18.2 Spontaneous Symmetry Breaking
125
because i) one does not have to identify a small asymmetric term in the Hamiltonian and one may use a fully symmetric Hamiltonian, ii) the symmetry breaking is
accounted for by the instability of the physical world or phase chosen to describe the
states of the system.
This mechanism also shows up in the classical case, where symmetric equations
of motion may nevertheless lead to an asymmetric physical description, due to the
existence of disjoint physical worlds or phases in which the symmetry is broken
(see Part I). As we shall see also in the quantum case different phases of a system
(e.g. gas, liquid and solid) with rather different physical properties can nevertheless
be described by the same algebra of canonical variables and by the same dynamics, their differences being ascribed to the fact that they correspond to inequivalent
representations.
A crucial role for the implementation of the above mechanism is played by the
concept of algebraic symmetry of an algebra A of observables or of canonical variables, defined as an invertible mapping β of the algebra into itself, which preserves
all the algebraic relations, including the
∗ (
∗ -automorphism of A). Clearly, if ω is a
state on A, also β
∗
ω defined by
(β
∗
ω)(A) ≡ ω(β(A))
(18.11)
is a state on A and the corresponding GNS representations are isomorphic and physically equivalent if β commutes with the dynamics α t . They may however yield
(mathematically) inequivalent representations of A. In this case, the corresponding
vector states cannot belong to the same Hilbert space, i.e. they describe disjoint (even
if equivalent) physical worlds. In a very similar way, one may introduce algebraic
symmetries defined by antiautomorphisms σ of A:
σ(λA + μB) = λσ(A) + μσ(B),
σ(AB) =σ(B)σ(A), σ(A
∗
) = σ(A)
∗
, ∀A, B ∈ A.
They correspond to the Wigner symmetries described by antiunitary operators. For
simplicity, we shall not consider this case in the sequel.
Given a representation π ω of A, the algebraic symmetry β gives rise to a Wigner
symmetry in H ω if there exists a unitary operator U β such that
U β π ω (A)U
−1
β = π ω (β(A)) = π β ∗ ω (A).
(18.12)
The above equation implies that π β ∗ ω is unitarily equivalent to π ω . In this case, the
physical description of the system in the phase (π ω , H ω ) is β-symmetric (briefly
the symmetry β is unbroken or exact). On the other hand, if π ω and π β ∗ ω are not
unitarily equivalent, there is no unitary operator U β which implements β in H ω and
the corresponding physical description is not β-symmetric. In this case the symmetry
is said to be spontaneously broken. The name should stress the fact that one has a
symmetry at the algebraic level and that the lack of symmetry of the matrix elements
125
because i) one does not have to identify a small asymmetric term in the Hamiltonian and one may use a fully symmetric Hamiltonian, ii) the symmetry breaking is
accounted for by the instability of the physical world or phase chosen to describe the
states of the system.
This mechanism also shows up in the classical case, where symmetric equations
of motion may nevertheless lead to an asymmetric physical description, due to the
existence of disjoint physical worlds or phases in which the symmetry is broken
(see Part I). As we shall see also in the quantum case different phases of a system
(e.g. gas, liquid and solid) with rather different physical properties can nevertheless
be described by the same algebra of canonical variables and by the same dynamics, their differences being ascribed to the fact that they correspond to inequivalent
representations.
A crucial role for the implementation of the above mechanism is played by the
concept of algebraic symmetry of an algebra A of observables or of canonical variables, defined as an invertible mapping β of the algebra into itself, which preserves
all the algebraic relations, including the
∗ (
∗ -automorphism of A). Clearly, if ω is a
state on A, also β
∗
ω defined by
(β
∗
ω)(A) ≡ ω(β(A))
(18.11)
is a state on A and the corresponding GNS representations are isomorphic and physically equivalent if β commutes with the dynamics α t . They may however yield
(mathematically) inequivalent representations of A. In this case, the corresponding
vector states cannot belong to the same Hilbert space, i.e. they describe disjoint (even
if equivalent) physical worlds. In a very similar way, one may introduce algebraic
symmetries defined by antiautomorphisms σ of A:
σ(λA + μB) = λσ(A) + μσ(B),
σ(AB) =σ(B)σ(A), σ(A
∗
) = σ(A)
∗
, ∀A, B ∈ A.
They correspond to the Wigner symmetries described by antiunitary operators. For
simplicity, we shall not consider this case in the sequel.
Given a representation π ω of A, the algebraic symmetry β gives rise to a Wigner
symmetry in H ω if there exists a unitary operator U β such that
U β π ω (A)U
−1
β = π ω (β(A)) = π β ∗ ω (A).
(18.12)
The above equation implies that π β ∗ ω is unitarily equivalent to π ω . In this case, the
physical description of the system in the phase (π ω , H ω ) is β-symmetric (briefly
the symmetry β is unbroken or exact). On the other hand, if π ω and π β ∗ ω are not
unitarily equivalent, there is no unitary operator U β which implements β in H ω and
the corresponding physical description is not β-symmetric. In this case the symmetry
is said to be spontaneously broken. The name should stress the fact that one has a
symmetry at the algebraic level and that the lack of symmetry of the matrix elements