6.4 Operator Formalism and Radial Quantization
123
where the OPE coefficients c k
ij are some constants and the sum runs over all
operators. When O k is primary, the coefficients c k
ij are related to the structure
constants and the field metric by
C ij k = g kk c
ij .
(6.82)
The radius of convergence for the OPE is given by the distance to the nearest
operators in the correlation function. The OPE defines an associative algebra
(commutative for bosonic operators), and the holomorphic sector forms a subalgebra
(called the chiral algebra).
Example 6.1: OPE with the Identity
The OPE of a field φ(z) with the identity 1 is found by a direct series expansion
φ(z)1 =
n∈N
(z − w) n
n!
∂
n φ(w).
(6.83)
Obviously there are no singular terms.
Starting from this point we consider only the holomorphic sector except when
stated otherwise. The formula for the OPE (6.81) can be rewritten as
A(z)B(w) :=
N
n=−∞
{AB} n (z)
(z − w) n
(6.84)
to simplify the manipulations. N is an integer and there are singular terms if N > 0.
Generally, only the terms singular as w → z are necessary in the computations (for
example, to use the Cauchy–Riemann formula (B.1)): equality up to non-singular
terms is denoted by a tilde
A(z)B(w) ∼
N
n=1
{AB} n (z)
(z − w) n =: A(z)B(w).
(6.85)
The RHS of this expression defines the contraction of the operators A and B.
While, most of the time, only singular terms are kept
φ i (z i )φ j (z j ) ∼
k
θ(h i + h j − h k )
c k
ij
(z − w) h i +h j −h k
φ k (w)
(6.86)
(with θ(x) the Heaviside step function), it can happen that one keeps also nonsingular terms (the product of two OPE have singular terms coming from non-