16
GAUGE FIELDS AND STRINGS
The invariance properties of (1.49) are given by:
or
(Px- (P x
(1.50)
(with arbitrary {(Pjc})- The formal continuum limit of (1.49) would give a
familar expression:
1
= const. + -
Fafi —
(1.51)
Should we conclude from this that the continuum limit of our model is
just the free Maxwell field? To understand what goes on, let us compare
the situation with the one described by (1.32). In the latter case the
energy in the formal continuum limit is:
= const. + i(^^a)^
(1-52)
It describes a massless scalar field, which is nothing but the Goldstone
field associated with symmetry breaking. In this Abelian case the
perturbative interactions (described by the omitted terms ^(5^a)"^) are
irrelevant as we saw. But, in the chapter devoted to instantons, we shall
show that due to nonperturbative effects, associated with vortices, there
exists a phase transition after which the field a acquires a mass. So, we
conclude that for P > Pc
system is indeed described by the massless
free Goldstone field, that in the critical region \P — PJ < Pc
have
some complicated interacting continuum theory with both massless
and massive particles, and that for P < Pc the massless particles
disappear. All these effects are nonperturbative. Notice also an interesting phenomenon for ^ = 2: we do not strictly speaking have spontaneous symmetry breaking, and the 0-field from (1.35) has the property:
<0> = O
(1.53)
Nevertheless we do have massless Goldstone modes, described by a(x),
which disappear at the phase transition point. One of the possible ways
to understand this is to introduce the decomposition
It is possible to show that
^
0(x) = p(x)e'
r
^ const.
(1.54)
(1.55)
GAUGE FIELDS AND STRINGS
The invariance properties of (1.49) are given by:
or
(Px- (P x
(1.50)
(with arbitrary {(Pjc})- The formal continuum limit of (1.49) would give a
familar expression:
1
= const. + -
Fafi —
(1.51)
Should we conclude from this that the continuum limit of our model is
just the free Maxwell field? To understand what goes on, let us compare
the situation with the one described by (1.32). In the latter case the
energy in the formal continuum limit is:
= const. + i(^^a)^
(1-52)
It describes a massless scalar field, which is nothing but the Goldstone
field associated with symmetry breaking. In this Abelian case the
perturbative interactions (described by the omitted terms ^(5^a)"^) are
irrelevant as we saw. But, in the chapter devoted to instantons, we shall
show that due to nonperturbative effects, associated with vortices, there
exists a phase transition after which the field a acquires a mass. So, we
conclude that for P > Pc
system is indeed described by the massless
free Goldstone field, that in the critical region \P — PJ < Pc
have
some complicated interacting continuum theory with both massless
and massive particles, and that for P < Pc the massless particles
disappear. All these effects are nonperturbative. Notice also an interesting phenomenon for ^ = 2: we do not strictly speaking have spontaneous symmetry breaking, and the 0-field from (1.35) has the property:
<0> = O
(1.53)
Nevertheless we do have massless Goldstone modes, described by a(x),
which disappear at the phase transition point. One of the possible ways
to understand this is to introduce the decomposition
It is possible to show that
0(x) = p(x)e'
r
^ const.
(1.54)
(1.55)