3.3 Classical Mechanical Projections of Quantal Dynamics
47
The functions f B for any z-classical B are defined in the both cases by the
formula
f B (x) := T r(P x B).
(3.3.26)
The ‘classical f B ’ is the restriction of the ‘quantal f B ’ to the manifold M := M z .
The classical flow is the specific kind of restriction of the flow U A onto M
(compare (3.3.18) and the note in the last sentence of 3.3.8).
Although the rules for computation of the functions
t → f X (F
A
t [x]), [x] ∈ M z , X = X
∗
∈ U (g),
(3.3.27)
and
t → f X (U A (t)x), x ∈ [x] ∈ M z , X = X
∗
∈ U (g),
(3.3.28)
seem to be very similar, the mutually corresponding functions from (3.3.27) and
(3.3.28) might be radically different for an abstractly defined selfadjoint (z-classical)
operator A. We shall give in the next chapter an example, in which both the functions
from (3.3.27) and (3.3.28) (given by the same X ∈ U (g) and with the same initial
condition x ∈ O z ) are periodic with different periods (and, moreover, with mutually
different dependence of these periods on the initial condition x); the corresponding
orbits in g
∗ :
{F
cl
[x] (t) : t ∈ R} ⊂ g
∗
, with F
cl
[x] (t) : ξ → f X ξ (F
A
t [x]), ξ ∈ g,
(3.3.29)
and the orbit
{F
q
x (t) : t ∈ R} ⊂ g
∗
, where F
q
x (t) : g → R, ξ → F
q
x (t)(ξ ) := f X ξ (U A (t)x),
(3.3.30)
are mutually different closed curves in g
∗
, see 4.1.10.
3.3.10 We expect, contrary to the above mentioned example, that in certain situations
the parametrized curves in g
∗ defined in (3.3.27) and (3.3.28) will be in some sense
close one to another, at least for not too large times t ∈ R. We mean namely such
situations, in which A is the Hamiltonian operator of a ‘realistic’ quantal model and
the initial condition x leads to subsequent evolution U A (t)x, which is sufficiently
well approximated by laws of CM. For some estimates in these directions, they might
be useful Taylor expansions of the functions in (3.3.27) and (3.3.28) in the initial
point t = 0. Set, as usual,
{ f A , f X }
(n)
:= { f A , { f A , f X }
(n−1)
}, { f A , f X }
(0)
:= f X , for n ∈ Z + , (3.3.31)
47
The functions f B for any z-classical B are defined in the both cases by the
formula
f B (x) := T r(P x B).
(3.3.26)
The ‘classical f B ’ is the restriction of the ‘quantal f B ’ to the manifold M := M z .
The classical flow is the specific kind of restriction of the flow U A onto M
(compare (3.3.18) and the note in the last sentence of 3.3.8).
Although the rules for computation of the functions
t → f X (F
A
t [x]), [x] ∈ M z , X = X
∗
∈ U (g),
(3.3.27)
and
t → f X (U A (t)x), x ∈ [x] ∈ M z , X = X
∗
∈ U (g),
(3.3.28)
seem to be very similar, the mutually corresponding functions from (3.3.27) and
(3.3.28) might be radically different for an abstractly defined selfadjoint (z-classical)
operator A. We shall give in the next chapter an example, in which both the functions
from (3.3.27) and (3.3.28) (given by the same X ∈ U (g) and with the same initial
condition x ∈ O z ) are periodic with different periods (and, moreover, with mutually
different dependence of these periods on the initial condition x); the corresponding
orbits in g
∗ :
{F
cl
[x] (t) : t ∈ R} ⊂ g
∗
, with F
cl
[x] (t) : ξ → f X ξ (F
A
t [x]), ξ ∈ g,
(3.3.29)
and the orbit
{F
q
x (t) : t ∈ R} ⊂ g
∗
, where F
q
x (t) : g → R, ξ → F
q
x (t)(ξ ) := f X ξ (U A (t)x),
(3.3.30)
are mutually different closed curves in g
∗
, see 4.1.10.
3.3.10 We expect, contrary to the above mentioned example, that in certain situations
the parametrized curves in g
∗ defined in (3.3.27) and (3.3.28) will be in some sense
close one to another, at least for not too large times t ∈ R. We mean namely such
situations, in which A is the Hamiltonian operator of a ‘realistic’ quantal model and
the initial condition x leads to subsequent evolution U A (t)x, which is sufficiently
well approximated by laws of CM. For some estimates in these directions, they might
be useful Taylor expansions of the functions in (3.3.27) and (3.3.28) in the initial
point t = 0. Set, as usual,
{ f A , f X }
(n)
:= { f A , { f A , f X }
(n−1)
}, { f A , f X }
(0)
:= f X , for n ∈ Z + , (3.3.31)