76
11 Quantum Mechanics. Algebraic Structure and States
Quite generally a pure state on a C
∗ -algebra is a state which cannot be decomposed
as a convex sum
Ω = λΩ 1 + (1 − λ)Ω 2 , 0 < λ < 1,
(11.8)
of two other states. Otherwise the state is called a mixed state.
The above definition of state is particularly useful for the description of systems
with infinite degrees of freedom, briefly infinite systems, for which the elementary
concept of wave function is not available and even the standard Schrödinger representation of the algebra of canonical variables may not be allowed (as we shall
see below). Whereas in the case of a quantum system with N degrees of freedom
one may choose, e.g. the maximal abelian algebra generated by the N coordinates
q 1 , . . . , q N and describe the states of the system by wave functions of such variables, for an infinite system one should consider an infinite set of coordinates and it
is problematic to define the corresponding wave functions. Moreover, the von Neumann uniqueness theorem does not apply to infinite systems and there are in general
physically relevant representations of the algebra of canonical variables which are
not equivalent to the Schrödinger representations (the physical meaning of this problem shall be discussed below). The virtue of the above definition of state is that it
applies in general, without involving the concept of wave function.
It is a deep result of Gelfand, Naimark and Segal,
54 also called the G N S construction, that the knowledge of a state Ω in the above sense, namely in terms of
its expectations on A, uniquely determine (up to isometries) a representation
55
π Ω
of the canonical variables, or more generally of the observables, as operators in a
Hilbert space H Ω which contains a reference vector Ψ Ω , whose matrix elements
reproduce the given expectations
(Ψ Ω , π Ω (A)Ψ Ω ) = Ω(A), ∀A ∈ A.
(11.9)
The idea of the proof is to associate to each element A ∈ A a vector Ψ A , which
will have the meaning of a vector obtained by applying A to the “reference” vector
Ψ 1 = Ψ Ω . If such an association is done in a way which preserves the linear structure
of A, (i.e. Ψ A + Ψ B = Ψ A+B ), one gets a vector space D A isomorphic to A, which
is naturally equipped with a non-negative inner product
(Ψ A , Ψ B ) = Ω(A
∗ B).
The null elements are those corresponding to the set
J = {A ∈ A; Ω(B
∗ A) = 0, ∀B ∈ A}.
(11.10)
54 M.A. Naimark, Normed Rings, Noordhoff 1964.
55 We recall that a representation π of a C ∗ -algebra in a Hilbert space H is ∗− homomorphism π
of A into the C ∗ -algebra of bounded (linear) operators in H, i.e. a mapping which preserves all the
algebraic operations, including the ∗.
11 Quantum Mechanics. Algebraic Structure and States
Quite generally a pure state on a C
∗ -algebra is a state which cannot be decomposed
as a convex sum
Ω = λΩ 1 + (1 − λ)Ω 2 , 0 < λ < 1,
(11.8)
of two other states. Otherwise the state is called a mixed state.
The above definition of state is particularly useful for the description of systems
with infinite degrees of freedom, briefly infinite systems, for which the elementary
concept of wave function is not available and even the standard Schrödinger representation of the algebra of canonical variables may not be allowed (as we shall
see below). Whereas in the case of a quantum system with N degrees of freedom
one may choose, e.g. the maximal abelian algebra generated by the N coordinates
q 1 , . . . , q N and describe the states of the system by wave functions of such variables, for an infinite system one should consider an infinite set of coordinates and it
is problematic to define the corresponding wave functions. Moreover, the von Neumann uniqueness theorem does not apply to infinite systems and there are in general
physically relevant representations of the algebra of canonical variables which are
not equivalent to the Schrödinger representations (the physical meaning of this problem shall be discussed below). The virtue of the above definition of state is that it
applies in general, without involving the concept of wave function.
It is a deep result of Gelfand, Naimark and Segal,
54 also called the G N S construction, that the knowledge of a state Ω in the above sense, namely in terms of
its expectations on A, uniquely determine (up to isometries) a representation
55
π Ω
of the canonical variables, or more generally of the observables, as operators in a
Hilbert space H Ω which contains a reference vector Ψ Ω , whose matrix elements
reproduce the given expectations
(Ψ Ω , π Ω (A)Ψ Ω ) = Ω(A), ∀A ∈ A.
(11.9)
The idea of the proof is to associate to each element A ∈ A a vector Ψ A , which
will have the meaning of a vector obtained by applying A to the “reference” vector
Ψ 1 = Ψ Ω . If such an association is done in a way which preserves the linear structure
of A, (i.e. Ψ A + Ψ B = Ψ A+B ), one gets a vector space D A isomorphic to A, which
is naturally equipped with a non-negative inner product
(Ψ A , Ψ B ) = Ω(A
∗ B).
The null elements are those corresponding to the set
J = {A ∈ A; Ω(B
∗ A) = 0, ∀B ∈ A}.
(11.10)
54 M.A. Naimark, Normed Rings, Noordhoff 1964.
55 We recall that a representation π of a C ∗ -algebra in a Hilbert space H is ∗− homomorphism π
of A into the C ∗ -algebra of bounded (linear) operators in H, i.e. a mapping which preserves all the
algebraic operations, including the ∗.