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

2.8 The CFT Hilbert Space 41
together with Eq. (2.63), we can write
T (z)T (w) = c/2 2 T (w) �T (w)
+ +
(z − w) 4 (z − w) 2 z − w + N � TT � (w) + ... . (2.70)
However, using the general expression for the OPE of two quasi-primary fields
shown in Eq. (2.53), we observe that there is a � 2 T term at (z − w) 0 with coefficient
C T TT
a 2 222
2!
where a 2 222 =
� �
5
−1� �
3
2 2
= 3
10
and C T TT = 2 .
(2.71)
But, since the index k in Eq. (2.53) runs over all quasi-primary fields of the theory,
we expect also other terms at order (z − w) 0 . If we denote these by N (TT), we find
from Eq. (2.70) that
N � TT � = N � TT � + 3
10 �2 T .
One can easily check that � 2 T is not a quasi-primary field, and by computing for
instance [Lm, N(TT)n] and comparing with Eq. (2.45), one arrives at the same conclusion
for N(TT). However, we note that
N � TT � = N � TT � − 3
10 �2 T (2.72)
actually is a quasi-primary normal ordered product. Moreover, it turns out that this
procedure can be iterated which allows one to write the entire field space in terms
of quasi-primary fields and derivatives thereof.
2.8 The CFT Hilbert Space
In this section, we are going summarise some general properties of the Hilbert space
of a conformal field theory.
The Verma Module
Let us consider again the chiral energy–momentum tensor. For the Laurent expansions
of T (z) and �T (z) as well as for the corresponding asymptotic in-states, we
find