172
7 Some Models of “Quantum Measurement”
Here, the (unbounded) operator H can be written in the evident form (its obvious
definition and a proof of selfadjointness is given in [36, Proposition II.1])
H :=
n∈Z
a
∗
n a n (a
∗
n+1 + a n+1 )a n+2 a
∗
n+2 .
(7.3.8)
This evolution is time-reflection invariant, but it is not invariant with respect to the
space reflection n → −n. Let us introduce the operators
g j := a j a
∗
j a
∗
j+1 a j+1 .
These quantities are integrals of motion. One can also prove that the Hilbert space
H vac can be decomposed into H -invariant orthogonal subspaces and on each of them
the restriction of the Hamiltonian H is a bounded operator.
Let X ⊂ Z be of finite cardinality, and let X :=
j∈X a
∗
j 0 . The vectors
X with all mutually distinct finite X ⊂ Z, with ∅ := 0 , form an orthonormal
basis in H vac . Each finite X ⊂ Z is of the form Y 1
Y 2
. . .
Y r , where all Y k ⊂ Z
are nonempty finite, mutually disjoint and of the form { j k + 1, j k + 2, . . . , j k + m k },
with j k+1 > j k + m k , |Y k | ≡ m k , i.e the sets Y k ⊂ X (k = 1, 2, . . . , r ) form mutually
separated “connected islands” consisting of “pointing up” spins. All the vectors X
are eigenvectors of all the operators g j . For the set X of the just described structure
we have
g j X =
X for j = j k , k = 1, 2, . . . , r
0
otherwise.
(7.3.9)
This implies that the time evolution of the vectors X conserves the number
of islands, leaving the initial (“left”) points j k + 1 of each Y k (k = 1, 2, . . . , r )
unchanged (“occupied”, or “pointing up”), and the places j k , k = 2, 3, . . . , r as
well as j 1 − n (n ∈ Z + ) remain all the time “unoccupied” (i.e. spins are there “pointing down”). Hence, the subspaces H {j} spanned by all such vectors with a fixed set
{j} := { j 1 , j 2 , . . . , j r } are left invariant with respect to the action of the Hamiltonian
H . Then the space H vac decomposes as
H vac =
{j}
H {j} ,
(7.3.10)
where the orthogonal sum is taken over all mutually different {j}; note that the
stationary subspace H {∅} := {λ 0 : λ ∈ C} is one dimensional.
The structure of the Hamiltonian H shows, moreover, that each H {j} can be written
as (i.e. it is isomorphic to) the tensor product of a vector (resp. of a one-dimensional
subspace) and a finite number of Hilbert spaces corresponding to restricted subchains
of spins:
H {j} =
0
(−∞, j 1 ] ⊗ H ( j 1 , j 2 ) ⊗ H ( j 2 , j 3 ) ⊗ · · · ⊗ H ( j r ,+∞) ,
(7.3.11)
7 Some Models of “Quantum Measurement”
Here, the (unbounded) operator H can be written in the evident form (its obvious
definition and a proof of selfadjointness is given in [36, Proposition II.1])
H :=
n∈Z
a
∗
n a n (a
∗
n+1 + a n+1 )a n+2 a
∗
n+2 .
(7.3.8)
This evolution is time-reflection invariant, but it is not invariant with respect to the
space reflection n → −n. Let us introduce the operators
g j := a j a
∗
j a
∗
j+1 a j+1 .
These quantities are integrals of motion. One can also prove that the Hilbert space
H vac can be decomposed into H -invariant orthogonal subspaces and on each of them
the restriction of the Hamiltonian H is a bounded operator.
Let X ⊂ Z be of finite cardinality, and let X :=
j∈X a
∗
j 0 . The vectors
X with all mutually distinct finite X ⊂ Z, with ∅ := 0 , form an orthonormal
basis in H vac . Each finite X ⊂ Z is of the form Y 1
Y 2
. . .
Y r , where all Y k ⊂ Z
are nonempty finite, mutually disjoint and of the form { j k + 1, j k + 2, . . . , j k + m k },
with j k+1 > j k + m k , |Y k | ≡ m k , i.e the sets Y k ⊂ X (k = 1, 2, . . . , r ) form mutually
separated “connected islands” consisting of “pointing up” spins. All the vectors X
are eigenvectors of all the operators g j . For the set X of the just described structure
we have
g j X =
X for j = j k , k = 1, 2, . . . , r
0
otherwise.
(7.3.9)
This implies that the time evolution of the vectors X conserves the number
of islands, leaving the initial (“left”) points j k + 1 of each Y k (k = 1, 2, . . . , r )
unchanged (“occupied”, or “pointing up”), and the places j k , k = 2, 3, . . . , r as
well as j 1 − n (n ∈ Z + ) remain all the time “unoccupied” (i.e. spins are there “pointing down”). Hence, the subspaces H {j} spanned by all such vectors with a fixed set
{j} := { j 1 , j 2 , . . . , j r } are left invariant with respect to the action of the Hamiltonian
H . Then the space H vac decomposes as
H vac =
{j}
H {j} ,
(7.3.10)
where the orthogonal sum is taken over all mutually different {j}; note that the
stationary subspace H {∅} := {λ 0 : λ ∈ C} is one dimensional.
The structure of the Hamiltonian H shows, moreover, that each H {j} can be written
as (i.e. it is isomorphic to) the tensor product of a vector (resp. of a one-dimensional
subspace) and a finite number of Hilbert spaces corresponding to restricted subchains
of spins:
H {j} =
0
(−∞, j 1 ] ⊗ H ( j 1 , j 2 ) ⊗ H ( j 2 , j 3 ) ⊗ · · · ⊗ H ( j r ,+∞) ,
(7.3.11)