Introduction to Conformal Field Theory: With Applications to String ...

2.7 Normal Ordered Products 39
The notation for normal ordering we will mainly employ is N(χφ), however, it is
also common to use : φχ :, (φχ) or[φχ]0. In the following, we will verify the
statement above for the case n = 0.
Let us first use Eq. (2.63) to obtain an expression for the normal ordered product
of two operators. To do so, we apply 1
�
−1 dz(z − w) to both sides of Eq. (2.63)
2πi
which picks out the n = 0 term on the right-hand side leading to
N � χφ � �
(w) =
C(w)
dz
2πi
φ(z)χ(w)
z − w
. (2.64)
However, we can also perform a Laurent expansion of N(χφ) in the usual way
which gives us
N � χφ � (w) = �
N � χφ �
n =
n∈Z
�
C(0)
w −n−hφ −h χ
N � χφ �
n ,
dw
2πi wn+hφ +h χ −1 N � χφ � (w) , (2.65)
where we also include the expression for the Laurent modes N(χ φ)n. Let us now
employ the relation (2.64) in Eq. (2.65) for which we find
N � χφ �
�
dw
C(0) 2πi wn+hφ +hχ �
−1
�
dw
=
2πi wn+hφ +hχ � �
−1
n =
I2
C(w)
dz
2πi
φ(z)χ(w)
z − w
(2.66)
�
�
dz φ(z)χ(w) dz χ(w)φ(z)
−
|z|>|w| 2πi z − w |z||w|
�
=
�
=
|z|>|w|
|z|>|w|
dz
2πi
dz
2πi
dz
2πi
Note that we employed 1
z−w =
1 �
z
z − w
r,s
−r−hφ
w −s−hχ
φr χs
1 �
� �p w �
z
z z
−r−hφ
w −s−hχ
φr χs
p≥0
r,s
� �
z −r−hφ −p−1 −s−h
w χ +p
φr χs .
p≥0
r,s
1
as well as the geometric series to go from
z(1−w/z)
the first to the second line and that only the z −1 term gives a non-zero contribution.
Thus, performing the integral over dz leads to a δ-function setting r =−h φ − p and