12
GAUGE FIELDS AND STRINGS
{Iq here is a Bessel function). Repeating the arguments of the previous
section we find that the theory (1.33) lies in the same universality class
as the theory described by the Lagrangian:
if = d^(j)* d^cj) + (/>*(/> -f ^Ào((t)*(l)Ÿ
(1.35)
Again, we have no phase transition for ^ = 1, and we have two
different phases for ^ > 2. The crucial difference from the Ising case is
the existence of gapless excitations for all P >
dictated by Goldstone’s theorem. The physical origin of these excitations is very simple.
Suppose that we have a broken symmetry, <(/>> ^ 0 (we shall see that
this is true for ^ > 3). Then the states with different orientation:
<0> = e‘“<(/)> must have the same energy. If we form a state with slowly
varying a(x), its energy will go to the vacuum value as the wave vector
tends to zero. Hence, there should be no gap in the spectrum. To see this
more formally let us introduce the conserved current
dx,
(1.36)
for which we have the Ward identity:
C 7X,
= i<5(x - y)<(/>(y)>
If we pass to the momentum representation
(1.37)
(1.38)
we conclude by taking q ^ O that ( —^)> must be singular in
this limit, having a singularity:
(-^)>^-^o = i{^)yqjq^ +
(1.39)
For ^ = 2 the situation is more tricky. It is quite clear in this case that
the propagator of the /i-field of the model (1.33) cannot have a
Goldstone pole. Indeed, since
1 = =
in {q )n {-q )')
(1.40)
such a pole in the right hand side would lead to an infrared contradiction. The answer is that the pole is softened and replaced by some
powerlife singularity. There is also no naive order parameter and no
true symmetry breaking in this case: = 0. Nevertheless a phase
transition at some P^ does take place and the observable properties of
the phases are quite different from each other. We shall discuss them in
GAUGE FIELDS AND STRINGS
{Iq here is a Bessel function). Repeating the arguments of the previous
section we find that the theory (1.33) lies in the same universality class
as the theory described by the Lagrangian:
if = d^(j)* d^cj) + (/>*(/> -f ^Ào((t)*(l)Ÿ
(1.35)
Again, we have no phase transition for ^ = 1, and we have two
different phases for ^ > 2. The crucial difference from the Ising case is
the existence of gapless excitations for all P >
dictated by Goldstone’s theorem. The physical origin of these excitations is very simple.
Suppose that we have a broken symmetry, <(/>> ^ 0 (we shall see that
this is true for ^ > 3). Then the states with different orientation:
<0> = e‘“<(/)> must have the same energy. If we form a state with slowly
varying a(x), its energy will go to the vacuum value as the wave vector
tends to zero. Hence, there should be no gap in the spectrum. To see this
more formally let us introduce the conserved current
dx,
(1.36)
for which we have the Ward identity:
C 7X,
= i<5(x - y)<(/>(y)>
If we pass to the momentum representation
(1.37)
(1.38)
we conclude by taking q ^ O that ( —^)> must be singular in
this limit, having a singularity:
(-^)>^-^o = i
(1.39)
For ^ = 2 the situation is more tricky. It is quite clear in this case that
the propagator of the /i-field of the model (1.33) cannot have a
Goldstone pole. Indeed, since
1 = =
in {q )n {-q )')
(1.40)
such a pole in the right hand side would lead to an infrared contradiction. The answer is that the pole is softened and replaced by some
powerlife singularity. There is also no naive order parameter and no
true symmetry breaking in this case: = 0. Nevertheless a phase
transition at some P^ does take place and the observable properties of
the phases are quite different from each other. We shall discuss them in