7.3 Quantum Domino
171
7.3.2 Let the C
∗ -algebra of observables A be the C
∗ -tensor product of countably infinite set of copies of the algebra of complex 2 × 2 matrices generated by
the spin creation and annihilation operators a
∗
j , a j , j ∈ Z satisfying the following
(anti)commutation relations
a i a j − a j a i =: [a i , a j ] = [a
∗
i , a j ] = 0, i = j
(7.3.1)
a i a i = 0, a
∗
i a i + a i a
∗
i = 1,
for all i, j ∈ Z. The algebra A is simple, hence each its nonzero representation is
faithful. We shall describe the dynamics in A in the “vacuum” representation, i.e. in
the GNS representation corresponding to the “vacuum state” ω 0 ∈ A
∗
+1 ≡ S(A) that
is given by the relation
ω 0 (a
∗
j a j ) = 0, for all j ∈ Z.
(7.3.2)
This state is pure, hence the GNS representation is irreducible. We shall call the
spins in this state to be “pointing down”, to be specific in verbal expression. Let the
cyclic vector (“vacuum” in the lattice gas terminology) of this representation be
denoted by 0 , i.e. for all elements x ∈ A it is
ω 0 (x) = = 0 |x| 0 , for all x ∈ A.
(7.3.3)
Here and in the following we shall denote the elements of A and their operator
representatives in the considered irreducible Hilbert space representation by the same
symbols. Let us denote this Hilbert space by H vac .
Let us define a “finite-subchain Hamiltonian” H ( j,k) :
H ( j,k) :=
k−2
n= j+1
a
∗
n a n (a
∗
n+1 + a n+1 ) a n+2 a
∗
n+2 .
(7.3.4)
Local time evolution automorphisms of A are given by
τ
n
t (x) := exp(it H (−n,n) ) x exp(−it H (−n,n) ),
(7.3.5)
and the norm limits
τ t (x) := norm- lim
n→∞
τ
n
t (x)
(7.3.6)
determine the time evolution in A (in the “Heisenberg picture”).
In our vacuum representation, this evolution is determined by a selfadjoint Hamiltonian H ,
τ t (x) = e
it H x e
−it H
.
(7.3.7)
171
7.3.2 Let the C
∗ -algebra of observables A be the C
∗ -tensor product of countably infinite set of copies of the algebra of complex 2 × 2 matrices generated by
the spin creation and annihilation operators a
∗
j , a j , j ∈ Z satisfying the following
(anti)commutation relations
a i a j − a j a i =: [a i , a j ] = [a
∗
i , a j ] = 0, i = j
(7.3.1)
a i a i = 0, a
∗
i a i + a i a
∗
i = 1,
for all i, j ∈ Z. The algebra A is simple, hence each its nonzero representation is
faithful. We shall describe the dynamics in A in the “vacuum” representation, i.e. in
the GNS representation corresponding to the “vacuum state” ω 0 ∈ A
∗
+1 ≡ S(A) that
is given by the relation
ω 0 (a
∗
j a j ) = 0, for all j ∈ Z.
(7.3.2)
This state is pure, hence the GNS representation is irreducible. We shall call the
spins in this state to be “pointing down”, to be specific in verbal expression. Let the
cyclic vector (“vacuum” in the lattice gas terminology) of this representation be
denoted by 0 , i.e. for all elements x ∈ A it is
ω 0 (x) = = 0 |x| 0 , for all x ∈ A.
(7.3.3)
Here and in the following we shall denote the elements of A and their operator
representatives in the considered irreducible Hilbert space representation by the same
symbols. Let us denote this Hilbert space by H vac .
Let us define a “finite-subchain Hamiltonian” H ( j,k) :
H ( j,k) :=
k−2
n= j+1
a
∗
n a n (a
∗
n+1 + a n+1 ) a n+2 a
∗
n+2 .
(7.3.4)
Local time evolution automorphisms of A are given by
τ
n
t (x) := exp(it H (−n,n) ) x exp(−it H (−n,n) ),
(7.3.5)
and the norm limits
τ t (x) := norm- lim
n→∞
τ
n
t (x)
(7.3.6)
determine the time evolution in A (in the “Heisenberg picture”).
In our vacuum representation, this evolution is determined by a selfadjoint Hamiltonian H ,
τ t (x) = e
it H x e
−it H
.
(7.3.7)