25.2 A Critical Look at the Hypotheses of Goldstone Theorem
173
25.2 A Critical Look at the Hypotheses of Goldstone
Theorem
The importance and usefulness of the Goldstone theorem is mainly that of providing
non-perturbative information on the energy spectrum of an infinite system. For this
purpose, it is crucial to be able to verify its assumptions without having to solve
the full dynamical problem. We shall therefore critically discuss the hypotheses of
the theorem and their possible verification; as a result, we shall discover the general
mechanism which is at the basis of the phenomenon of spontaneous breaking of a
continuous symmetry accompanied by an energy gap, in all the examples mentioned
above.
I. Symmetry of the Dynamics
At a formal level, the existence of an internal symmetry is inferred from the invariance
of the formal Hamiltonian (or Lagrangian) which (formally) defines the model.
Now, the commutation of the symmetry β
λ with the space translations α x is a
kinematical property which is easily checked, once the action of β
λ on the canonical
variables (or on the observables) is specified.
Less obvious is the check of the commutation of β
λ with the time translations α t ,
since in general the infinite volume dynamics is not explicitly known.
Proposition 25.2 If the finite volume dynamics α
V
t , defined by the finite volume
Hamiltonian H V , converges to the infinite volume dynamics α t in the norm topology,
then
β
λ
α
V
t = α
V
t β
λ
(25.10)
implies
β
λ
α t = α t β
λ
.
(25.11)
Proof. In fact,
∗ -automorphisms of a C
∗ -algebra are norm preserving and therefore
continuous in the norm topology
β
λ
α t (A) = β
λ
(α t − α
V
t )(A) + α
V
t β
λ
(A) −→
V →∞
α t β
λ
(A).
Thus, the check of (25.11) is reduced to the invariance of the finite volume Hamiltonian and the current wisdom is essentially correct.
173
25.2 A Critical Look at the Hypotheses of Goldstone
Theorem
The importance and usefulness of the Goldstone theorem is mainly that of providing
non-perturbative information on the energy spectrum of an infinite system. For this
purpose, it is crucial to be able to verify its assumptions without having to solve
the full dynamical problem. We shall therefore critically discuss the hypotheses of
the theorem and their possible verification; as a result, we shall discover the general
mechanism which is at the basis of the phenomenon of spontaneous breaking of a
continuous symmetry accompanied by an energy gap, in all the examples mentioned
above.
I. Symmetry of the Dynamics
At a formal level, the existence of an internal symmetry is inferred from the invariance
of the formal Hamiltonian (or Lagrangian) which (formally) defines the model.
Now, the commutation of the symmetry β
λ with the space translations α x is a
kinematical property which is easily checked, once the action of β
λ on the canonical
variables (or on the observables) is specified.
Less obvious is the check of the commutation of β
λ with the time translations α t ,
since in general the infinite volume dynamics is not explicitly known.
Proposition 25.2 If the finite volume dynamics α
V
t , defined by the finite volume
Hamiltonian H V , converges to the infinite volume dynamics α t in the norm topology,
then
β
λ
α
V
t = α
V
t β
λ
(25.10)
implies
β
λ
α t = α t β
λ
.
(25.11)
Proof. In fact,
∗ -automorphisms of a C
∗ -algebra are norm preserving and therefore
continuous in the norm topology
β
λ
α t (A) = β
λ
(α t − α
V
t )(A) + α
V
t β
λ
(A) −→
V →∞
α t β
λ
(A).
Thus, the check of (25.11) is reduced to the invariance of the finite volume Hamiltonian and the current wisdom is essentially correct.