25.2 A Critical Look at the Hypotheses of Goldstone Theorem
175
Such an integrability condition is satisfied if j 0 and A satisfy the relativistic locality
condition, since the smearing with g(t) ∈ S can at worse change the compact support
in x of J (x, t) to a fast decrease.
152
More generally, the condition is satisfied by systems with short range dynamics,
namely if ∀A, B ∈ A L , as distributions in t,
lim
|x|→∞
|x|
s+ε
< [A x , α t (B)] > 0 = 0,
(25.15)
where s = space dimensions, ε > 0. This is the case of spin systems with short range
interactions (see Sect. 17.3).
It is worthwhile to note that the charge integrability condition, (25.14), is much
weaker than (25.15), since it involves a special operator j 0 ; as we shall see below,
(25.15) fails in models with long range interactions, whereas there are indications
that the charge integrability condition holds.
In conclusion, the charge integrability should be taken as part of the definition
that β
λ is locally generated by a charge density; the physical meaning is that β
λ can
be reasonably well approximated by
∗ -automorphisms β
λ
R with good localization
properties. As we shall see below, such a condition leads to the existence of quasiparticles with infinite lifetime in the limit of zero momentum (Goldstone quasiparticles).
ii) Local generation by a charge and time evolution
The really delicate issue (not sufficiently emphasized in the literature), which crucially enters into the proof of the theorem, is the condition that the local generation
by a charge, (25.6), holds on an algebra stable under time evolution. An equivalent
condition is that Q R and Q R (t) = α t (Q R ) generate the same automorphism.
Equation (25.6) can be easily checked on the time zero algebra generated by the
canonical variables at time t = 0, since this is a purely kinematical question which
involves the CCR or the ACR. The problem is whether (25.6) remains true when A is
replaced by α t (A). This is not trivial to check, because the infinite volume dynamics
α t is not explicitly known and the limit R → ∞ involved in (25.6) may not commute
with the infinite volume limit of α
V
t .
The heuristic argument that “since the Hamiltonian commutes with β
λ the charge
which generates β
λ is independent of time, i.e. Q(t) = Q(0),” and therefore
lim
R→∞
[Q R (t), A] = lim
R→∞
[Q R (0), A]
is not correct, because it overlooks the following important points. A global charge as
algebraic generator of β
λ does not exist if the symmetry is broken (we have already
remarked that the formal integral of j 0 does not define an operator), and one can only
speak of a local generation of β
λ in terms of local charges, so that a limit R → ∞ is
152 This can be seen, e.g. by using the Jost–Lehmann–Dyson representation.
175
Such an integrability condition is satisfied if j 0 and A satisfy the relativistic locality
condition, since the smearing with g(t) ∈ S can at worse change the compact support
in x of J (x, t) to a fast decrease.
152
More generally, the condition is satisfied by systems with short range dynamics,
namely if ∀A, B ∈ A L , as distributions in t,
lim
|x|→∞
|x|
s+ε
< [A x , α t (B)] > 0 = 0,
(25.15)
where s = space dimensions, ε > 0. This is the case of spin systems with short range
interactions (see Sect. 17.3).
It is worthwhile to note that the charge integrability condition, (25.14), is much
weaker than (25.15), since it involves a special operator j 0 ; as we shall see below,
(25.15) fails in models with long range interactions, whereas there are indications
that the charge integrability condition holds.
In conclusion, the charge integrability should be taken as part of the definition
that β
λ is locally generated by a charge density; the physical meaning is that β
λ can
be reasonably well approximated by
∗ -automorphisms β
λ
R with good localization
properties. As we shall see below, such a condition leads to the existence of quasiparticles with infinite lifetime in the limit of zero momentum (Goldstone quasiparticles).
ii) Local generation by a charge and time evolution
The really delicate issue (not sufficiently emphasized in the literature), which crucially enters into the proof of the theorem, is the condition that the local generation
by a charge, (25.6), holds on an algebra stable under time evolution. An equivalent
condition is that Q R and Q R (t) = α t (Q R ) generate the same automorphism.
Equation (25.6) can be easily checked on the time zero algebra generated by the
canonical variables at time t = 0, since this is a purely kinematical question which
involves the CCR or the ACR. The problem is whether (25.6) remains true when A is
replaced by α t (A). This is not trivial to check, because the infinite volume dynamics
α t is not explicitly known and the limit R → ∞ involved in (25.6) may not commute
with the infinite volume limit of α
V
t .
The heuristic argument that “since the Hamiltonian commutes with β
λ the charge
which generates β
λ is independent of time, i.e. Q(t) = Q(0),” and therefore
lim
R→∞
[Q R (t), A] = lim
R→∞
[Q R (0), A]
is not correct, because it overlooks the following important points. A global charge as
algebraic generator of β
λ does not exist if the symmetry is broken (we have already
remarked that the formal integral of j 0 does not define an operator), and one can only
speak of a local generation of β
λ in terms of local charges, so that a limit R → ∞ is
152 This can be seen, e.g. by using the Jost–Lehmann–Dyson representation.