QUANTUM STRINGS AND RANDOM SURFACES
157
(9.7) will become just the integration over T. This is indeed the case, but
it is worthwhile for future generalizations to work out this fact in detail.
9.2 Measures in the Space of Metrics and Diifeomorphisms
In order to define an invariant measure for the h integration, we shall
start from the definition of the metric in the space of metrics h, and then
use the “metric tensor” in this functional space for finding the volume
element. The only local expression for the “distance” \\Sh\\ between the
metrics /i(t) and h(z) +
has the form:
\\Shf = dT(Sh(r)fh-^^\x)
(9.23)
It is quite obvious that (9.23) is invariant under reparametrizations.
Let us notice now that any metric h(T) can be made r-independent by
means of a properly chosen gauge transformation / (t). Due to (9.9) this
implies that any h can be represented as:
(9.24)
with some T and / . Our strategy will be to pass from the integration
over S>h(T) to integration over the new variables T and / . According to
general rules:
Q >h{x) = dT^f(x) X (Jacobian)
(9.25)
If we manage to find the Jacobian, our problem will be solved, since
instead of ^h{x)/^f{x) we shall obtain a properly defined integral over
dT. The easiest way for solving this problem is to substitute the
decomposition (9.24) into (9.23) so as to find the “distance” in terms of
the new coordinates.
The computation is greatly simplified if we use a general geometrical
formula, which we derive now in its n-dimensional form (needed for
future applications). Suppose, that we have two metric tensors Qabi^)
and h^biO^ where are coordinates of some n-dimensional Riemannian
manifold. Let these two metrics be connected by the coordinate
transformation (or diffeomorphism) ^ /(^):
g =
(9.26a)
157
(9.7) will become just the integration over T. This is indeed the case, but
it is worthwhile for future generalizations to work out this fact in detail.
9.2 Measures in the Space of Metrics and Diifeomorphisms
In order to define an invariant measure for the h integration, we shall
start from the definition of the metric in the space of metrics h, and then
use the “metric tensor” in this functional space for finding the volume
element. The only local expression for the “distance” \\Sh\\ between the
metrics /i(t) and h(z) +
has the form:
\\Shf = dT(Sh(r)fh-^^\x)
(9.23)
It is quite obvious that (9.23) is invariant under reparametrizations.
Let us notice now that any metric h(T) can be made r-independent by
means of a properly chosen gauge transformation / (t). Due to (9.9) this
implies that any h can be represented as:
(9.24)
with some T and / . Our strategy will be to pass from the integration
over S>h(T) to integration over the new variables T and / . According to
general rules:
Q >h{x) = dT^f(x) X (Jacobian)
(9.25)
If we manage to find the Jacobian, our problem will be solved, since
instead of ^h{x)/^f{x) we shall obtain a properly defined integral over
dT. The easiest way for solving this problem is to substitute the
decomposition (9.24) into (9.23) so as to find the “distance” in terms of
the new coordinates.
The computation is greatly simplified if we use a general geometrical
formula, which we derive now in its n-dimensional form (needed for
future applications). Suppose, that we have two metric tensors Qabi^)
and h^biO^ where are coordinates of some n-dimensional Riemannian
manifold. Let these two metrics be connected by the coordinate
transformation (or diffeomorphism) ^ /(^):
g =
(9.26a)