7.5 The X-Y Chain as a Measuring Device
197
corresponding comments; the reader could easily add the necessary specifications
on his own.
We shall use the known formula to express the automorphism (7.5.3):
e
it H Ae
−it H
=
∞
m=0
(it)
m
m!
[H, A]
(m)
,
(7.5.5)
where [H, A]
(0)
:= A, and higher elements are recurrently defined with a help of the
commutator [H, A]
(1)
:= [H, A] ≡ H A − AH:
[H, A]
(m+1)
:= [H, [H, A]
(m)
].
(7.5.6)
The application of (7.5.5) to norm-bounded elements A (with also H → H n ) makes
no principal problems, but calculations of time evolved elements in (7.5.5) of e.g.
A → a j is technically complicated and it is much easier to work, instead with the
spin operators a j , with elements b j ∈ A satisfying the Fermi canonical anticommutation relations (CAR). This can be reached by the Jordan-Wigner transformation
([171], and also [106, Chaps. 3, 2]):
b j := a j
j−1
k=−n−1
(1 − 2a
∗
k a k ), b
∗
j := (b j )
∗
,
(7.5.7)
for | j| ≤ n. Although these elements become to be nonlocal with n → ∞, their
bilinear combinations remain local, and this is sufficient for our calculations. Note
also that there is the inverse transformation expressing a j in terms of b j , which has
the same form as (7.5.7) after the exchange a j,k ↔ b j,k .
The elements b j , b k , j, k ∈ [−n, n] satisfy CAR:
[b j , b k ] + ≡ 0, b j b
∗
k + b
∗
k b j =: [b j , b
∗
k ] + = δ jk .
(7.5.8)
The local Hamiltonians H n from (7.5.1) can be written now as
H n =
κ
2
n−1
j=−n
(b
∗
j b j+1 + b
∗
j+1 b j ).
(7.5.9)
We can calculate now the time evolution of the elements b j ∈ A. We shall need
later the estimates for τ t (a
∗
j a j ), and due to equality a
∗
j a j = b
∗
j b j the explicit expressions for τ t (b j ) will be sufficient for us. We can use (7.5.5) to calculate τ t (b j ). One
easily checks that the multiple commutators have the form:
[H, b j ]
(m)
=
p
c
(m)
j ( p)b p ,
(7.5.10)
197
corresponding comments; the reader could easily add the necessary specifications
on his own.
We shall use the known formula to express the automorphism (7.5.3):
e
it H Ae
−it H
=
∞
m=0
(it)
m
m!
[H, A]
(m)
,
(7.5.5)
where [H, A]
(0)
:= A, and higher elements are recurrently defined with a help of the
commutator [H, A]
(1)
:= [H, A] ≡ H A − AH:
[H, A]
(m+1)
:= [H, [H, A]
(m)
].
(7.5.6)
The application of (7.5.5) to norm-bounded elements A (with also H → H n ) makes
no principal problems, but calculations of time evolved elements in (7.5.5) of e.g.
A → a j is technically complicated and it is much easier to work, instead with the
spin operators a j , with elements b j ∈ A satisfying the Fermi canonical anticommutation relations (CAR). This can be reached by the Jordan-Wigner transformation
([171], and also [106, Chaps. 3, 2]):
b j := a j
j−1
k=−n−1
(1 − 2a
∗
k a k ), b
∗
j := (b j )
∗
,
(7.5.7)
for | j| ≤ n. Although these elements become to be nonlocal with n → ∞, their
bilinear combinations remain local, and this is sufficient for our calculations. Note
also that there is the inverse transformation expressing a j in terms of b j , which has
the same form as (7.5.7) after the exchange a j,k ↔ b j,k .
The elements b j , b k , j, k ∈ [−n, n] satisfy CAR:
[b j , b k ] + ≡ 0, b j b
∗
k + b
∗
k b j =: [b j , b
∗
k ] + = δ jk .
(7.5.8)
The local Hamiltonians H n from (7.5.1) can be written now as
H n =
κ
2
n−1
j=−n
(b
∗
j b j+1 + b
∗
j+1 b j ).
(7.5.9)
We can calculate now the time evolution of the elements b j ∈ A. We shall need
later the estimates for τ t (a
∗
j a j ), and due to equality a
∗
j a j = b
∗
j b j the explicit expressions for τ t (b j ) will be sufficient for us. We can use (7.5.5) to calculate τ t (b j ). One
easily checks that the multiple commutators have the form:
[H, b j ]
(m)
=
p
c
(m)
j ( p)b p ,
(7.5.10)