126
18 Symmetry Breaking in Quantum Systems
between states of a given representation π is due to the impossibility of describing
the given algebraic symmetry by a unitary operator which maps the states of H π into
themselves.
It should be clear from the above discussion that the concept of algebraic symmetry disentangles the concept of symmetry from a concrete representation, and
it is particularly useful for the description of infinite systems, for which there are
generically several inequivalent representations of the algebra of observables or of
canonical variables. As we shall see below, it also allows for the mechanism by
which a symmetry of the dynamics may fail to be a symmetry of the physical world
associated to a given description of the system.
Perhaps, one of the reasons why the mechanism of spontaneous symmetry breaking has been realized so late after the foundations of quantum mechanics is that, as
in the classical case, its realization crucially involves infinite degrees of freedom.
For this purpose, we consider an algebraic symmetry γ of the Weyl algebra with
the property that it can be extended to the Heisenberg algebra, namely to the canonical
variables q, p; from a technical point of view such a property can be formalized by
the condition that γ preserves the regularity of the Weyl operators, i.e. if π(U (α)),
π(V (β)) are weakly continuous in α, β, so are π(γ(U (α))), π(γ(V (β))). Indeed,
by Stone’s theorem such a property allows to define γ(q), γ( p) as the generators
of the one-parameter groups π(γ(U (α))), π(γ(V (β))). The algebraic symmetries of
A W which have this property shall be called regular (or regular
∗ -automorphisms of
A W ).
Proposition 18.1 If π is a regular irreducible representation of the Weyl algebra
A W (for finite degrees of freedom), then any regular algebraic symmetry γ of A W is
implemented by a unitary operator in the representation space H π (no spontaneous
symmetry breaking).
Proof. In fact, if π and π γ are the GNS representations defined by the states ω and
γ
∗
ω, respectively, by (18.11) one has
(( γ ∗ ω , π γ (A) ) γ ∗ ω ) = (γ
∗
ω)(A) = ω(γ(A)) = (( ω , π(γ(A)) ) ω ).
Now, if ω is pure so must be γ
∗
ω since γ
∗ is invertible, and therefore if π is irreducible
so is also π γ . Finally, if π is regular, so is π γ by the regularity of γ and therefore the
two representations are unitarily equivalent by von Neumann’s uniqueness theorem.
This means that (18.12) holds and γ is unitarily implemented.
18 Symmetry Breaking in Quantum Systems
between states of a given representation π is due to the impossibility of describing
the given algebraic symmetry by a unitary operator which maps the states of H π into
themselves.
It should be clear from the above discussion that the concept of algebraic symmetry disentangles the concept of symmetry from a concrete representation, and
it is particularly useful for the description of infinite systems, for which there are
generically several inequivalent representations of the algebra of observables or of
canonical variables. As we shall see below, it also allows for the mechanism by
which a symmetry of the dynamics may fail to be a symmetry of the physical world
associated to a given description of the system.
Perhaps, one of the reasons why the mechanism of spontaneous symmetry breaking has been realized so late after the foundations of quantum mechanics is that, as
in the classical case, its realization crucially involves infinite degrees of freedom.
For this purpose, we consider an algebraic symmetry γ of the Weyl algebra with
the property that it can be extended to the Heisenberg algebra, namely to the canonical
variables q, p; from a technical point of view such a property can be formalized by
the condition that γ preserves the regularity of the Weyl operators, i.e. if π(U (α)),
π(V (β)) are weakly continuous in α, β, so are π(γ(U (α))), π(γ(V (β))). Indeed,
by Stone’s theorem such a property allows to define γ(q), γ( p) as the generators
of the one-parameter groups π(γ(U (α))), π(γ(V (β))). The algebraic symmetries of
A W which have this property shall be called regular (or regular
∗ -automorphisms of
A W ).
Proposition 18.1 If π is a regular irreducible representation of the Weyl algebra
A W (for finite degrees of freedom), then any regular algebraic symmetry γ of A W is
implemented by a unitary operator in the representation space H π (no spontaneous
symmetry breaking).
Proof. In fact, if π and π γ are the GNS representations defined by the states ω and
γ
∗
ω, respectively, by (18.11) one has
(( γ ∗ ω , π γ (A) ) γ ∗ ω ) = (γ
∗
ω)(A) = ω(γ(A)) = (( ω , π(γ(A)) ) ω ).
Now, if ω is pure so must be γ
∗
ω since γ
∗ is invertible, and therefore if π is irreducible
so is also π γ . Finally, if π is regular, so is π γ by the regularity of γ and therefore the
two representations are unitarily equivalent by von Neumann’s uniqueness theorem.
This means that (18.12) holds and γ is unitarily implemented.