62
GAUGE FIELDS AND STRINGS
lattice the energy must be a periodic function of displacements. Therefore, we have to account for jumps in uj^x). If we have a branch point at
jC q, and a loop L surrounding this point then:
dpu^ix) dx^ =
(4.52)
For such uj^x) it is said that we have a dislocation located at Xq. We see
that the properties of dislocations almost coincide with those of
vortices. In particular, at some P we have a condensation of dislocations
which can be interpreted as melting.
4.3. Compact QED (0(2) Gauge Model)
In this section we shall examine the case of Abelian gauge theories. This
case is nontrivial, in spite of the fact that the naive continuum limit of
the action is given by:
(4.53)
and describes apparently free photons. The nontriviality, as in the
preceeding sections, comes from the fact that the vector potential has
certain angular properties which force us to account for the analogues
of vortices or dislocations in the functional integral.
Before we do this, let us explain why the vector potential is supposed
to be an angular variable.
A priori, we can define on a lattice two different models. The first one
is (1.49) with the action:
S = i I (1 -c o s f,.,,)
^^0 x,0 ip
(4.54)
- ^x.a + ^x + ot.p ^x + p,at
-n <
< n
The second option is:
1
5 = 7^2 Z f'L»; - 00 s
< + 00
^ ^ 0 x.«P
(4.55)
In the naive continuum limit both of these actions lead to (4.53), but the
physics of these models is different. An analogous situation arises in
GAUGE FIELDS AND STRINGS
lattice the energy must be a periodic function of displacements. Therefore, we have to account for jumps in uj^x). If we have a branch point at
jC q, and a loop L surrounding this point then:
dpu^ix) dx^ =
(4.52)
For such uj^x) it is said that we have a dislocation located at Xq. We see
that the properties of dislocations almost coincide with those of
vortices. In particular, at some P we have a condensation of dislocations
which can be interpreted as melting.
4.3. Compact QED (0(2) Gauge Model)
In this section we shall examine the case of Abelian gauge theories. This
case is nontrivial, in spite of the fact that the naive continuum limit of
the action is given by:
(4.53)
and describes apparently free photons. The nontriviality, as in the
preceeding sections, comes from the fact that the vector potential has
certain angular properties which force us to account for the analogues
of vortices or dislocations in the functional integral.
Before we do this, let us explain why the vector potential is supposed
to be an angular variable.
A priori, we can define on a lattice two different models. The first one
is (1.49) with the action:
S = i I (1 -c o s f,.,,)
^^0 x,0 ip
(4.54)
- ^x.a + ^x + ot.p ^x + p,at
-n <
< n
The second option is:
1
5 = 7^2 Z f'L»; - 00 s
< + 00
^ ^ 0 x.«P
(4.55)
In the naive continuum limit both of these actions lead to (4.53), but the
physics of these models is different. An analogous situation arises in