CHAPTER 7
Analogies Between Gauge and
Chiral Fields. Loop Dynamics
In the previous chapters we indicated some similarities between chiral
fields and gauge fields. We saw that both theories—the one for ^ = 2
and the other for ^ = 4 share the property of asymptotic freedom.
Their strong coupling expansions also looked similar, provided that we
substitute point-like excitations of chiral fields by closed flux lines for
the gauge fields. Recall also the similarity of the instanton structures of
/i-fields and Yang-Mills fields.
In this chapter we shall explore the reasons for these analogies.
Roughly speaking, it appears that small pieces of the flux lines in the
gauge case move as if they were point-like excitations of a chiral field.
On the other hand, the gauge field itself can be considered as a chiral
field defined on the space of all possible loops.
These ideas are not well developed. But they provide us with a bridge
between gauge theories, chiral fields and the theory of strings (discussed
in Chapter 9). They may also help in a search for hidden symmetries in
many-dimensional systems. One cannot help feeling that many beautiful secrets are concealed in loop space.
This space is the main topic of discussion below.
7.1 Non-Abelian Phase Factor
In the previous chapter we have already been dealing with a field:
111
DOI: 10.1201/9780203755082-7
Analogies Between Gauge and
Chiral Fields. Loop Dynamics
In the previous chapters we indicated some similarities between chiral
fields and gauge fields. We saw that both theories—the one for ^ = 2
and the other for ^ = 4 share the property of asymptotic freedom.
Their strong coupling expansions also looked similar, provided that we
substitute point-like excitations of chiral fields by closed flux lines for
the gauge fields. Recall also the similarity of the instanton structures of
/i-fields and Yang-Mills fields.
In this chapter we shall explore the reasons for these analogies.
Roughly speaking, it appears that small pieces of the flux lines in the
gauge case move as if they were point-like excitations of a chiral field.
On the other hand, the gauge field itself can be considered as a chiral
field defined on the space of all possible loops.
These ideas are not well developed. But they provide us with a bridge
between gauge theories, chiral fields and the theory of strings (discussed
in Chapter 9). They may also help in a search for hidden symmetries in
many-dimensional systems. One cannot help feeling that many beautiful secrets are concealed in loop space.
This space is the main topic of discussion below.
7.1 Non-Abelian Phase Factor
In the previous chapter we have already been dealing with a field:
111
DOI: 10.1201/9780203755082-7