40
3 Classical Mechanical Projections of QM
F x ([ξ, η]) = 0, ∀η ∈ g.
(3.2.6)
A trivial consequence of this is, according to (3.2.2), the
3.2.5 Proposition.
◦
x (σ ξ , σ η ) = 0 for all η ∈ g iff ξ ∈ k x .
3.2.6 We can decompose O z into equivalence classes
[x] := {g · x : g ∈ K x }, x ∈ O z (z ∈ PA G ).
(3.2.7)
The action of G on O z is analytic, and [x] are analytic submanifolds of O z (if
O z is endowed with the topology of G/K
◦
x ) which are mutually diffeomorphic for
all x ∈ O z . Hence O z can-be considered as a fibred manifold with a typical fibre
diffeomorphic to K z · z = [z], which is in turn diffeomorphic to K x /K
◦
x (x ∈ O z ).
Let us denote the base space by M = M z :
M := M z := {[x] : x ∈ O z },
(3.2.8)
which is endowed with the natural factor topology given by the continuity and openness condition on the projection
p M : O z → M z , x → p M (x) := [x].
(3.2.9)
From the definitions (3.1.5) of F x and of the action of G on F x in (3.2.4), we see
that [x] are exactly those subsets of O z , on which expectations of all the observables
X ξ (ξ ∈ g) remain constant.
3.2.7 Lemma.
◦
h·x (σ ξ , σ η ) =
◦
x (σ ξ , σ η ) for all h ∈ K x and all η, ξ ∈ g.
Proof. Immediate from (3.2.2) and the definition of K x .
3.2.8 Let p M∗ := T p M : T O z → T M z be the tangent mapping corresponding to
the natural projection (3.2.9). For a general vector field σ on O z , the vectors T p M σ (x)
are mutually different for various choices of x ∈ [z]. Let, however, t → g(t) be any
differentiable curve in G. Then curves t → g(t) · x and c h : t → g(t)h · x for any
h ∈ K x are projected by p M onto the same curve t → [g(t) · x] in M z . This is true
due to the validity of
[g · x] = K g·x g · x = gK x g
−1 g · x = gK x · x,
(3.2.10)
for all g ∈ G,
[gh · x] = gh K x · x = gK x · x = [g · x], ∀h ∈ K x , g ∈ G.
(3.2.11)
Hence tangent vectors ˙
c h ∈ T h·x O z corresponding to the curves c h with g(t =
0) := e have identical projections T p M ( ˙
c h ) = T p M ( ˙
c e ) ∈ T [x] M z for all h ∈ K x . If
we set g(t) := exp(tξ), i.e. ˙
c h = σ ξ (h · x), then we have obtained:
3 Classical Mechanical Projections of QM
F x ([ξ, η]) = 0, ∀η ∈ g.
(3.2.6)
A trivial consequence of this is, according to (3.2.2), the
3.2.5 Proposition.
◦
x (σ ξ , σ η ) = 0 for all η ∈ g iff ξ ∈ k x .
3.2.6 We can decompose O z into equivalence classes
[x] := {g · x : g ∈ K x }, x ∈ O z (z ∈ PA G ).
(3.2.7)
The action of G on O z is analytic, and [x] are analytic submanifolds of O z (if
O z is endowed with the topology of G/K
◦
x ) which are mutually diffeomorphic for
all x ∈ O z . Hence O z can-be considered as a fibred manifold with a typical fibre
diffeomorphic to K z · z = [z], which is in turn diffeomorphic to K x /K
◦
x (x ∈ O z ).
Let us denote the base space by M = M z :
M := M z := {[x] : x ∈ O z },
(3.2.8)
which is endowed with the natural factor topology given by the continuity and openness condition on the projection
p M : O z → M z , x → p M (x) := [x].
(3.2.9)
From the definitions (3.1.5) of F x and of the action of G on F x in (3.2.4), we see
that [x] are exactly those subsets of O z , on which expectations of all the observables
X ξ (ξ ∈ g) remain constant.
3.2.7 Lemma.
◦
h·x (σ ξ , σ η ) =
◦
x (σ ξ , σ η ) for all h ∈ K x and all η, ξ ∈ g.
Proof. Immediate from (3.2.2) and the definition of K x .
3.2.8 Let p M∗ := T p M : T O z → T M z be the tangent mapping corresponding to
the natural projection (3.2.9). For a general vector field σ on O z , the vectors T p M σ (x)
are mutually different for various choices of x ∈ [z]. Let, however, t → g(t) be any
differentiable curve in G. Then curves t → g(t) · x and c h : t → g(t)h · x for any
h ∈ K x are projected by p M onto the same curve t → [g(t) · x] in M z . This is true
due to the validity of
[g · x] = K g·x g · x = gK x g
−1 g · x = gK x · x,
(3.2.10)
for all g ∈ G,
[gh · x] = gh K x · x = gK x · x = [g · x], ∀h ∈ K x , g ∈ G.
(3.2.11)
Hence tangent vectors ˙
c h ∈ T h·x O z corresponding to the curves c h with g(t =
0) := e have identical projections T p M ( ˙
c h ) = T p M ( ˙
c e ) ∈ T [x] M z for all h ∈ K x . If
we set g(t) := exp(tξ), i.e. ˙
c h = σ ξ (h · x), then we have obtained: