120
6 Conformal Field Theory on the Plane
corresponds to dilatation on the plane
z −→ e
T z.
(6.69)
Thus, time evolution on the cylinder and radial evolution (from the origin to the
complex infinity) are identified. In particular, the Hamiltonian of the system of the
plane is
H =
2π
L
(L 0 + ¯
L 0 ),
(6.70)
since the RHS is the dilatation operator. The cylinder length L was defined in (6.9).
The theory is quantized according to this Hamiltonian. In the string theory language,
a state with H = 0 is said to be on-shell:
on-shell state:
h + ¯
h = 0.
(6.71)
6.4.1 Radial Ordering and Commutators
Time-ordering in τ becomes radial ordering in the plane:
R
A(z)B(w)
=
A(z)B(w)
|z| > |w|,
(−1) F B(w)A(z) |w| > |z|,
(6.72)
where F = 0 (F = 1) for bosonic (fermionic) operators. Radial ordering will often
be kept implicit.
The equal-time (anti-)commutator becomes an equal radius commutator defined
by point-splitting:
[A(z), B(w)] ±,|z|=|w| = lim
δ→0
A(z)B(w)| |z|=|w|+δ ± B(w)A(z)| |z|=|w|−δ
.
(6.73)
If A and B are two operators which can be written as the contour integrals of a(z)
and b(z) (corresponding to integral over closed curves on the cylinder)
A =
C 0
dz
2π i
a(z),
B =
C 0
dz
2π i
b(z),
(6.74)
then one finds the following commutators:
[A, B] ± =
C 0
dw
2π i
C w
dz
2π i
a(z)b(w),
(6.75a)
[A, b(w)] ± =
C w
dz
2π i
a(z)b(w).
(6.75b)