18.3 Symmetry Breaking Order Parameter
127
18.3 Symmetry Breaking Order Parameter
The above characterization of spontaneous symmetry breaking as non-existence of
a unitary operator implementing a given algebraic symmetry in the given phase,
although simple and general is not easy to check. It is therefore convenient to have
a practically simpler criterion. For simplicity, we restrict our attention to the case
of algebraic symmetries which commute with space and time translations, briefly
called internal symmetries.
Proposition 18.2 Given a representation π of the algebra A of observables or of
canonical variables, satisfying conditions I-III of Chap. 15, and an internal symmetry
β, a necessary and sufficient condition for β being unbroken in π is that all the ground
state correlation functions are invariant under β, namely
ω(β(A)) ≡< β(A) > 0 =< A > 0 = ω(A), ∀A ∈ A,
(18.13)
where ω denotes the ground state.
Proof. In fact, if β is unitarily implementable the state β
∗
ω (see (18.11)) is described
by a vector of H π , and it is translationally invariant, since β commutes with the space
translations. By the uniqueness of the translationally invariant state, it follows that
β
∗
ω must coincide with ω and (18.13) follows. The converse has essentially been
proved in the remark after the GNS construction at the end of Chap. 11.
A ground state expectation value < A > 0 , such that
< β(A) > 0 =< A > 0 ,
will be called a symmetry breaking order parameter.
The above criterion of symmetry breaking in a phase Γ ω applies also to the general
case in which ω is invariant under space translations (but not necessarily under time
translations) and satisfies the cluster property.
The above Proposition makes clear the mechanism by which a symmetry of the
dynamics may nevertheless give rise to an asymmetrical physical description of the
system: the point is that the states of the system are described by essentially local
modifications of the ground state and states of the form A 0 , β(A) ) 0 describing
modifications of 0 related by the algebraic symmetry β, cannot be unitarily related if
0 is not invariant. Even if the two representations π ω and π β ∗ ω are physically equivalent (in the sense that they are related by a physically indistinguishable relabeling
of the observables or of the coordinates (A → β(A)), β is not a Wigner symmetry
in either of them.
It is worthwhile to stress that two ingredients play a crucial role: due to the infinite
number of degrees of freedom, two ground states define two disjoint worlds or phases
of the system and therefore, in contrast with the case of ordinary quantum mechanics,
127
18.3 Symmetry Breaking Order Parameter
The above characterization of spontaneous symmetry breaking as non-existence of
a unitary operator implementing a given algebraic symmetry in the given phase,
although simple and general is not easy to check. It is therefore convenient to have
a practically simpler criterion. For simplicity, we restrict our attention to the case
of algebraic symmetries which commute with space and time translations, briefly
called internal symmetries.
Proposition 18.2 Given a representation π of the algebra A of observables or of
canonical variables, satisfying conditions I-III of Chap. 15, and an internal symmetry
β, a necessary and sufficient condition for β being unbroken in π is that all the ground
state correlation functions are invariant under β, namely
ω(β(A)) ≡< β(A) > 0 =< A > 0 = ω(A), ∀A ∈ A,
(18.13)
where ω denotes the ground state.
Proof. In fact, if β is unitarily implementable the state β
∗
ω (see (18.11)) is described
by a vector of H π , and it is translationally invariant, since β commutes with the space
translations. By the uniqueness of the translationally invariant state, it follows that
β
∗
ω must coincide with ω and (18.13) follows. The converse has essentially been
proved in the remark after the GNS construction at the end of Chap. 11.
A ground state expectation value < A > 0 , such that
< β(A) > 0 =< A > 0 ,
will be called a symmetry breaking order parameter.
The above criterion of symmetry breaking in a phase Γ ω applies also to the general
case in which ω is invariant under space translations (but not necessarily under time
translations) and satisfies the cluster property.
The above Proposition makes clear the mechanism by which a symmetry of the
dynamics may nevertheless give rise to an asymmetrical physical description of the
system: the point is that the states of the system are described by essentially local
modifications of the ground state and states of the form A 0 , β(A) ) 0 describing
modifications of 0 related by the algebraic symmetry β, cannot be unitarily related if
0 is not invariant. Even if the two representations π ω and π β ∗ ω are physically equivalent (in the sense that they are related by a physically indistinguishable relabeling
of the observables or of the coordinates (A → β(A)), β is not a Wigner symmetry
in either of them.
It is worthwhile to stress that two ingredients play a crucial role: due to the infinite
number of degrees of freedom, two ground states define two disjoint worlds or phases
of the system and therefore, in contrast with the case of ordinary quantum mechanics,