138
6 Conformal Field Theory on the Plane
=
n≥0
A −n B m+n +
n>0
B m−n A n +
h A −1
n=0
[B m+n , A −n ]
=
AB
m +
h A −1
n=0
[B m+n , A −n ].
The choice of the normal ordering for the operators is related to the ordering
ambiguity when quantizing the system: when the product of two non-commuting
modes appears in the classical composite field, the corresponding quantum operator
is ambiguous (generally up to a constant). In practice, one starts with the conformal
ordering since it is invariant under conformal transformations and because one
can compute with contour integrals. Then, the expression can be translated in the
energy ordering using (6.163). But, knowing how the conformal and energy vacua
are related, it is often simpler to find the difference between the two orderings by
applying the operator on the vacua.
6.4.6 CFT on the Cylinder
According to (6.44), the relation between the field on the cylinder and on the plane
is
φ(z) =
L
2π
h
z
−h φ cyl (w)
(6.164)
(quantities without indices are on the plane by definition). The mode expansion on
the cylinder is
φ cyl =
2π
L
h
n∈Z
φ n e
−
2π
L w
=
2π
L
h
n∈Z
φ n
z n .
(6.165)
Using the finite transformation (6.90) for the energy–momentum tensor T , one
finds the relation
T cyl (w) =
2π
L
2
T (z)z
2
−
c
24
.
(6.166)
For the L 0 mode, one finds
(L 0 ) cyl = L 0 −
c
24
,
(6.167)
6 Conformal Field Theory on the Plane
=
n≥0
A −n B m+n +
n>0
B m−n A n +
h A −1
n=0
[B m+n , A −n ]
=
AB
m +
h A −1
n=0
[B m+n , A −n ].
The choice of the normal ordering for the operators is related to the ordering
ambiguity when quantizing the system: when the product of two non-commuting
modes appears in the classical composite field, the corresponding quantum operator
is ambiguous (generally up to a constant). In practice, one starts with the conformal
ordering since it is invariant under conformal transformations and because one
can compute with contour integrals. Then, the expression can be translated in the
energy ordering using (6.163). But, knowing how the conformal and energy vacua
are related, it is often simpler to find the difference between the two orderings by
applying the operator on the vacua.
6.4.6 CFT on the Cylinder
According to (6.44), the relation between the field on the cylinder and on the plane
is
φ(z) =
L
2π
h
z
−h φ cyl (w)
(6.164)
(quantities without indices are on the plane by definition). The mode expansion on
the cylinder is
φ cyl =
2π
L
h
n∈Z
φ n e
−
2π
L w
=
2π
L
h
n∈Z
φ n
z n .
(6.165)
Using the finite transformation (6.90) for the energy–momentum tensor T , one
finds the relation
T cyl (w) =
2π
L
2
T (z)z
2
−
c
24
.
(6.166)
For the L 0 mode, one finds
(L 0 ) cyl = L 0 −
c
24
,
(6.167)