 1 the central extension term will vanish. This singles out L1 , L0 , and L 1 which form a closed subalgebra. We call this the SL (2, R) algebra. The Virasoro operators can be used to eliminate unphysical states (i.e., negative norm states) from the theory by requiring that the expectation value of L0 a vanishes for a physical state . That is, we impose the constraint
Lm a m ,0 = 0
for m 0. The term a m ,0 takes care of the fact that we only need the normal ordering constant a in the case of L0. To eliminate negative norm states, speci c conditions must be put on a and D, which is the origin of the extra dimensions in string theory. In particular, it can be shown that negative norm states can be eliminated if a =1 D = 26 (4.21)
The reason that a = 1is chosen is a bit beyond the scope we want to cover in this book, see the references if interested in the proof. We can proceed further to obtain a mass operator. First recall that Einstein s equation tells us p p + m 2 = 0 m 2 = p p
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