QUANTUM STRINGS AND RANDOM SURFACES
197
We have proved that its correlation functions are invariant under the
transformation:
with
/dw\''2/dw\^2
---(xz
B
w(z) = -
(XÔ ~ yP = 1
(9.215)
yz + S'
We shall refer to such a situation by saying that the operator has
anomalous dimension
and that we are dealing with a conformal
quantum field theory.
How did it happen, that starting from the field x which has dimension
zero we obtained by exponentiating it a field with nonzero dimension?
The “leak” of dimensionality occurs through the ultraviolet cut-off
which we excluded by taking the normal product : :. Otherwise we
would get coefficients containing
(where s is the small distance
cut-off) and the correlation functions are formally dimensionless,
provided that we change the cut-off scale simultaneously. This scale
transformation is not interesting. The physical rescaling does not
change the cut-off. Generically, there could be no scale invariance at all
in this case. But in many important cases, like the one we have
discussed, it still exists although the transformation properties of the
fields are renormalized.
Let us turn now to the first factor in (9.208). By construction it has
invariance under the transformation (see (9.203))
z ) -
dw\^/ dw\^
----(9.216)
This conformal invariance, as we have already noted, is a remnant
of the general covariance in our gauge. The anomalous dimension
A in (9.207) is such that the vertex operator (9.208) has dimension
2(A -h p^/l) = 2. Therefore, the mass of the ground state of the string is
determined by the anomalous dimension of the field
the quantity
being completely defined in terms of the Liouville field theory. Also, the
integral (9.208) is invariant under SL(2, C) transformations, and hence,
in order to avoid integrating over the infinite volume of SL(2, C) one
has to factor this volume out of the integral. Practically, this consists of
simply shifting three points
^3 0, 1, 00 by the use of SL(2, C).
Different choices of SL(2, C)—“gauge fixing” would lead to apparently
197
We have proved that its correlation functions are invariant under the
transformation:
with
/dw\''2/dw\^2
---(xz
B
w(z) = -
(XÔ ~ yP = 1
(9.215)
yz + S'
We shall refer to such a situation by saying that the operator has
anomalous dimension
and that we are dealing with a conformal
quantum field theory.
How did it happen, that starting from the field x which has dimension
zero we obtained by exponentiating it a field with nonzero dimension?
The “leak” of dimensionality occurs through the ultraviolet cut-off
which we excluded by taking the normal product : :. Otherwise we
would get coefficients containing
(where s is the small distance
cut-off) and the correlation functions are formally dimensionless,
provided that we change the cut-off scale simultaneously. This scale
transformation is not interesting. The physical rescaling does not
change the cut-off. Generically, there could be no scale invariance at all
in this case. But in many important cases, like the one we have
discussed, it still exists although the transformation properties of the
fields are renormalized.
Let us turn now to the first factor in (9.208). By construction it has
invariance under the transformation (see (9.203))
z ) -
dw\^/ dw\^
----(9.216)
This conformal invariance, as we have already noted, is a remnant
of the general covariance in our gauge. The anomalous dimension
A in (9.207) is such that the vertex operator (9.208) has dimension
2(A -h p^/l) = 2. Therefore, the mass of the ground state of the string is
determined by the anomalous dimension of the field
the quantity
being completely defined in terms of the Liouville field theory. Also, the
integral (9.208) is invariant under SL(2, C) transformations, and hence,
in order to avoid integrating over the infinite volume of SL(2, C) one
has to factor this volume out of the integral. Practically, this consists of
simply shifting three points
^3 0, 1, 00 by the use of SL(2, C).
Different choices of SL(2, C)—“gauge fixing” would lead to apparently