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GAUGE FIELDS AND STRINGS
coupling problem. By counting powers of N, we can easily show that
multiloop corrections to the (8.76) would lead to the replacement:
%(nnp)f
with some unknown / , which can be computed for small fip loop by
loop. The use of the approximation /(x ) ~ x for x ^ 1 is only qualitatively justified.
So, the final conclusion is the following. At all values of N we have
nontrivial topological fluctuations of the fields. They can be efficiently
described by the 1 /N approximation. There could be another, complementary, description in which they are represented as a collection of
melted instantons. Unfortunately rigorous quantitative methods for the
second description are not known at present. It seems that the one-loop
WKB approximation is qualitatively sensible. But before the development of quantitative methods, the possibility of describing topological
excitations in terms of instantons remains a semantic question.
8.4 Non-Abelian Gauge Theory
This is the case most interesting for us. Its l/N properties are to some
extent similar to those of principal chiral fields. It is convenient to
consider not SU(N) but the U(N) = SU(N)® U(l) case. Here again,
the 1/(1) part trivially decouples. The field is described by antiHermitian matrices:
(Fji =
- e,AF - A%Ai, + AUU
(8.77)
N
N
Ael
Tr
d*x
iFji(Fjjd*x
Here we have changed the normalization for the bare coupling constant
(in comparison with the previous chapters) so as to make the action
have its natural scale
One of the N-factors come from the trace; the
scale is natural since the Lagrangian (8.77) describes
interacting
gluons. Their vacuum fluctuations provide the energy or effective action
of the same scale as the classical action (8.77). Another check is