162
GAUGE FIELDS AND STRINGS
(where the exponential smallness of the correction follows from the
Poisson summation formula). We obtain:
-logdetRi - ,
1 '
di
Q O
"ItI,
exp(-7T^M^T/r^)
r dx ®
=
— Z exp(-7t^/i^x)
J ^ n= I
(£IT)2
J
n=l
e x p ( — TT^/I^x) -Idx
^ e x p ( — TT^M^x)
(e/T)2
(e/T)2
0
1
1
2
2 ^ { n x )
+
dx
—
^ e x p ( — Tc^M^x)
(9.43)
Now, the last two terms in the last formula are 7-independent
constants. Therefore:
,
d^\
T
T
- l o g d e t -
= — /------ lo g -
' d f7 ,,o « V ’'
«
(9.44)
If we substitute this expression into (9.39) we obtain:
| | = c o n st.e x p (-T /2 a V .)^ .r‘/^
= const • exp( — T/2e^n) dT
(9.45)
The divergences are condensed into the constant factor in front and the
term exp(—T/2e^n).
GAUGE FIELDS AND STRINGS
(where the exponential smallness of the correction follows from the
Poisson summation formula). We obtain:
-logdetRi - ,
1 '
di
Q O
"ItI,
exp(-7T^M^T/r^)
r dx ®
=
— Z exp(-7t^/i^x)
J ^ n= I
(£IT)2
J
n=l
e x p ( — TT^/I^x) -Idx
^ e x p ( — TT^M^x)
(e/T)2
(e/T)2
0
1
1
2
2 ^ { n x )
+
dx
—
^ e x p ( — Tc^M^x)
(9.43)
Now, the last two terms in the last formula are 7-independent
constants. Therefore:
,
d^\
T
T
- l o g d e t -
= — /------ lo g -
' d f7 ,,o « V ’'
«
(9.44)
If we substitute this expression into (9.39) we obtain:
| | = c o n st.e x p (-T /2 a V .)^ .r‘/^
= const • exp( — T/2e^n) dT
(9.45)
The divergences are condensed into the constant factor in front and the
term exp(—T/2e^n).