Chapter 3
Classical Mechanical Projections of QM
3.1 Orbits of Lie Group Actions on P(H)
3.1.1 Let U be a weakly continuous unitary representation of a connected Lie group
G in the Hilbert space H, and X ξ be the selfadjoint generator of the one-parameter
subgroup of U (G) corresponding to an arbitrary element ξ of the Lie algebra g of
G, as it was defined in (2.3.20). Let D G ⊂ H be the Gårding domain of U (G) [13,
11.1.8.], i.e. a dense U (G)-invariant set of vectors x ∈ H, for which the functions
g → U (g)x (g ∈ G) are infinitely differentiable. We shall denote by A G (⊂ H) the
dense set of analytic vectors of U (G) invariant with respect to the action of U (G).
For x ∈ A G , not only the functions g → U (g)x are real analytic (resp. the functions
t → U (exp(tξ))x are complex analytic in a neighbourhood
1 of real axis for any
ξ ∈ g) in the norm of H, but also A G is invariant and analytic with respect to the Lie
algebra U (g) of generators X ξ (ξ ∈ g); for x ∈ A G , also X ξ x ∈ A G (∀ξ ∈ U (g))
and for any basis {X j ∈ U (g) : j = 1, 2, . . . d := dim G} ⊂ U (g) and x ∈ A G there
is some t = 0 such that
∞
n=0
|t|
n
n!
d
j 1 ,... j n =1
X j 1 . . . X j n x < ∞,
(3.1.1)
compare [13, Chap. 11, §3].
Let U(G) be the projection of U (G) onto P(H), i.e. U(G) is a realization of G
in a continuous group of symplectic isometries of (P(H), ,).
2 For any x ∈ P(H),
define the orbit O x := G · x (we shall use also the notation g · x := U(g)x):
O x = O g·x := {z ∈ P(H) : z = g · x, g ∈ G}.
(3.1.2)
1 In the following, if not explicitly mentioned different, the word ‘neighbourhood’ in a topological
space will mean ‘an open neighbourhood’.
2 Remember that if x ≡ P x ∈ P(H), then U(g)x ≡ P U (g)x ≡ U (g)P x U (g −1 ).
© Springer Nature Switzerland AG 2020, corrected publication 2020
P. Bóna, Classical Systems in Quantum Mechanics,
https://doi.org/10.1007/978-3-030-45070-0_3
35
Classical Mechanical Projections of QM
3.1 Orbits of Lie Group Actions on P(H)
3.1.1 Let U be a weakly continuous unitary representation of a connected Lie group
G in the Hilbert space H, and X ξ be the selfadjoint generator of the one-parameter
subgroup of U (G) corresponding to an arbitrary element ξ of the Lie algebra g of
G, as it was defined in (2.3.20). Let D G ⊂ H be the Gårding domain of U (G) [13,
11.1.8.], i.e. a dense U (G)-invariant set of vectors x ∈ H, for which the functions
g → U (g)x (g ∈ G) are infinitely differentiable. We shall denote by A G (⊂ H) the
dense set of analytic vectors of U (G) invariant with respect to the action of U (G).
For x ∈ A G , not only the functions g → U (g)x are real analytic (resp. the functions
t → U (exp(tξ))x are complex analytic in a neighbourhood
1 of real axis for any
ξ ∈ g) in the norm of H, but also A G is invariant and analytic with respect to the Lie
algebra U (g) of generators X ξ (ξ ∈ g); for x ∈ A G , also X ξ x ∈ A G (∀ξ ∈ U (g))
and for any basis {X j ∈ U (g) : j = 1, 2, . . . d := dim G} ⊂ U (g) and x ∈ A G there
is some t = 0 such that
∞
n=0
|t|
n
n!
d
j 1 ,... j n =1
X j 1 . . . X j n x < ∞,
(3.1.1)
compare [13, Chap. 11, §3].
Let U(G) be the projection of U (G) onto P(H), i.e. U(G) is a realization of G
in a continuous group of symplectic isometries of (P(H), ,).
2 For any x ∈ P(H),
define the orbit O x := G · x (we shall use also the notation g · x := U(g)x):
O x = O g·x := {z ∈ P(H) : z = g · x, g ∈ G}.
(3.1.2)
1 In the following, if not explicitly mentioned different, the word ‘neighbourhood’ in a topological
space will mean ‘an open neighbourhood’.
2 Remember that if x ≡ P x ∈ P(H), then U(g)x ≡ P U (g)x ≡ U (g)P x U (g −1 ).
© Springer Nature Switzerland AG 2020, corrected publication 2020
P. Bóna, Classical Systems in Quantum Mechanics,
https://doi.org/10.1007/978-3-030-45070-0_3
35