176
25 Breaking of Continuous Symmetries: Goldstone’s Theorem
unavoidably involved in (25.6). The commutation of β
λ with α t does not imply that
the limit of the commutator [Q R (t), A] is time independent; in fact, the symmetry
of the finite volume Hamiltonian
lim
R→∞
[Q R , H V ] = 0
implies the time independence of the limit of the commutator [Q R (t), A], provided
the two limits R → ∞, V → ∞ commute. In fact, in this case one has
lim
R→∞
[ ˙
Q R (0), A] = i lim
R→∞
lim
V →∞
[ [H V , Q R (0)], A] =
= i lim
V →∞
lim
R→∞
[ [H V , Q R (0)], A] = 0.
These remarks may look pedantic and with little physical relevance, but they
actually identify the crucial point which invalidates the heuristic argument and is
at the basis of the apparent evasion of the Goldstone theorem by the physically
relevant examples mentioned above. As a matter of fact, the interchangeability of
the two limits depends on the localization properties of the dynamics, which in
turn are governed by the range of the potential. Indeed, for short range interactions
the dynamics essentially preserves the localization of the operators, so that the limit
R → ∞ is essentially reached for finite R and the interchange of the limits is allowed.
The role of the delocalization effects of the time evolution can be explicitly displayed by working out the implications of the current conservation on the time
dependence of the charge commutator
[ ˙
Q R (t), A] = −
dx f R (x)[div j(x, t), A] =
dx ∇ f R (x) [j(x, t), A].
(25.16)
Now, since supp ∇ f R (x) ⊂ {R ≤ |x| ≤ R(1 + ε)} the time independence of the
charge commutator in the limit R → ∞ is governed by the fall off of the commutator
[j(x, t), A] for |x| → ∞.
As pointed out by Swieca,
153 the time independence of the charge commutators
holds if the time evolution is sufficiently local, namely if in s space dimensions
∀A, B ∈ A L
lim
|x|→∞
|x|
s−1
[A x , α t (B)] = 0,
(25.17)
(Swieca condition). In fact, if (25.17) holds, ∀δ > 0, ∃L such that for |x| > L, t in
a compact set,
|x|
s−1
| < [j(x, t), A] > 0 | < δ
and therefore the r.h.s. of (25.16) is bounded by (y ≡ x/R)
153 J.A. Swieca, Comm. Math. Phys. 4, 1 (1967).
25 Breaking of Continuous Symmetries: Goldstone’s Theorem
unavoidably involved in (25.6). The commutation of β
λ with α t does not imply that
the limit of the commutator [Q R (t), A] is time independent; in fact, the symmetry
of the finite volume Hamiltonian
lim
R→∞
[Q R , H V ] = 0
implies the time independence of the limit of the commutator [Q R (t), A], provided
the two limits R → ∞, V → ∞ commute. In fact, in this case one has
lim
R→∞
[ ˙
Q R (0), A] = i lim
R→∞
lim
V →∞
[ [H V , Q R (0)], A] =
= i lim
V →∞
lim
R→∞
[ [H V , Q R (0)], A] = 0.
These remarks may look pedantic and with little physical relevance, but they
actually identify the crucial point which invalidates the heuristic argument and is
at the basis of the apparent evasion of the Goldstone theorem by the physically
relevant examples mentioned above. As a matter of fact, the interchangeability of
the two limits depends on the localization properties of the dynamics, which in
turn are governed by the range of the potential. Indeed, for short range interactions
the dynamics essentially preserves the localization of the operators, so that the limit
R → ∞ is essentially reached for finite R and the interchange of the limits is allowed.
The role of the delocalization effects of the time evolution can be explicitly displayed by working out the implications of the current conservation on the time
dependence of the charge commutator
[ ˙
Q R (t), A] = −
dx f R (x)[div j(x, t), A] =
dx ∇ f R (x) [j(x, t), A].
(25.16)
Now, since supp ∇ f R (x) ⊂ {R ≤ |x| ≤ R(1 + ε)} the time independence of the
charge commutator in the limit R → ∞ is governed by the fall off of the commutator
[j(x, t), A] for |x| → ∞.
As pointed out by Swieca,
153 the time independence of the charge commutators
holds if the time evolution is sufficiently local, namely if in s space dimensions
∀A, B ∈ A L
lim
|x|→∞
|x|
s−1
[A x , α t (B)] = 0,
(25.17)
(Swieca condition). In fact, if (25.17) holds, ∀δ > 0, ∃L such that for |x| > L, t in
a compact set,
|x|
s−1
| < [j(x, t), A] > 0 | < δ
and therefore the r.h.s. of (25.16) is bounded by (y ≡ x/R)
153 J.A. Swieca, Comm. Math. Phys. 4, 1 (1967).