134
6 Conformal Field Theory on the Plane
Given a basis of states {φ i } (i can run over both discrete and continuous indices),
the conjugate or dual states {φ c
i } are defined by
φ
c
i |φ j = δ ij
(6.145)
(the delta function is discrete and/or continuous according to the indices).
Verma Modules
If φ(z) is a weight h primary, then the associated state |φ satisfies:
L 0 |φ = h|φ,
∀n ≥ 1 : L n |φ = 0.
(6.146)
Such a state is also called a highest-weight state. The descendant states are defined
by all possible states of the form
|φ {n i } :=
i
L −n i |φ,
(6.147)
where the same L −n i can appear multiple times and n i > 0. The set of states φ {n i }
is called a Verma module V (h, c). One finds that the L 0 eigenvalues of this state is
L 0 = h +
i
n i .
(6.148)
Normal Ordering
The normal ordering of an operator with respect to a vacuum corresponds to placing
all creation (resp. annihilation) operators of this vacuum on the left (resp. right).
From this definition, the expectation value of a normal ordered operator in the
vacuum vanishes identically. The main reason for normal ordering is to remove
singularities in expectation values.
Given an operator φ(z), we define two normal orderings:
• The conformal normal order (CNO) : O : is defined with respect to the conformal
vacuum (6.121):
0|: O :|0 = 0.
(6.149)
• The energy normal order (ENO)
O
is defined with respect to the energy vacuum
(6.129):
|
O
| = 0.
(6.150)
We first discuss the conformal normal ordering before explaining how to relate it to
the energy normal ordering.
6 Conformal Field Theory on the Plane
Given a basis of states {φ i } (i can run over both discrete and continuous indices),
the conjugate or dual states {φ c
i } are defined by
φ
c
i |φ j = δ ij
(6.145)
(the delta function is discrete and/or continuous according to the indices).
Verma Modules
If φ(z) is a weight h primary, then the associated state |φ satisfies:
L 0 |φ = h|φ,
∀n ≥ 1 : L n |φ = 0.
(6.146)
Such a state is also called a highest-weight state. The descendant states are defined
by all possible states of the form
|φ {n i } :=
i
L −n i |φ,
(6.147)
where the same L −n i can appear multiple times and n i > 0. The set of states φ {n i }
is called a Verma module V (h, c). One finds that the L 0 eigenvalues of this state is
L 0 = h +
i
n i .
(6.148)
Normal Ordering
The normal ordering of an operator with respect to a vacuum corresponds to placing
all creation (resp. annihilation) operators of this vacuum on the left (resp. right).
From this definition, the expectation value of a normal ordered operator in the
vacuum vanishes identically. The main reason for normal ordering is to remove
singularities in expectation values.
Given an operator φ(z), we define two normal orderings:
• The conformal normal order (CNO) : O : is defined with respect to the conformal
vacuum (6.121):
0|: O :|0 = 0.
(6.149)
• The energy normal order (ENO)
O
is defined with respect to the energy vacuum
(6.129):
|
O
| = 0.
(6.150)
We first discuss the conformal normal ordering before explaining how to relate it to
the energy normal ordering.