130
6 Conformal Field Theory on the Plane
Finally, every holomorphic current j (z) has a conformal weight h = 1 and can
be expanded as
j (z) =
n
j n
z n+1 .
(6.119)
By definition, the zero-mode is equal to the holomorphic charge
Q L = j 0 .
(6.120)
6.4.5 Hilbert Space
The Hilbert space of the CFT is denoted by H. The SL(2, C) (or conformal)
vacuum 15 |0 is defined by the state which is invariant under the global conformal
transformations:
L 0 |0 = 0,
L ±1 |0 = 0.
(6.121)
Expectation value of an operator O in the SL(2, C) vacuum is denoted as:
O :==0|O|0.
(6.122)
If the fields are expressed in terms of creation and annihilation operators (which
happens e.g. for free scalars, free fermions and ghosts), then the Hilbert space has
the structure of a Fock space.
State-Operator Correspondence
The state–operator correspondence identifies every state |O of the CFT Hilbert
space with an operator O(z, ¯
z) through
|O = lim
z,¯ z→0
O(z, ¯
z)|0 = O(0, 0)|0.
(6.123)
Such a state can be interpreted as an “in” state since it is located at τ → −∞ on the
cylinder. Focusing now on a holomorphic field φ(z), the state is defined as
|φ = lim
z→0
φ(z)|0 = φ(0)|0.
(6.124)
15 There are different notions of “vacuum”, see (6.129). However, the SL(2, C) vacuum is unique.
Indeed, it is mapped to the unique identity operator under the state–operator correspondence
(however, there can be other states of weight 0, see Remark 6.6).