ANALOGIES BETWEEN GAUGE AND CHIRAL FIELDS
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String (which, according to Chapter 3 are elementary excitations in the
gauge case) is precisely the same as scattering of point-like excitations
of a chiral field. This could be checked in the frame of the strong
coupling expansion, but never has been. Another problem is to derive
asymptotic freedom directly in the loop space, following the pattern we
had for chiral fields in Chapter 2.
Another set of questions is the following. We derived the loop
representation starting from local gauge fields. However, another view
is possible (and seems even more natural to me). Let us suppose that the
primary quantities are the fields on the loop space. Then we can
consider gauge fields as something like Goldstone fields on the loop
space, efficiently described by the chiral equations of motion (7.29). This
point of view leads to many new options. For example we can consider,
instead of 'F fields which belong to a gauge group G, fields belonging to
different coset spaces G/H. These seem to be interesting objects, though
I do not know either their local representation or their role (if any) in
Nature.
7.2 Quantum Theory of Loops
It appears more practical to use a set of equations slightly different from
(7.29). It exploits the fact that, according to the transformation law
(7.2), only (p(C) = Tr 'F(C) is a gauge-invariant quantity. Therefore the
only quantities which make sense in quantum theory are:
M^(C)=<(P(C)>; W2(C,C) = WiCMC)y
etc. Using formulas (7.24), (7.25) we derive:
Sx/s)Sx/s') ^
j
dx'‘j^x,(s)x,(s')
+ ¿(s - s')x,(s){^Tr P^V^f^,(x(s)) exp^
dx"
Let us now introduce a local derivative;
(7.33)
(7.34)
def,.
= lim d f 5x„(s + tl2)6xJs - t/2)
(7.35)