214
17. Spontaneously Broken Global Symmetry
of the magnetization M at temperature T , and make an expansion of the
form
F ≈ F 0 (T ) + μ
2 (T )M
2 +
λ M
4 + · · ·
(17.83)
4
valid for weak and slowly varying magnetization. If the parameter μ
2 is positive, it is clear that F has a simple ‘bowl’ shape as a function of |M |, with a
minimum at |M | = 0. This is the case for T greater than the ferromagnetic
transition temperature T C . However, if one assumes that μ
2 (T ) changes sign
at T C , becoming negative for T < T C , then F will now resemble a vertical
section of figure 17.2, the minimum being at |M | / 0. Any direction of M
=
is possible (only |M | is specified); but the system must choose one particular
direction (e.g. via the influence of a very weak external field, as discussed in
section 17.3.1), and when it does so the rotational invariance exhibited by F
of (17.83) is lost. This symmetry has been broken ‘spontaneously’ – though
this is still only a classical analogue. Nevertheless, the model is essentially
the Landau mean field theory of ferromagnetism, and suggests that we should
think of the ‘symmetric’ and ‘broken symmetry’ situations as different phases
of the same system. It may also be the case in particle physics, that parameters such as μ
2 change sign as a function of T , or some other variable, thereby
effectively precipitating a phase change.
If we maintain the idea that the vacuum expectation value of the quantum
field should equal the ground state value of the classical field, the vacuum in
this μ
2 < 0 case must therefore be |0› B such that B ‹0|φ ˆ |0› B does not vanish,
in contrast to (17.76). It is clear that this is exactly the situation met in the
superfluid (but ‘B’ here will stand for ‘broken symmetry’), and is moreover
the condition for the existence of massless (Goldstone) modes. Let us see how
they emerge in this model.
In quantum field theory, particles are thought of as excitations from a
ground state, which is the vacuum. Figure 17.2 strongly suggests that if we
want a sensible quantum interpretation of a theory with the potential (17.77),
we had better expand the fields about a point on the circle of minima, about
which stable oscillations are likely, rather than about the obviously unstable
point φ ˆ = 0. Let us pick the point ρ = v, θ = 0 in the classical case. We might
well guess that ‘radial’ oscillations in ˆ
ρ would correspond to a conventional
massive field (having a parabolic restoring potential), while ‘angle’ oscillations
in θ ˆ – which pass through all the degenerate vacuua – have no restoring force
and are massless. Accordingly, we set
φ ˆ (x) = √
1 (v + h ˆ (x)) exp(−iθ ˆ (x)/v)
(17.84)
2
and find (problem 17.5) that L ˆ G (with V ˆ = V ˆ SB of (17.77) with hats on)
becomes
ˆ
ˆ h − μ
2 h ˆ 2
ˆ
L G =
1 ∂ μ h∂
μ ˆ
+
1 ∂ μ θ∂
μ θ ˆ + μ
4 /λ
2
2
17. Spontaneously Broken Global Symmetry
of the magnetization M at temperature T , and make an expansion of the
form
F ≈ F 0 (T ) + μ
2 (T )M
2 +
λ M
4 + · · ·
(17.83)
4
valid for weak and slowly varying magnetization. If the parameter μ
2 is positive, it is clear that F has a simple ‘bowl’ shape as a function of |M |, with a
minimum at |M | = 0. This is the case for T greater than the ferromagnetic
transition temperature T C . However, if one assumes that μ
2 (T ) changes sign
at T C , becoming negative for T < T C , then F will now resemble a vertical
section of figure 17.2, the minimum being at |M | / 0. Any direction of M
=
is possible (only |M | is specified); but the system must choose one particular
direction (e.g. via the influence of a very weak external field, as discussed in
section 17.3.1), and when it does so the rotational invariance exhibited by F
of (17.83) is lost. This symmetry has been broken ‘spontaneously’ – though
this is still only a classical analogue. Nevertheless, the model is essentially
the Landau mean field theory of ferromagnetism, and suggests that we should
think of the ‘symmetric’ and ‘broken symmetry’ situations as different phases
of the same system. It may also be the case in particle physics, that parameters such as μ
2 change sign as a function of T , or some other variable, thereby
effectively precipitating a phase change.
If we maintain the idea that the vacuum expectation value of the quantum
field should equal the ground state value of the classical field, the vacuum in
this μ
2 < 0 case must therefore be |0› B such that B ‹0|φ ˆ |0› B does not vanish,
in contrast to (17.76). It is clear that this is exactly the situation met in the
superfluid (but ‘B’ here will stand for ‘broken symmetry’), and is moreover
the condition for the existence of massless (Goldstone) modes. Let us see how
they emerge in this model.
In quantum field theory, particles are thought of as excitations from a
ground state, which is the vacuum. Figure 17.2 strongly suggests that if we
want a sensible quantum interpretation of a theory with the potential (17.77),
we had better expand the fields about a point on the circle of minima, about
which stable oscillations are likely, rather than about the obviously unstable
point φ ˆ = 0. Let us pick the point ρ = v, θ = 0 in the classical case. We might
well guess that ‘radial’ oscillations in ˆ
ρ would correspond to a conventional
massive field (having a parabolic restoring potential), while ‘angle’ oscillations
in θ ˆ – which pass through all the degenerate vacuua – have no restoring force
and are massless. Accordingly, we set
φ ˆ (x) = √
1 (v + h ˆ (x)) exp(−iθ ˆ (x)/v)
(17.84)
2
and find (problem 17.5) that L ˆ G (with V ˆ = V ˆ SB of (17.77) with hats on)
becomes
ˆ
ˆ h − μ
2 h ˆ 2
ˆ
L G =
1 ∂ μ h∂
μ ˆ
+
1 ∂ μ θ∂
μ θ ˆ + μ
4 /λ
2
2