196
7 Some Models of “Quantum Measurement”
a nonzero interval γ
2
∈ [0, γ
2
0 ] ⊂ R. For an arbitrary normalized ψ ∈ H B , it is
ψ|W γ=0 |ψ = 0, and it is 0 < ψ|W γ |ψ < 1 for all sufficiently small |γ| > 0 and
all normalized ψ ∈ H B . Hence, at least for sufficiently small nonzero γ ∈ R, it is
ψ|W γ |ψ = 1 for normalized ψ ∈ H W , so that W
2
γ = W γ , i.e. the positive operator
W γ is not a projector. The theorem is proved.
7.5 The X-Y Chain as a Measuring Device
7.5.1 The X-Y chain
Let us formulate first what we understand here under the “X-Y chain” (cf. [267], and
also [35, 107, 271])—a special case of the Heisenberg spin chains:
It is again a model of one-dimensional spin chain with C
∗ -algebra of observables A generated by spin creation-annihilation operators a
∗
j , a j ( j ∈ Z), as it was
introduced in 7.3.2. The algebra A is the C
∗ -inductive limit of the sequence of its
local subalgebras A n (n ∈ N), each generated by a
∗
j , a j (| j| ≤ n). The dynamics in
any subalgebra A n is given by the local Hamiltonian H n (without interaction with
external magnetic field):
H n :=
κ
2
n−1
j=−n
(a
∗
j a j+1 + a
∗
j+1 a j ),
(7.5.1)
where κ ∈ R. These local Hamiltonians define the time-evolution (t; x) → τ
(n)
t (x)
of local elements x ∈ A n :
τ
(n)
t (x) := e
it H n xe
−it H n , x ∈ A n , n ∈ N, t ∈ R.
(7.5.2)
The evolution in the whole algebra A is obtained by taking first the limit n → ∞
in norm of A for any fixed t ∈ R and any local x ∈ A, and afterwards obtaining the
result by the norm-continuity, extending it to all x ∈ A:
τ t (x) := n- lim
n→∞
τ
(n)
t (x).
(7.5.3)
Note that the term “X-Y model” comes from the form of the hamiltonian if it is
rewritten in the terms of Pauli σ-matrices : σ
x
j := a
∗
j + a j , σ
y
j := ia j − ia
∗
j , σ
z
j :=
2a
∗
j a j − 1, i.e.
H =
κ
4
j
(σ
x
j σ
x
j+1 + σ
y
j σ
y
j+1 ).
(7.5.4)
We shall write often H instead of H n , also without specifying the local characters
of the entering algebraic elements x, or A ∈ A, …, to simplify the notation and the
7 Some Models of “Quantum Measurement”
a nonzero interval γ
2
∈ [0, γ
2
0 ] ⊂ R. For an arbitrary normalized ψ ∈ H B , it is
ψ|W γ=0 |ψ = 0, and it is 0 < ψ|W γ |ψ < 1 for all sufficiently small |γ| > 0 and
all normalized ψ ∈ H B . Hence, at least for sufficiently small nonzero γ ∈ R, it is
ψ|W γ |ψ = 1 for normalized ψ ∈ H W , so that W
2
γ = W γ , i.e. the positive operator
W γ is not a projector. The theorem is proved.
7.5 The X-Y Chain as a Measuring Device
7.5.1 The X-Y chain
Let us formulate first what we understand here under the “X-Y chain” (cf. [267], and
also [35, 107, 271])—a special case of the Heisenberg spin chains:
It is again a model of one-dimensional spin chain with C
∗ -algebra of observables A generated by spin creation-annihilation operators a
∗
j , a j ( j ∈ Z), as it was
introduced in 7.3.2. The algebra A is the C
∗ -inductive limit of the sequence of its
local subalgebras A n (n ∈ N), each generated by a
∗
j , a j (| j| ≤ n). The dynamics in
any subalgebra A n is given by the local Hamiltonian H n (without interaction with
external magnetic field):
H n :=
κ
2
n−1
j=−n
(a
∗
j a j+1 + a
∗
j+1 a j ),
(7.5.1)
where κ ∈ R. These local Hamiltonians define the time-evolution (t; x) → τ
(n)
t (x)
of local elements x ∈ A n :
τ
(n)
t (x) := e
it H n xe
−it H n , x ∈ A n , n ∈ N, t ∈ R.
(7.5.2)
The evolution in the whole algebra A is obtained by taking first the limit n → ∞
in norm of A for any fixed t ∈ R and any local x ∈ A, and afterwards obtaining the
result by the norm-continuity, extending it to all x ∈ A:
τ t (x) := n- lim
n→∞
τ
(n)
t (x).
(7.5.3)
Note that the term “X-Y model” comes from the form of the hamiltonian if it is
rewritten in the terms of Pauli σ-matrices : σ
x
j := a
∗
j + a j , σ
y
j := ia j − ia
∗
j , σ
z
j :=
2a
∗
j a j − 1, i.e.
H =
κ
4
j
(σ
x
j σ
x
j+1 + σ
y
j σ
y
j+1 ).
(7.5.4)
We shall write often H instead of H n , also without specifying the local characters
of the entering algebraic elements x, or A ∈ A, …, to simplify the notation and the