204
7 Some Models of “Quantum Measurement”
We shall keep in mind such an idea to be able to believe that also our finite system
described in this section can be considered as a model of “quantum measurement”
process.
7.6.2 Let us look at the Quantum Domino from Sect. 7.3. We shall restrict here that
model to finite number of degrees of freedom, hence the spin chain will be of finite
length and its algebra of observables A (with unity I A ) is generated by the spin-1/2
creation and annihilation operators a
∗
j , a j ( j = 0, 1, . . . , N ) satisfying (7.3.1). This
system will interact with the (nonrelativistic scalar) Fermi field, the algebra F (with
unity I F ) of which is generated by the particle creation-annihilation operators
b
∗
(ϕ), b(ϕ) satisfying the relations
b(ϕ)
2
= 0, b(ϕ)b
∗
(ψ) + b
∗
(ψ)b(ϕ) = (ϕ, ψ)I F , (for all ϕ, ψ ∈ L
2
(R
3
, d
3 x)),
(7.6.1)
with the linear dependence ψ → b
∗
(ψ).
The dynamics is given by the Hamiltonian H := H 0 + V , where
H 0 :=
N −2
n=0
a
∗
n a n (a
∗
n+1 + a n+1 )a n+2 a
∗
n+2 − ε 0 a
∗
N a N
⊗ I F + I A ⊗ d(h),
(7.6.2a)
V := v
2
a
∗
N −1 a N −1 a
∗
N ⊗ b
∗
(σ) + a
∗
N −1 a N −1 a N ⊗ b(σ)
.
(7.6.2b)
We can consider these algebras A and F as algebras of operators acting on the
Hilbert space H S := (C
2
)
N +1 , and on the Fermi Fock space H F respectively,
resp. on their tensor product H := H S ⊗ H F . In the above written formulas, the
symbol d(h) means the “second quantization” (cf. [54, Sect. 5.2.1]
14 ) of the
operator h ∈ L(h) := L(L
2
(R
3
, d
3 x)) given by the function p → ε(p) of oneparticle momentum p, hence acting on the vectors of h := L
2
(R
3
, d
3 x) “in the
p-representation” as multiplication by ε(p) : (hψ)(p) ≡ ε(p)ψ(p). The nonnegative
function ε(p), as well as the parameters ε 0 > 0, v ∈ R, σ ∈ L
2
(R
3
, d
3 x), will be
specified later. In our expressions of action of elements of A, resp. F, on vectors of
H S ⊗ H F , the unity operators of the other algebra will be usually omitted, e.g. for
a ∈ A, |s ⊗ |ϕ ∈ H S ⊗ H F , we shall write a ⊗ I F (|s ⊗ |ϕ) ≡ a(|s ⊗ |ϕ) ≡
a|s ⊗ |ϕ.
Let
F
0 be the Fermi vacuum in H F , and
S
0 ∈ H S be the state of the spin
chain “with all spins pointing down”: a n
S
0 = 0, ∀n. Notice also that here |n :=
a
∗
0 a
∗
1 . . . a
∗
n
S
0 , n = 0, 1, . . . N . Let the Hilbert subspace H min ⊂ H be generated
by the vectors
14 The “second quantization” d(h) of the ‘one-Fermi-particle-operator’ h is the linear operator
acting in the Fermi Fock space H F := ⊕ ∞
n=0 P − ⊗ n
1 h, where P − is the antisymmetrization operator,
such that d(h)P − ⊗ n
k=1 ψ k := P −
n
j=1 ψ 1 ⊗ ψ 2 ⊗ · · · ⊗ hψ j ⊗ · · · ⊗ ψ n for all n ∈ Z + .
7 Some Models of “Quantum Measurement”
We shall keep in mind such an idea to be able to believe that also our finite system
described in this section can be considered as a model of “quantum measurement”
process.
7.6.2 Let us look at the Quantum Domino from Sect. 7.3. We shall restrict here that
model to finite number of degrees of freedom, hence the spin chain will be of finite
length and its algebra of observables A (with unity I A ) is generated by the spin-1/2
creation and annihilation operators a
∗
j , a j ( j = 0, 1, . . . , N ) satisfying (7.3.1). This
system will interact with the (nonrelativistic scalar) Fermi field, the algebra F (with
unity I F ) of which is generated by the particle creation-annihilation operators
b
∗
(ϕ), b(ϕ) satisfying the relations
b(ϕ)
2
= 0, b(ϕ)b
∗
(ψ) + b
∗
(ψ)b(ϕ) = (ϕ, ψ)I F , (for all ϕ, ψ ∈ L
2
(R
3
, d
3 x)),
(7.6.1)
with the linear dependence ψ → b
∗
(ψ).
The dynamics is given by the Hamiltonian H := H 0 + V , where
H 0 :=
N −2
n=0
a
∗
n a n (a
∗
n+1 + a n+1 )a n+2 a
∗
n+2 − ε 0 a
∗
N a N
⊗ I F + I A ⊗ d(h),
(7.6.2a)
V := v
2
a
∗
N −1 a N −1 a
∗
N ⊗ b
∗
(σ) + a
∗
N −1 a N −1 a N ⊗ b(σ)
.
(7.6.2b)
We can consider these algebras A and F as algebras of operators acting on the
Hilbert space H S := (C
2
)
N +1 , and on the Fermi Fock space H F respectively,
resp. on their tensor product H := H S ⊗ H F . In the above written formulas, the
symbol d(h) means the “second quantization” (cf. [54, Sect. 5.2.1]
14 ) of the
operator h ∈ L(h) := L(L
2
(R
3
, d
3 x)) given by the function p → ε(p) of oneparticle momentum p, hence acting on the vectors of h := L
2
(R
3
, d
3 x) “in the
p-representation” as multiplication by ε(p) : (hψ)(p) ≡ ε(p)ψ(p). The nonnegative
function ε(p), as well as the parameters ε 0 > 0, v ∈ R, σ ∈ L
2
(R
3
, d
3 x), will be
specified later. In our expressions of action of elements of A, resp. F, on vectors of
H S ⊗ H F , the unity operators of the other algebra will be usually omitted, e.g. for
a ∈ A, |s ⊗ |ϕ ∈ H S ⊗ H F , we shall write a ⊗ I F (|s ⊗ |ϕ) ≡ a(|s ⊗ |ϕ) ≡
a|s ⊗ |ϕ.
Let
F
0 be the Fermi vacuum in H F , and
S
0 ∈ H S be the state of the spin
chain “with all spins pointing down”: a n
S
0 = 0, ∀n. Notice also that here |n :=
a
∗
0 a
∗
1 . . . a
∗
n
S
0 , n = 0, 1, . . . N . Let the Hilbert subspace H min ⊂ H be generated
by the vectors
14 The “second quantization” d(h) of the ‘one-Fermi-particle-operator’ h is the linear operator
acting in the Fermi Fock space H F := ⊕ ∞
n=0 P − ⊗ n
1 h, where P − is the antisymmetrization operator,
such that d(h)P − ⊗ n
k=1 ψ k := P −
n
j=1 ψ 1 ⊗ ψ 2 ⊗ · · · ⊗ hψ j ⊗ · · · ⊗ ψ n for all n ∈ Z + .