QUANTUM STRINGS AND RANDOM SURFACES
199
condition is equivalent to the operator product expansion of the
underlying two dimensional field theory, and that there is a correspondence between 2d-operators and resonances.
The operator product expansion is a fundamental property inherent
in any reasonable field theory. It states that two local operators in the
theory, placed close together, can be viewed as a set of local operators.
“Close together” here means that we are considering some multi-point
correlation function, including our pair of operators, as well as many
others, and the distance between the pair is smaller than all the other
distances. Roughly speaking, when viewed from a large distance, our
two points coalesce into one. Formally, if the field theory possesses a set
of local operators {0„((^)} the following relation holds:
(9.218)
which should be understood as a prescription to be inserted into any
correlation function, say:
(9.219)
We shall also assume that it is possible to find a complete set of
operators {0„}, so that any quantum state can be generated by acting
on the vacuum by their linear combination. This implies, that the
relation (9.218), when the whole set {0„} is inserted, becomes an exact
relation. Now, what can we say about the “structure functions”, Cl,^(0?
Suppose that we are dealing with conformal field theory, invariant
under SL(2, C) transformations. Then any operator
is characterized
by its anomalous dimension A„. This means that the correlation
functions are invariant under the transformations:
0„(Í)-A^"0„(AÍ)
(9.220)
In order that (9.218) be consistent with this, we have to require:
=
(9.221)
For scalar operators that means simply:
(9.222)
while in general some tensor structures appear. All this already looks
similar to the factorization property, but a few more steps are needed to
199
condition is equivalent to the operator product expansion of the
underlying two dimensional field theory, and that there is a correspondence between 2d-operators and resonances.
The operator product expansion is a fundamental property inherent
in any reasonable field theory. It states that two local operators in the
theory, placed close together, can be viewed as a set of local operators.
“Close together” here means that we are considering some multi-point
correlation function, including our pair of operators, as well as many
others, and the distance between the pair is smaller than all the other
distances. Roughly speaking, when viewed from a large distance, our
two points coalesce into one. Formally, if the field theory possesses a set
of local operators {0„((^)} the following relation holds:
(9.218)
which should be understood as a prescription to be inserted into any
correlation function, say:
(9.219)
We shall also assume that it is possible to find a complete set of
operators {0„}, so that any quantum state can be generated by acting
on the vacuum by their linear combination. This implies, that the
relation (9.218), when the whole set {0„} is inserted, becomes an exact
relation. Now, what can we say about the “structure functions”, Cl,^(0?
Suppose that we are dealing with conformal field theory, invariant
under SL(2, C) transformations. Then any operator
is characterized
by its anomalous dimension A„. This means that the correlation
functions are invariant under the transformations:
0„(Í)-A^"0„(AÍ)
(9.220)
In order that (9.218) be consistent with this, we have to require:
=
(9.221)
For scalar operators that means simply:
(9.222)
while in general some tensor structures appear. All this already looks
similar to the factorization property, but a few more steps are needed to