6.4 Operator Formalism and Radial Quantization
129
= (∓1)
h
dz
2π i
z
n−h−1 φ
±
1
z
= (∓1)
h
dw
2π i
±
1
w
n−h
w
−1 φ(w)
= (∓1)
h (±1)
n−h
dw
2π i
w
−n+h−1 φ(w),
where we have set w = ±1/z such that
dz
z
= ∓
dw
w 2 z
= −
dw
w
,
(6.112)
and the minus sign disappears upon reversing the contour orientation.
The mode expansion of the energy–momentum tensor is
T (z) =
n∈Z
L n
z n+2 ,
L n =
dz
2π i
T (z)z
n+1 ,
(6.113)
where one recognizes the Virasoro operators as the modes. In most situations, the
Virasoro operators are Hermitian
L
†
n = L −n .
(6.114)
The OPE (6.88) and (6.87) together with (6.75a) help to reconstruct the Virasoro
algebra (6.58) and the commutation relations between the L m and the modes φ n of
a weight h primary:
[L m , φ n ] =
m(h − 1) − n
φ m+n .
(6.115)
This easily gives the commutation relation for the complete field:
[L m , φ(z)] = z
m
z∂ + (n + 1)h
φ(z).
(6.116)
We will often use (6.58) and (6.115) for m = 0:
[L 0 , L −n ] = nL −n ,
[L 0 , φ −n ] = nφ −n .
(6.117)
This means that both φ n and L n act as raising operators for L 0 if n < 0, and
as lowering operators if n > 0 (remember that L 0 is the Hamiltonian in the
holomorphic sector). When both the holomorphic and anti-holomorphic sectors
enter, it is convenient to introduce the combinations
L
±
n = L n ± ¯
L n ,
(6.118)
such that L
+
0 is the Hamiltonian.