INSTANTONS IN ABELIAN SYSTEMS
69
Taking the surface integral over the surface of the cube with direction y
we obtain the analogue of eq. (4.62)
-d^(j)y +
= Q y
(4.85)
Here we have used a continuum notation in the hope that after previous
discussions it will not be misleading. Formulas with explicit lattice
notations would look rather clumsy for ^ = 4.
From eq. (4.85) we see that Q y are subjected to the constraint:
= (‘¡X. Ï - 9 . - Y, y) = 0
(4 -8 6 )
The instanton part of the partition function has the form:
7^ = 4 _
V
^IN ST —
A
{qxyl àyQy =
exp - ¿ z
(4.87)
Let us explain the meaning of these results, which could have been
anticipated without the derivation given. The instanton for ^ = 3 was
described by a magnetic monopole solution (4.65). Suppose, we add one
more dimension (“time”). Then this solution, being point-like for ^ = 3
is represented by a line for ^ = 4 (the world-line of the point-like
object). We can take the line to be arbitrarily curved. Then we shall
have for each shape a different classical solution, minimizing locally the
action (4.54). Since magnetic flux is conserved these lines have to be
closed or infinite. The contribution to the partition function can be
expressed as a sum over all possible magnetic flux lines, which have
Coulomb attraction between them. This picture is quantitatively reflected in formulas (4.83), (4.86), (4.87).
It is easy to deduce that for large ^ those flux lines have negligible
influence on the system in the infrared limit. The reason is that lines of
length L have action
and their contribution to (4.87) is given by:
^(l) ^
(4 gg)
(Here (c')^ is the number of loops of the length L). We conclude that for
large enough C q ^ magnetic loops are small and we have no disordering
in the system, which is effectively described in this case by means of
massless free photons. At the same time there is a critical coupling el^ at
which condensation of magnetic lines begins. It is to be expected that
for el > el^ we obtain a strong coupling phase with confinement of
charge, while for el < el, we obtain the photon phase.
The picture described above can be understood in yet another way.
Magnetic monopoles which were instantons for ^ = 3 become particles
69
Taking the surface integral over the surface of the cube with direction y
we obtain the analogue of eq. (4.62)
-d^(j)y +
= Q y
(4.85)
Here we have used a continuum notation in the hope that after previous
discussions it will not be misleading. Formulas with explicit lattice
notations would look rather clumsy for ^ = 4.
From eq. (4.85) we see that Q y are subjected to the constraint:
= (‘¡X. Ï - 9 . - Y, y) = 0
(4 -8 6 )
The instanton part of the partition function has the form:
7^ = 4 _
V
^IN ST —
A
{qxyl àyQy =
exp - ¿ z
(4.87)
Let us explain the meaning of these results, which could have been
anticipated without the derivation given. The instanton for ^ = 3 was
described by a magnetic monopole solution (4.65). Suppose, we add one
more dimension (“time”). Then this solution, being point-like for ^ = 3
is represented by a line for ^ = 4 (the world-line of the point-like
object). We can take the line to be arbitrarily curved. Then we shall
have for each shape a different classical solution, minimizing locally the
action (4.54). Since magnetic flux is conserved these lines have to be
closed or infinite. The contribution to the partition function can be
expressed as a sum over all possible magnetic flux lines, which have
Coulomb attraction between them. This picture is quantitatively reflected in formulas (4.83), (4.86), (4.87).
It is easy to deduce that for large ^ those flux lines have negligible
influence on the system in the infrared limit. The reason is that lines of
length L have action
and their contribution to (4.87) is given by:
^(l) ^
(4 gg)
(Here (c')^ is the number of loops of the length L). We conclude that for
large enough C q ^ magnetic loops are small and we have no disordering
in the system, which is effectively described in this case by means of
massless free photons. At the same time there is a critical coupling el^ at
which condensation of magnetic lines begins. It is to be expected that
for el > el^ we obtain a strong coupling phase with confinement of
charge, while for el < el, we obtain the photon phase.
The picture described above can be understood in yet another way.
Magnetic monopoles which were instantons for ^ = 3 become particles