visual basic print barcode label Conformal Field Theory Part I in Java

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CHAPTER 5 Conformal Field Theory Part I
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= , taking the derivative with respect to we have, on the left-hand + = + ( )
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Hence, taking the same derivative on the right side and equating results we obtain 2 + 1 ( ) = 0 d Notice that this equation singles out the case of two dimensions. Setting d = 2 we obtain
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We can obtain a second equation which highlights the importance of d = 2 by operating on * with = . This gives { + (d 2) }( ) = 0 The in nitesimal parameter can represent four different types of transformations: translations, scale transformations, rotations, and special conformal transformations. A translation takes the form
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where a is a constant. A scale transformation is one of the form:
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For a rotation, we write
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= x
where we require that is antisymmetric, that is, = . Finally, a special conformal transformation assumes the form
= b x 2 2 x (b x )
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These operations can be combined with the Poincar group to form the conformal group. We incorporate two generators from the Poincar group, the generator of translations P and the generator of rotations J . Denoting the generator of a scale transformation by D and the generator of a special conformal transformation by K , the generators of the conformal group are
P = i D = i x J = i ( x x ) K = i[ x 2 2 x ( x )]
(5.19)
The Two-Dimensional Conformal Group
We now simplify the discussion somewhat and consider the special case of interest to us, the conformal group in two dimensions. In Eq. (5.18) we found that + = where we have set d = 2. Proceeding with the two-dimensional case, take coordinates ( x1 , x 2 ) . Then when = 1, = 2 we have = 0 and we obtain 1 2 + 2 1 = 0 Dispensing with the shorthand notation for a moment, this might look more familiar as 2 = 1 1 x x 2 This is nothing other than one of the Cauchy-Riemann equations when we take = 1 + i 2 and x = x1 + ix2 . Similarly you can also show that 1 2 = x1 x 2 In the theory of complex variables we learned that a function that satis es the Cauchy-Riemann equations in a given region R is called analytic. An analytic function is one that is a function of z only. So, labeling our coordinates with the usual complex coordinates ( z, z ) conformal transformations in two dimensions are implemented using analytic functions: z f (z) z f (z ) (5.20)
CHAPTER 5 Conformal Field Theory Part I
where f = f = 0. To obtain the generators, we consider a coordinate transformation of the form: z z = z n z n+1 z z = z n z n+1 (5.21)
To obtain an expression for the generators of a conformal transformation in two dimensions, we take the derivatives of the transformed coordinates z , z and look for terms containing the derivatives n and n , respectively. In the rst case we obtain z = ( z n z n+1 ) = 1 n (n + 1) z n z n+1 z n z
This allows us to identify the generator:
= z n +1 z
(5.22)
A similar procedure applied to the complex conjugate coordinate gives
= z n +1 z
(5.23)
In the classical case, the generators [Eqs. (5.22) and (5.23)] satisfy the Virasoro algebra: [
] = (m n)
= (m n)
(5.24)
EXAMPLE 5.1 Show that the in nitesimal generator [ m , n ] = ( m n ) m +n.
= z n +1 satis es the Virasoro algebra
SOLUTION We apply the generator, which is an operator, to a test function f. So we obtain [ , ]f =( )f
m n m +1
m n +1
= z
( z ) f ( z n +1 )( z m +1 ) f
+ = z m +1[ (n + 1)z n f z n +1 2 f ] + z n+1[ ( m + 1)z m f z m +1 2 f ]
= (n + 1)z m +n +1 f + z m +n +2 2 f ( m + 1)z m +n +1 f z m +n +2 2 f = ( m n )[ z m +n +1 ] f = (m n ) m+n f
Hence we conclude that [ ,
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