78
5 Macroscopic Limits
U z :=
j∈
(z j ϕ j ).
(5.1.18)
Let {ϕ
n
: n ∈ Z + } be an orthonormal basis in H. Let a, b ∈ Z
+ with components
a j , b j ∈ Z + ( j ∈ ), and set
ϕ
a
j := u j (ϕ
a j ) ∈ H j , ,
a
:=
j∈
ϕ
a
j .
(5.1.19)
For a = b, the vectors
a and
b are mutually orthogonal: ((
a
, ,
b
) = 0. Let
:=
a for some a (this can be done so for any normalized product-vector ∈
H by a choice of the identifications u j , j ∈ , of H j with H). The vectors
b
,
for which b j = a j for all j ∈ \J b , b j ∈ Z + for all j ∈ J b , where J b runs over
all finite subsets of ,
1 form an orthonormal basis in a closed subspace of H
denoted by H
and called ITPS (incomplete tensor product space). Let P be the
orthogonal projector in H onto H
. For two arbitrary product vectors , , ∈ H
the projectors P and P are either orthogonal or equal. For any U z from (5.1.18)
we have
U z P U
∗
z = P U z ,
(5.1.20)
and the product vectors and U z are weakly equivalent, cf. [227]. If P =
(hence P = ), then and are (strongly) equivalent. The set of all product
vectors weakly equivalent to a product vector form a total set in a closed
subspace of H with the orthogonal projector P
w
. Clearly, P
w
is the sum of all
such P , which correspond to mutually strongly inequivalent product vectors , all
of them being weakly equivalent to . The sum of all mutually strongly inequivalent
P (we use an obvious licence in language) is the unit operator in H .
2
Let A
denotes the C
∗ -subalgebra of the algebra of all bounded operators on H
(denoted by L(H )), generated by the elements
{π j (A) ∈ L(H ) : A ∈ L(H), j ∈ },
(5.1.21)
where L(H) is the algebra of all bounded operators on the Hilbert space H.
For any x ∈ A
, the following relations are valid, [227]:
[x, P ] = [x, U z ] = 0 for all U z , and for all P ,
(5.1.22)
with U z from (5.1.18). If p is another orthogonal projector in L(H ), and for some
product-vector it is p P = p, then
if [x, p] = 0 for all x ∈ A
⇒ p = P or p = 0,
(5.1.23)
1 i.e. all the vectors b for which b j = a j for finite number of indices j ∈ only.
2 We shall use sometimes projectors instead of the corresponding subspaces.
5 Macroscopic Limits
U z :=
j∈
(z j ϕ j ).
(5.1.18)
Let {ϕ
n
: n ∈ Z + } be an orthonormal basis in H. Let a, b ∈ Z
+ with components
a j , b j ∈ Z + ( j ∈ ), and set
ϕ
a
j := u j (ϕ
a j ) ∈ H j , ,
a
:=
j∈
ϕ
a
j .
(5.1.19)
For a = b, the vectors
a and
b are mutually orthogonal: ((
a
, ,
b
) = 0. Let
:=
a for some a (this can be done so for any normalized product-vector ∈
H by a choice of the identifications u j , j ∈ , of H j with H). The vectors
b
,
for which b j = a j for all j ∈ \J b , b j ∈ Z + for all j ∈ J b , where J b runs over
all finite subsets of ,
1 form an orthonormal basis in a closed subspace of H
denoted by H
and called ITPS (incomplete tensor product space). Let P be the
orthogonal projector in H onto H
. For two arbitrary product vectors , , ∈ H
the projectors P and P are either orthogonal or equal. For any U z from (5.1.18)
we have
U z P U
∗
z = P U z ,
(5.1.20)
and the product vectors and U z are weakly equivalent, cf. [227]. If P =
(hence P = ), then and are (strongly) equivalent. The set of all product
vectors weakly equivalent to a product vector form a total set in a closed
subspace of H with the orthogonal projector P
w
. Clearly, P
w
is the sum of all
such P , which correspond to mutually strongly inequivalent product vectors , all
of them being weakly equivalent to . The sum of all mutually strongly inequivalent
P (we use an obvious licence in language) is the unit operator in H .
2
Let A
denotes the C
∗ -subalgebra of the algebra of all bounded operators on H
(denoted by L(H )), generated by the elements
{π j (A) ∈ L(H ) : A ∈ L(H), j ∈ },
(5.1.21)
where L(H) is the algebra of all bounded operators on the Hilbert space H.
For any x ∈ A
, the following relations are valid, [227]:
[x, P ] = [x, U z ] = 0 for all U z , and for all P ,
(5.1.22)
with U z from (5.1.18). If p is another orthogonal projector in L(H ), and for some
product-vector it is p P = p, then
if [x, p] = 0 for all x ∈ A
⇒ p = P or p = 0,
(5.1.23)
1 i.e. all the vectors b for which b j = a j for finite number of indices j ∈ only.
2 We shall use sometimes projectors instead of the corresponding subspaces.