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puts all destruction operators to the right annihilating the vacuum. So the vacuum expectation value of the normal ordered piece reduces to 0 : X ( , ) X ( , ) : 0 = 0 x x 0 Hence the two-point function is X ( , ), X ( , ) = T ( X ( , ), X ( , )) : X ( , ) X ( , ) :
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Now that we have this expression, we can easily calculate other quantities that involve derivatives, say. For example, to calculate the two-point function z X ( z, z ), X ( z , z ) , we just differentiate the result: 2 s z X ( z, z ), X ( z , z ) = z ln( z z ) 2 2 2 s 1 = 2 z z And z X ( z, z ), z X ( z , z ) =
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Operator Product Expansion
A key concept we need to continue forward is known as an operator product expansion. This is often given the abbreviation OPE. An operator product expansion is a series expansion of a product of two operator-valued elds. Let s denote these elds by Ai and consider two space-time points z and w. Then in a region R that does not contain w Ai ( z ) A j (w ) = cijk ( z w ) Ak (w )
(5.36)
CHAPTER 5 Conformal Field Theory Part I
The cijk ( z w ) are analytic functions in R and the Ak (w ) are operator-valued elds. Now, de ne a conformal transformation z w( z ). A conformal eld or primary eld is one that transforms as w w ( z, z ) = (w, w ) z z
(5.37)
We say that (h, h ) are the conformal weights or conformal dimension of the eld. In particular, h + h is the dimension which describes how the eld behaves under scaling, while h h is the spin of , which describes how the eld is transformed under a rotation. If Eq. (5.37) is satis ed, then it follows that ( z , z )(dz )h (d z )h = ( w, w)(dw)h (dw)h (5.38)
That is, ( z , z )(dz )h (d z )h is invariant under a conformal transformation. Working in the complex plane, time ordering is transformed into radial ordering because as we mentioned above, the radial direction encodes the ow of time in the z plane. Consider two operators de ned in the complex plane A( z ) and B(w ) . The radial-ordering operator R xes the order of the operators based on which one has the larger radius in the complex plane. That is, A( z ) B( w), z > w R[ A( z ) B( w)] = f ( 1) B( w) A( z ), w > z If the operators are fermionic, then f = 1. One operator product expansion of particular interest involves the energymomentum tensor. In the complex plane Tzz ( z ) = : z X z X : (5.39)
EXAMPLE 5.4 Find the operator product expansion of the radially ordered product Tzz ( z ) w X (w ) . SOLUTION Using Eq. (5.39) we have R(Tzz ( z ) w X ( w)) = R(: z X ( z ) z X ( z ) : w X ( w)) = z X ( z ) w X ( w) z X ( z ) + z X ( z ) w X ( w) z X ( z )
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