4.1 The Heisenberg Group (CCR)
55
A
λ
:=
1
2
n
jk=1
a jk P
λ
j P
λ
k + V (Q
λ
),
(4.1.29)
we obtain:
f
λ
A (q, p) := T r( P
(λ)
x A
λ
) =
1
2
n
jk=1
a jk p j p k +
λ
∗ V (q) + b λ .
(4.1.30)
Here b λ is a constant depending on λ as O(λ
2
) and
λ
(q) := λ
−n
|ϕ(−
q
λ
)|
2
(4.1.31)
is a normalized density on R
n , which weakly converges to the Dirac δ-function with
λ → 0. A comparison of flows on O
λ
ϕ (= R
2n
) generated by f
λ
A for various λ is not
easy for given V and ϕ in general.
4.1.9 Let U
λ
A (t) := exp(−
it
λ 2 A
λ
) be the time evolution group corresponding to the
generator (4.1.29) (we set = 1). Let
x
λ
j (t, x) := T r(U
λ
A (t) P
(λ)
x U
λ
A (−t)X
λ
j ) = f
λ
X j
(U
λ
A (t)ϕ
λ
x )
(4.1.32)
be time-evolved quantal expectations of the ‘canonical’ observables X
λ
j with initial
values ϕ
λ
x ∈ O
λ
ϕ (the mapping f
λ from (4.1.28) is here extended to a mapping into
functions on P(H)). The well-known Ehrenfest’s equations are certain equalities
including the functions (4.1.32) and their time-derivatives, which have an analogous
form to that of equations of motion of CM, being in the same time exact consequences of QM. We can write them in the form (with x
λ
:= (q
λ
1 , . . . q
λ
n , p
λ
1 , . . . p
λ
n )
and summation is over 1, 2, . . . n):
d
dt
q
λ
j (t, x) = a jk p
λ
k (t, x) =
∂
∂ p j
f
λ
A (x
λ
(t, x)),
(4.1.33)
d
dt
p
λ
j (t, x) = −T r
U
λ
A (t) P
(λ)
x U
λ
A (−t)
∂V
∂q j
(Q
λ
)
.
(4.1.34)
Here f
λ
A is the classical Hamiltonian function corresponding to the quantal generator
A
λ . The corresponding equations for the classical projection on O
λ
ϕ are of the form:
d
dt
q
λ
j (t, x) cl = a jk p
λ
k (t, x) cl =
∂
∂ p j
f
λ
A (x
λ
(t, x) cl ),
(4.1.35)
d
dt
p
λ
j (t, x) cl = −
∂
∂q j
λ
∗ V (q
λ
(t, x) cl )
= −
∂
∂q j
f
λ
A (x
λ
(t, x) cl ), (4.1.36)
55
A
λ
:=
1
2
n
jk=1
a jk P
λ
j P
λ
k + V (Q
λ
),
(4.1.29)
we obtain:
f
λ
A (q, p) := T r( P
(λ)
x A
λ
) =
1
2
n
jk=1
a jk p j p k +
λ
∗ V (q) + b λ .
(4.1.30)
Here b λ is a constant depending on λ as O(λ
2
) and
λ
(q) := λ
−n
|ϕ(−
q
λ
)|
2
(4.1.31)
is a normalized density on R
n , which weakly converges to the Dirac δ-function with
λ → 0. A comparison of flows on O
λ
ϕ (= R
2n
) generated by f
λ
A for various λ is not
easy for given V and ϕ in general.
4.1.9 Let U
λ
A (t) := exp(−
it
λ 2 A
λ
) be the time evolution group corresponding to the
generator (4.1.29) (we set = 1). Let
x
λ
j (t, x) := T r(U
λ
A (t) P
(λ)
x U
λ
A (−t)X
λ
j ) = f
λ
X j
(U
λ
A (t)ϕ
λ
x )
(4.1.32)
be time-evolved quantal expectations of the ‘canonical’ observables X
λ
j with initial
values ϕ
λ
x ∈ O
λ
ϕ (the mapping f
λ from (4.1.28) is here extended to a mapping into
functions on P(H)). The well-known Ehrenfest’s equations are certain equalities
including the functions (4.1.32) and their time-derivatives, which have an analogous
form to that of equations of motion of CM, being in the same time exact consequences of QM. We can write them in the form (with x
λ
:= (q
λ
1 , . . . q
λ
n , p
λ
1 , . . . p
λ
n )
and summation is over 1, 2, . . . n):
d
dt
q
λ
j (t, x) = a jk p
λ
k (t, x) =
∂
∂ p j
f
λ
A (x
λ
(t, x)),
(4.1.33)
d
dt
p
λ
j (t, x) = −T r
U
λ
A (t) P
(λ)
x U
λ
A (−t)
∂V
∂q j
(Q
λ
)
.
(4.1.34)
Here f
λ
A is the classical Hamiltonian function corresponding to the quantal generator
A
λ . The corresponding equations for the classical projection on O
λ
ϕ are of the form:
d
dt
q
λ
j (t, x) cl = a jk p
λ
k (t, x) cl =
∂
∂ p j
f
λ
A (x
λ
(t, x) cl ),
(4.1.35)
d
dt
p
λ
j (t, x) cl = −
∂
∂q j
λ
∗ V (q
λ
(t, x) cl )
= −
∂
∂q j
f
λ
A (x
λ
(t, x) cl ), (4.1.36)