INSTANTONS IN ABELIAN SYSTEMS
67
where
P(x) = X
“ ^a)
ZM = I ^xexp{ - ( ^
[(V(z - rj)Ÿ -
cos x\
(4.74)
The simplest correlation functions for our problem are those of the
operator
=
(4.75)
At large distances this is just the electromagnetic field strength.
In the quasiclassical approximation, HJ^x) is connected with the
charge density as follows:
Hix) ■ ■
- y).
|at - y l '
piy)
2nik„
H,(x) = -j^p(k)
Using formula (4.74) we get:
/2n\V
{ — ]
Now the correlation function of the //-field is given by
(4.76)
(4.77)
= iH ^(k)H X-k)y°^ + (27t)^ '^ ^ p ^ k ) p ( - k ) y (4.78)
The first term in (4.78) is the bare (that is without monopoles) Green
function of the if-field. It has the form
«» = 6 ^ ( 5 , , - ^
(4.79)
Its singularity at fc = 0 reflects the existence of a massless photon in this
approximation. Using (4.77) and the previous comment about the small
coupling of the /-field we get
.
y2,j
+ e
e
M^k^
= 1 ^ 1
(4.80)
67
where
P(x) = X
“ ^a)
ZM = I ^xexp{ - ( ^
[(V(z - rj)Ÿ -
cos x\
(4.74)
The simplest correlation functions for our problem are those of the
operator
=
(4.75)
At large distances this is just the electromagnetic field strength.
In the quasiclassical approximation, HJ^x) is connected with the
charge density as follows:
Hix) ■ ■
- y).
|at - y l '
piy)
2nik„
H,(x) = -j^p(k)
Using formula (4.74) we get:
/2n\V
{ — ]
Now the correlation function of the //-field is given by
(4.76)
(4.77)
= iH ^(k)H X-k)y°^ + (27t)^ '^ ^ p ^ k ) p ( - k ) y (4.78)
The first term in (4.78) is the bare (that is without monopoles) Green
function of the if-field. It has the form
«» = 6 ^ ( 5 , , - ^
(4.79)
Its singularity at fc = 0 reflects the existence of a massless photon in this
approximation. Using (4.77) and the previous comment about the small
coupling of the /-field we get
y2,j
+ e
e
M^k^
= 1 ^ 1
(4.80)