TOPOLOGY OF GAUGE FIELDS AND RELATED PROBLEMS
91
instanton-anti-instanton configuration can be unambiguously defined
only for well-separated objects, because otherwise it is difficult to
distinguish it from other fluctuations with zero topological charge. This
ambiguity is closely connected with another source of errors, coming
from the fact that the quantum fluctuations above were treated in one
loop approximation. For each separate <^-instanton contribution this
would be disastrous because these terms are infrared divergent and the
effective charge becomes unboundedly large. [However, we see that
after summation over q these divergences are cut off at the Debye length
At this length ^ 1 and so are higher quantum corrections. That
gives hope that at least qualitatively the system is properly described by
the above approach. Moreover, it often happens in integrable systems
that the one loop approximation turns out to be exact. So an optimistic
view of the situation is the following. One has to introduce instantons,
described by a massive Dirac field
and anti-instantons described by
if/2 - Next one must find a certain extrapolation of the instanton-antiinstanton interaction to small distance. Then it might be hoped that
there exists such an extrapolation that the resulting system describes
the /i-field exactly and not only in one loop approximation. Whether
this is true can in principle be checked by the use of exact solutions, but
up to now this has not yet been done.
So at present we do not know whether exact properties of the /i-field
can be formulated in terms of instantons, but this possibility seems to be
open.
The same, and even more difficult, problems exist in nonabelian
gauge theories which we describe in the next section. Before coming to
that let us mention briefly what kind of instanton structure is present in
other versions of chiral models. First of all, /i-fields with the group
0(iV), N > 4 do not have any nontrivial topology; that is to say any
map 5^ ^
^ for A > 4 is contractible. The reason for this is easy to
understand if we consider the case of mapping
that is a circle
mapped on to an ordinary sphere, say to its equator. It is obvious, that
by moving this circle to the north pole we can contract it to a point. For
similar reasons any map
can be deformed to the trivial one.
A slightly more complicated argument shows that the map of onto
any Non-Abelian Lie group G, described by the principal chiral field
g(x) is also contractible. These theories do not have stable instantons.
The chiral theories which do have them are described by the coset
spaces G/H in which H contains G(l) as a factor. Let us explain how
this comes about. In the above theories the fields can be represented by
91
instanton-anti-instanton configuration can be unambiguously defined
only for well-separated objects, because otherwise it is difficult to
distinguish it from other fluctuations with zero topological charge. This
ambiguity is closely connected with another source of errors, coming
from the fact that the quantum fluctuations above were treated in one
loop approximation. For each separate <^-instanton contribution this
would be disastrous because these terms are infrared divergent and the
effective charge becomes unboundedly large. [However, we see that
after summation over q these divergences are cut off at the Debye length
At this length ^ 1 and so are higher quantum corrections. That
gives hope that at least qualitatively the system is properly described by
the above approach. Moreover, it often happens in integrable systems
that the one loop approximation turns out to be exact. So an optimistic
view of the situation is the following. One has to introduce instantons,
described by a massive Dirac field
and anti-instantons described by
if/2 - Next one must find a certain extrapolation of the instanton-antiinstanton interaction to small distance. Then it might be hoped that
there exists such an extrapolation that the resulting system describes
the /i-field exactly and not only in one loop approximation. Whether
this is true can in principle be checked by the use of exact solutions, but
up to now this has not yet been done.
So at present we do not know whether exact properties of the /i-field
can be formulated in terms of instantons, but this possibility seems to be
open.
The same, and even more difficult, problems exist in nonabelian
gauge theories which we describe in the next section. Before coming to
that let us mention briefly what kind of instanton structure is present in
other versions of chiral models. First of all, /i-fields with the group
0(iV), N > 4 do not have any nontrivial topology; that is to say any
map 5^ ^
^ for A > 4 is contractible. The reason for this is easy to
understand if we consider the case of mapping
that is a circle
mapped on to an ordinary sphere, say to its equator. It is obvious, that
by moving this circle to the north pole we can contract it to a point. For
similar reasons any map
can be deformed to the trivial one.
A slightly more complicated argument shows that the map of onto
any Non-Abelian Lie group G, described by the principal chiral field
g(x) is also contractible. These theories do not have stable instantons.
The chiral theories which do have them are described by the coset
spaces G/H in which H contains G(l) as a factor. Let us explain how
this comes about. In the above theories the fields can be represented by