Chapter 12
Fock Representation
The general lesson from the GNS theorem is that a state Ω on the algebra of observables, namely a set of expectations, defines a realization of the system in terms of a
Hilbert space H Ω of states with a reference vector Ψ Ω which represents Ω as a cyclic
vector (so that all the other vectors of H Ω can be obtained by applying the observables
to Ψ Ω ). In this sense, a state identifies the family of states related to it by observables,
equivalently accessible from it by means of physically realizable operations. Thus,
one may say that H Ω describes a closed world, or phase, to which Ω belongs. An
interesting physical and mathematical question is how many closed worlds or phases
are associated to a quantum system. In the mathematical language this amounts to
investigating how many inequivalent (physically acceptable) representations of the
observable algebra, which defines the system, exist.
For this purpose we remark that, given a pure state Ω, all the states defined by
vectors of the Hilbert space H Ω of the GNS construction define (unitarily) equivalent
representations; in fact, the corresponding GNS Hilbert spaces can be identified, and
any element A ∈ A is represented by the same operator π Ω (A) in all cases. Also
the mixed states defined by density matrices in H Ω define essentially the same
representation. In fact, the equation
Ω ρ (A) ≡ Tr(ρπ Ω (A)) =
i
λ i (Ψ i , π Ω (A)Ψ i ) =
i
λ i Ω i (A),
(12.1)
where Ψ i ∈ H Ω , expresses Ω ρ as a convex linear combination of states which define
representations equivalent to π Ω . Technically one says that π Ω ρ is quasi-equivalent
to π Ω , meaning that it can be decomposed into a sum of representations equivalent
to π Ω .
The set of states of the form (12.1) is called the folium of the representation π Ω and
can be interpreted as the set of the states which are accessible from Ω by observable
“operations”, i.e. the closed world of states associated with Ω.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_12
79
Fock Representation
The general lesson from the GNS theorem is that a state Ω on the algebra of observables, namely a set of expectations, defines a realization of the system in terms of a
Hilbert space H Ω of states with a reference vector Ψ Ω which represents Ω as a cyclic
vector (so that all the other vectors of H Ω can be obtained by applying the observables
to Ψ Ω ). In this sense, a state identifies the family of states related to it by observables,
equivalently accessible from it by means of physically realizable operations. Thus,
one may say that H Ω describes a closed world, or phase, to which Ω belongs. An
interesting physical and mathematical question is how many closed worlds or phases
are associated to a quantum system. In the mathematical language this amounts to
investigating how many inequivalent (physically acceptable) representations of the
observable algebra, which defines the system, exist.
For this purpose we remark that, given a pure state Ω, all the states defined by
vectors of the Hilbert space H Ω of the GNS construction define (unitarily) equivalent
representations; in fact, the corresponding GNS Hilbert spaces can be identified, and
any element A ∈ A is represented by the same operator π Ω (A) in all cases. Also
the mixed states defined by density matrices in H Ω define essentially the same
representation. In fact, the equation
Ω ρ (A) ≡ Tr(ρπ Ω (A)) =
i
λ i (Ψ i , π Ω (A)Ψ i ) =
i
λ i Ω i (A),
(12.1)
where Ψ i ∈ H Ω , expresses Ω ρ as a convex linear combination of states which define
representations equivalent to π Ω . Technically one says that π Ω ρ is quasi-equivalent
to π Ω , meaning that it can be decomposed into a sum of representations equivalent
to π Ω .
The set of states of the form (12.1) is called the folium of the representation π Ω and
can be interpreted as the set of the states which are accessible from Ω by observable
“operations”, i.e. the closed world of states associated with Ω.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_12
79