INSTANTONS IN ABELIAN SYSTEMS
59
go around such vortex line. The vortex lines carry energy and are
directly observable in "^He.
To conclude this section, we describe the continuum description of
0(2) systems. It is given by the Lagrangian:
(4.40)
where, as we explained in Chapter 1, i;(|(/>|^) can be taken without loss
of generality to be:
(4.41)
The relation of this theory to the one described by (4.21) is the same as
that of the quantum mechanics of a double well to the ^ = 1 Ising
model. Their long range properties are identical. This is most easily seen
in the present case if we introduce variables:
(/>(x) = M(x)e‘^<*>
s =
+ v{u^) +
dx.
(4.42)
For small values of Aq we have
(4.43)
and we see that the fluctuations of the modulus u(jc) are small and short
ranged (they have a mass fio). At the same time, the field 0(x) is massless
and the only one contributing at very large distances. It is worthwhile
demonstrating how the vortex contribution arises directly in (4.40). As
before, the vortex arises as a nontrivial classical minimum for the action
(4.40). In the two-dimensional case the equations have the form:
+
= o
(4.44)
or
(where 0 i_2 are defined hy < f> = {4> i +
This equation has 0(2) ® 0(2) invariance, one of the 0(2) being
rotation of x-space and the other rotation of the < f). Let us look for a
solution which breaks this 0(2) (g) 0(2) but preserves the single 0(2)
59
go around such vortex line. The vortex lines carry energy and are
directly observable in "^He.
To conclude this section, we describe the continuum description of
0(2) systems. It is given by the Lagrangian:
(4.40)
where, as we explained in Chapter 1, i;(|(/>|^) can be taken without loss
of generality to be:
(4.41)
The relation of this theory to the one described by (4.21) is the same as
that of the quantum mechanics of a double well to the ^ = 1 Ising
model. Their long range properties are identical. This is most easily seen
in the present case if we introduce variables:
(/>(x) = M(x)e‘^<*>
s =
+ v{u^) +
dx.
(4.42)
For small values of Aq we have
(4.43)
and we see that the fluctuations of the modulus u(jc) are small and short
ranged (they have a mass fio). At the same time, the field 0(x) is massless
and the only one contributing at very large distances. It is worthwhile
demonstrating how the vortex contribution arises directly in (4.40). As
before, the vortex arises as a nontrivial classical minimum for the action
(4.40). In the two-dimensional case the equations have the form:
+
= o
(4.44)
or
(where 0 i_2 are defined hy < f> = {4> i +
This equation has 0(2) ® 0(2) invariance, one of the 0(2) being
rotation of x-space and the other rotation of the < f). Let us look for a
solution which breaks this 0(2) (g) 0(2) but preserves the single 0(2)