The generalization of this formula for the gauge system is:f
74
GAUGE FIELDS AND STRINGS
=
^yl„(jr,T)exp|-| dTTr(i^ + F^Jdx^| (5.4)
A„(x,0) = An(x)
0
A „ (x,P ) = A n {x)
Here the dot means time derivative, indices refer to space directions,
is the magnetic part of the Yang-Mills field strength, and the sum
over eigenstates includes them all. Let us first compute the partition
function corresponding to the vacuum sector. In order to project it out
of (5.4), we use the following comment. According to (3.37),
T„(4“) = T„(T)
for the vacuum sector
and
T„(4«) = n
j
(A^ = n-'A„Q + Q-^d„il)
(5.5)
for the other sectors (where ® is the representation matrix for the SU(2)
group). These functions ^ have the property:
i
dfi &'(Q) = 0
for / # 0
(5.6)
(where the integral is taken over the SU(2) group measure). Using (5.5)
and (5.6) we conclude that:
Z o =
I
e
-fE„ _
n e v a cu u m
sector
^Q(x)
= ^Q(jc)
9A„{x, t)
/ln(x.^)=>C“ (x.O)
X exp^ —Tr
+ FL) dx dx
= @ fi(x)Z[ii(x)]
(5.7)
t We have fixed the gauge by /lo(x) = 0