116
GAUGE FIELDS AND STRINGS
From this picture one sees that ^ is nothing but the field strength,
transported along C to the beginning of the loop, x. It is now easy to
check that equations (7.17) imply that
X
- n J ) - h.c. = 0
(7.21)
Before discussing the meaning of these relations, let us present their
continuous version. Its form can be extracted from the preceding
formulas. Namely, in this case the field T(C) = 'F[x(s)] is a functional
depending on the shape of the contour but not on its parametrization.
Therefore:
T[x(5)] = n m s m
(7.22)
and
dx,(5)
ds SxAs)
= 0
(7.23)
Relation (7.23) follows from (7.22) if one takes a(s) = s + e(s) with
infinitesimal r(5). The variational derivative of any functional / is
defined through the relation:
SxJs)
(7.24)
Let us compute
using the definition (7.1). For that we shall
use the following general relations:
d
d i’’ " ' ’
M dt = ( P exp MdxjMit)
S P exp M(T)dT= J di|^Pexp J M(T,)dTi )¿AÍ(í)
0
0
2n
X ^P exp I M(X2)dX2
2n
2n
dt pf SM(t) exp I M(x)dx^
3^1 P exp
2n
2n
M(t) d i ) = JJ di, dí2 P^^M(U) <5M(í2) exp
0
0
¿n
I M(T)dtj
for an arbitrary matrix M(t).