QUANTUM STRINGS AND RANDOM SURFACES
193
The quantity
is singular and an explanation of its meaning will
be given shortly below. Before doing this let us rewrite (9.194) in the
form:
j
i
“ fpNy iu ' y ^n)
(9.195)
where F is given by a Liouville functional integral. We have to expect
that F has poles in pj corresponding to the physical spectrum of the
string. The on-shell scattering amplitude will be given by the residue at
these poles. Denoting
0 (i 1 , . , ^n) = reSp2 =
F(p I
i i , ..., i^)
(9.196)
(where
is the position of the presumed pole and the space-time
signature is chosen such that
= pf — pi = —m^) we obtain an
expression for the scattering amplitude ^(Pi, ...,/>iv)^(P,,...,P,)= i n
. ..,iN)
(9.197)
J7=l
i
(f) depends on fewer variables than A. But what is the meaning of (^? To
be sure it is some correlation function of the Liouville field theory, and
so in principle we have reduced the problem of finding the ^ -
dimensional scattering amplitude to two-dimensional theory. In order
to find the kind of the correlation functions we need let us go back to
(9.195) and try to decipher
in il- H is clear that we have to use a
cut-oif in this expression which does not destroy general covariance. A
suitable choice is
(9.198)
where il/„iO and are eigenfunctions and eigenvalues of the laplacian.
This choice is the same one which we used in defining determinants
(which were regularized by the Schwinger proper time cut-off) and is
equivalent to the use of Pauli-Villars regulators. There is a simple naive
way to compute (9.198). It consists of the observation that:
^ m ) = -iog(AftLn
(9.199)
where (A(^)^jn is the cut-off in the (^-space. However, this cut-off is
^-dependent, since we have to fix the minimal invariant interval (A5)^i„.
193
The quantity
is singular and an explanation of its meaning will
be given shortly below. Before doing this let us rewrite (9.194) in the
form:
j
i
“ fpNy iu ' y ^n)
(9.195)
where F is given by a Liouville functional integral. We have to expect
that F has poles in pj corresponding to the physical spectrum of the
string. The on-shell scattering amplitude will be given by the residue at
these poles. Denoting
0 (i 1 , . , ^n) = reSp2 =
F(p I
i i , ..., i^)
(9.196)
(where
is the position of the presumed pole and the space-time
signature is chosen such that
= pf — pi = —m^) we obtain an
expression for the scattering amplitude ^(Pi, ...,/>iv)^(P,,...,P,)= i n
. ..,iN)
(9.197)
J7=l
i
(f) depends on fewer variables than A. But what is the meaning of (^? To
be sure it is some correlation function of the Liouville field theory, and
so in principle we have reduced the problem of finding the ^ -
dimensional scattering amplitude to two-dimensional theory. In order
to find the kind of the correlation functions we need let us go back to
(9.195) and try to decipher
in il- H is clear that we have to use a
cut-oif in this expression which does not destroy general covariance. A
suitable choice is
(9.198)
where il/„iO and are eigenfunctions and eigenvalues of the laplacian.
This choice is the same one which we used in defining determinants
(which were regularized by the Schwinger proper time cut-off) and is
equivalent to the use of Pauli-Villars regulators. There is a simple naive
way to compute (9.198). It consists of the observation that:
^ m ) = -iog(AftLn
(9.199)
where (A(^)^jn is the cut-off in the (^-space. However, this cut-off is
^-dependent, since we have to fix the minimal invariant interval (A5)^i„.