176
GAUGE FIELDS AND STRINGS
More precisely, this means that they have a correlation length proportional to the inverse cut-off. Therefore their leading correction to the
effective action must be local, and on the basis of general covariance
must have the form (9.100). This is again the demonstration of our
general rule that fields with short range correlations can be replaced by
their mean values. In our case covariance dictates that these mean
values are given by (9.100).
So, by the /i-integration of (9.98) we have recovered the action (9.74),
provided that
= d^x • df,x. This proves the equivalence of (9.97) to
(9.76). Again, as in our first derivation of this equivalence, we have
assumed the generic situation, i.e. that no divergent constants are zero.
Whether this continuum limit is what we are interested in must be
investigated separately in each particular case. In the next section we
shall show how to compute (9.97) in the case n — 2 and what kind of
physics is described by it.
9.5 Two-Dimensional Surfaces. Geometrical Introduction
For n = 2 we can proceed much further with (9.97), by explicit
computation of the functional integrals.
To do this we shall first present the necessary geometrical properties
of the functionals involved in the game.
Let us consider first:
W =
d,x‘d,xd^i
(9.101)
where the function jc(0 satisfies the boundary condition:
JC(«5)) = c(s).
(9.102)
Variation of W with respect to h^b gives as we have seen in the
preceeding section:
SW =
(9.103)
where Tat can be considered as an energy-momentum tensor for the x
field:
Tat =
■ 8tX -
daX ■ daX
(9.104)
If we take hat
to minimise W, that is apply the condition
Tat = 0
(9.105)
GAUGE FIELDS AND STRINGS
More precisely, this means that they have a correlation length proportional to the inverse cut-off. Therefore their leading correction to the
effective action must be local, and on the basis of general covariance
must have the form (9.100). This is again the demonstration of our
general rule that fields with short range correlations can be replaced by
their mean values. In our case covariance dictates that these mean
values are given by (9.100).
So, by the /i-integration of (9.98) we have recovered the action (9.74),
provided that
= d^x • df,x. This proves the equivalence of (9.97) to
(9.76). Again, as in our first derivation of this equivalence, we have
assumed the generic situation, i.e. that no divergent constants are zero.
Whether this continuum limit is what we are interested in must be
investigated separately in each particular case. In the next section we
shall show how to compute (9.97) in the case n — 2 and what kind of
physics is described by it.
9.5 Two-Dimensional Surfaces. Geometrical Introduction
For n = 2 we can proceed much further with (9.97), by explicit
computation of the functional integrals.
To do this we shall first present the necessary geometrical properties
of the functionals involved in the game.
Let us consider first:
W =
d,x‘d,xd^i
(9.101)
where the function jc(0 satisfies the boundary condition:
JC(«5)) = c(s).
(9.102)
Variation of W with respect to h^b gives as we have seen in the
preceeding section:
SW =
(9.103)
where Tat can be considered as an energy-momentum tensor for the x
field:
Tat =
■ 8tX -
daX ■ daX
(9.104)
If we take hat
to minimise W, that is apply the condition
Tat = 0
(9.105)