3.3 Classical Mechanical Projections of Quantal Dynamics
43
converges absolutely for all x ∈ O z . In an analogy with the constructions of the
preceding sections, the function f A will be considered as a candidate for a classical
observable corresponding to the quantal observable A. We shall require that
f A ∈ C
∞
(O z ).
(3.3.3)
This requirement is fulfilled in the following situation:
3.3.3 Lemma. Let E(g) be the linear space of all polynomials in selfadjoint generators X ξ (ξ ∈ g) of U (G) with complex coefficients. Assume that for a fixed z ∈ PA G
and for any x ∈ O z and any E ∈ E(g) there is an open neighbourhood N (x, E) of
the identity e ∈ G such, that the function
g → →AEU (g)x (x ∈ x)
(3.3.4)
is uniformly bounded on N (x, E). Here A is a given symmetric operator on H
containing E(g)U (G)z := {EU (g)z : E ∈ E(g), g ∈ G} in its domain D(A), z ∈ z.
Set f A (x) := (x, Ax) for x = 1, x ∈ x ∈ O z . Then f A is infinitely differentiable
on O z .
Proof. It suffices to prove infinite differentiability of the function g → f A (g · x)
defined on G. For any E 1 , E 2 ∈ E(g) the functions g → E j U (g)x ( j = 1, 2) are
norm-analytic according to (3.1.1), see also [13]. Consequently, the function
(g 1 ; g 2 ) → (E 1 U (g 1 )x, AE 2 U (g 2 )x) from G × G to C
(3.3.5)
is infinitely differentiable in each variable g 1 , g 2 separately and any partial derivative
(in the direction of some one parameter subgroup of G) has the form (3.3.5) (with
some other E j ’s). To prove differentiability of
g → (E 1 U (g)x, AE 2 U (g)x),
(3.3.6)
it suffices to prove simultaneous continuity of all functions of the form (3.3.5) in both
variables g 1 , g 2 . It follows, however, from the assumption of uniform boundedness
on N (x, E 2 ), analyticity of U (g)x with respect to U (G) and continuity of U (g):
|(E 1 U (g 1 )x, AE 2 U (g 2 )x) − (E 1 x, AE 2 x)| ≤
E 1 (U (g 1 ) − I )x · ·AE 2 U (g 2 )x + +E 2 (U (g 2 ) − I )x · ·AE 1 x. (3.3.7)
This concludes the proof.
3.3.4 If the assumptions of the preceding lemma are valid for A, the explicit expressions for the partial derivatives ∂ ξ f A along the curves t → ex p(tξ) · x have the form
(x = 1, ξ, η ∈ g):
43
converges absolutely for all x ∈ O z . In an analogy with the constructions of the
preceding sections, the function f A will be considered as a candidate for a classical
observable corresponding to the quantal observable A. We shall require that
f A ∈ C
∞
(O z ).
(3.3.3)
This requirement is fulfilled in the following situation:
3.3.3 Lemma. Let E(g) be the linear space of all polynomials in selfadjoint generators X ξ (ξ ∈ g) of U (G) with complex coefficients. Assume that for a fixed z ∈ PA G
and for any x ∈ O z and any E ∈ E(g) there is an open neighbourhood N (x, E) of
the identity e ∈ G such, that the function
g → →AEU (g)x (x ∈ x)
(3.3.4)
is uniformly bounded on N (x, E). Here A is a given symmetric operator on H
containing E(g)U (G)z := {EU (g)z : E ∈ E(g), g ∈ G} in its domain D(A), z ∈ z.
Set f A (x) := (x, Ax) for x = 1, x ∈ x ∈ O z . Then f A is infinitely differentiable
on O z .
Proof. It suffices to prove infinite differentiability of the function g → f A (g · x)
defined on G. For any E 1 , E 2 ∈ E(g) the functions g → E j U (g)x ( j = 1, 2) are
norm-analytic according to (3.1.1), see also [13]. Consequently, the function
(g 1 ; g 2 ) → (E 1 U (g 1 )x, AE 2 U (g 2 )x) from G × G to C
(3.3.5)
is infinitely differentiable in each variable g 1 , g 2 separately and any partial derivative
(in the direction of some one parameter subgroup of G) has the form (3.3.5) (with
some other E j ’s). To prove differentiability of
g → (E 1 U (g)x, AE 2 U (g)x),
(3.3.6)
it suffices to prove simultaneous continuity of all functions of the form (3.3.5) in both
variables g 1 , g 2 . It follows, however, from the assumption of uniform boundedness
on N (x, E 2 ), analyticity of U (g)x with respect to U (G) and continuity of U (g):
|(E 1 U (g 1 )x, AE 2 U (g 2 )x) − (E 1 x, AE 2 x)| ≤
E 1 (U (g 1 ) − I )x · ·AE 2 U (g 2 )x + +E 2 (U (g 2 ) − I )x · ·AE 1 x. (3.3.7)
This concludes the proof.
3.3.4 If the assumptions of the preceding lemma are valid for A, the explicit expressions for the partial derivatives ∂ ξ f A along the curves t → ex p(tξ) · x have the form
(x = 1, ξ, η ∈ g):