11 Quantum Mechanics. Algebraic Structure and States
77
J is a left ideal of A, i.e. a linear subspace such that AJ ⊆ J and one may consider
the quotient A/J and correspondingly the equivalence classes of vectors Ψ [A+J ] =
Ψ [A] ∈ D A /D J ≡ D Ω . In this way, the inner product becomes strictly positive on
D Ω , which is therefore a pre-Hilbert space, and by completion one gets a Hilbert
space H Ω = D Ω .
The representation is then defined by
π Ω (A)Ψ [B] ≡ Ψ [AB]
(11.11)
(this equation is well defined since [B] = [C] implies [AB] = [AC]). By construction, the vector Ψ Ω ≡ Ψ [1] is cyclic with respect to π Ω (A) , namely π Ω (A)Ψ Ω is
dense in H Ω ; moreover ||π Ω (A)|| H Ω ≤ ||A||, thanks to the continuity of Ω.
The so constructed representation is unique up to unitary equivalence. In fact, if
π
is another representation in a Hilbert space H
with a cyclic vector Ψ
such that
(Ψ
, π
(A)Ψ
) = Ω(A),
then the mapping U : π Ω (A)Ψ Ω → π
(A)Ψ
and its inverse U
−1 are defined on
dense sets and preserve the scalar products, so that they are unitary and π
(A) =
U π(A)U
−1 .
The GNS representation π Ω defined by a state Ω is irreducible iff Ω is pure.
As a relevant application of the above result, we consider a ∗-automorphism α
of A, namely an invertible mapping of A into A, which preserves all the algebraic
operations including the ∗ (also called algebraic symmetry). If the state Ω is invariant
in the sense that
Ω(α(A)) = Ω(A) , ∀A ∈ A,
(11.12)
then, Ω and Ω α , Ω α (A) ≡ Ω(α
−1
(A)) define unitary equivalent GNS representations and the automorphism α is implemented by the unitary operator U α
U α Ψ Ω = Ψ Ω α , U α π Ω (A)Ψ Ω = π Ω (α(A))Ψ Ω α ,
(11.13)
where Ψ Ω α denotes a vector representative of the state Ω α . Therefore, all the matrix
elements are invariant under the operation which implements α, and briefly one says
that α gives rise to a symmetry of the states of the Hilbert space H Ω . Thus, the
invariance of Ω under α implies that α is a symmetry of the physical world or phase
defined by Ω through the GNS construction.
77
J is a left ideal of A, i.e. a linear subspace such that AJ ⊆ J and one may consider
the quotient A/J and correspondingly the equivalence classes of vectors Ψ [A+J ] =
Ψ [A] ∈ D A /D J ≡ D Ω . In this way, the inner product becomes strictly positive on
D Ω , which is therefore a pre-Hilbert space, and by completion one gets a Hilbert
space H Ω = D Ω .
The representation is then defined by
π Ω (A)Ψ [B] ≡ Ψ [AB]
(11.11)
(this equation is well defined since [B] = [C] implies [AB] = [AC]). By construction, the vector Ψ Ω ≡ Ψ [1] is cyclic with respect to π Ω (A) , namely π Ω (A)Ψ Ω is
dense in H Ω ; moreover ||π Ω (A)|| H Ω ≤ ||A||, thanks to the continuity of Ω.
The so constructed representation is unique up to unitary equivalence. In fact, if
π
is another representation in a Hilbert space H
with a cyclic vector Ψ
such that
(Ψ
, π
(A)Ψ
) = Ω(A),
then the mapping U : π Ω (A)Ψ Ω → π
(A)Ψ
and its inverse U
−1 are defined on
dense sets and preserve the scalar products, so that they are unitary and π
(A) =
U π(A)U
−1 .
The GNS representation π Ω defined by a state Ω is irreducible iff Ω is pure.
As a relevant application of the above result, we consider a ∗-automorphism α
of A, namely an invertible mapping of A into A, which preserves all the algebraic
operations including the ∗ (also called algebraic symmetry). If the state Ω is invariant
in the sense that
Ω(α(A)) = Ω(A) , ∀A ∈ A,
(11.12)
then, Ω and Ω α , Ω α (A) ≡ Ω(α
−1
(A)) define unitary equivalent GNS representations and the automorphism α is implemented by the unitary operator U α
U α Ψ Ω = Ψ Ω α , U α π Ω (A)Ψ Ω = π Ω (α(A))Ψ Ω α ,
(11.13)
where Ψ Ω α denotes a vector representative of the state Ω α . Therefore, all the matrix
elements are invariant under the operation which implements α, and briefly one says
that α gives rise to a symmetry of the states of the Hilbert space H Ω . Thus, the
invariance of Ω under α implies that α is a symmetry of the physical world or phase
defined by Ω through the GNS construction.