36
3 Classical Mechanical Projections of QM
Let K
◦
z := K
◦
:= {h ∈ G : U(h)z = z} be the stability (or ‘isotropy’) group
of the point z ∈ O x = O z . Because P(H) is a Hausdorff space and U is continuous,
the group K
◦ is closed, hence it is a Lie subgroup of G. The space G/K
◦ of left
cosets gK
◦
⊂ G is an analytic manifold (with the analytic structure coming from
G via the natural projection, [152, Ch.II, Theorem 4.2]) and it is bijectively and
continuously mapped onto O z by the mapping u : m := gK
◦
→ U(g)z. The orbit
is not, in general, closed in P(H) and it need not be a submanifold of P(H), cf. also
[37, Proposition 2.1.5] & [47]. The mapping u induces, however, a manifold structure
on O z from the analytic manifold G/K
◦
. This manifold structure is not in general
consistent with the relative topology of O z in P(H). If the map is differentiable, then
we have:
3.1.2 Proposition. Let u (defined as above) be continuously differentiable in a
neighbourhood of a point m ∈ G/K
◦
, where K
◦
:= {h ∈ G : U(h)z = z}. Then
there is a neighbourhood N m of m such, that the restriction of u on N m is a diffeomorphism of N m onto the submanifold u(N m ) of P(H). If z ∈ PD G (resp. z ∈ PA G ),
then each point m ∈ G/K
◦ has a neighbourhood N m , which is C
∞
−diffeomorphic
(resp. analytically diffeomorphic) to u(N m ), with the submanifold structure from
P(H); in this case, the orbit O z is an immersed submanifold of P(H).
Proof. Bijectivity of u : G/K
◦
→ O z and differentiability in a neighbourhood of m
imply, that the tangent mapping T m u : T m (G/K
◦
) → T u(m) P(H) is an isomorphism
onto a finite dimensional subspace of the tangent space of P(H) at u(m). Since each
finite dimensional real subspace of a Banach space is complementable, the restriction
of u to a neighbourhood is an immersion. Hence, there is a neighbourhood N m of m
satisfying the first statement, compare [74, p. 549]. The rest is a consequence of the
invariance of PD G and PA G as well as of the inverse mapping theorem, see also
[51].
3.1.3 We shall assume in the following that z ∈ PA G , for the orbit O z which we
shall consider. Many of the following considerations are valid, however, also for
orbits passing through z ∈ PD G . Let σ ξ (ξ ∈ g) be the (densely defined) vector field
on P(H) corresponding to the generator X ξ according to 2.3.9 and 2.3.5. According
to the definition of O z , for any x ∈ O z , the vectors σ ξ (x) (ξ ∈ g) are well defined,
they span T x O z and depend analytically on x ∈ O z . (Note: Here and in the following,
we use without comments the topology on O z inherited from G/K
◦ via the mapping
u introduced in 3.1.1) Let K
◦
x be the stability subgroup of G at the point x ∈ O z , and
let its Lie algebra be k
◦
x . Then the Lie algebra g of G is the direct sum
g = m
◦
x ⊕ k
◦
x
(3.1.3)
of two vector spaces (the choice of m
◦
x ⊂ g is nonunique). If {ξ j ∈ g : j = 1, 2, . . . n :=
dim O z } is a basis of m
◦
x , then σ ξ j span tangent spaces to O z in any point y lying in
some neighbourhood of x in O z . Then integral curves of σ ξ j ( j = 1, 2, . . . n) can be
used to introduce a natural coordinate system on O z in a neighbourhood of x (see
[152, Ch.II. Lemma 4.1]). In these coordinates, the point
3 Classical Mechanical Projections of QM
Let K
◦
z := K
◦
:= {h ∈ G : U(h)z = z} be the stability (or ‘isotropy’) group
of the point z ∈ O x = O z . Because P(H) is a Hausdorff space and U is continuous,
the group K
◦ is closed, hence it is a Lie subgroup of G. The space G/K
◦ of left
cosets gK
◦
⊂ G is an analytic manifold (with the analytic structure coming from
G via the natural projection, [152, Ch.II, Theorem 4.2]) and it is bijectively and
continuously mapped onto O z by the mapping u : m := gK
◦
→ U(g)z. The orbit
is not, in general, closed in P(H) and it need not be a submanifold of P(H), cf. also
[37, Proposition 2.1.5] & [47]. The mapping u induces, however, a manifold structure
on O z from the analytic manifold G/K
◦
. This manifold structure is not in general
consistent with the relative topology of O z in P(H). If the map is differentiable, then
we have:
3.1.2 Proposition. Let u (defined as above) be continuously differentiable in a
neighbourhood of a point m ∈ G/K
◦
, where K
◦
:= {h ∈ G : U(h)z = z}. Then
there is a neighbourhood N m of m such, that the restriction of u on N m is a diffeomorphism of N m onto the submanifold u(N m ) of P(H). If z ∈ PD G (resp. z ∈ PA G ),
then each point m ∈ G/K
◦ has a neighbourhood N m , which is C
∞
−diffeomorphic
(resp. analytically diffeomorphic) to u(N m ), with the submanifold structure from
P(H); in this case, the orbit O z is an immersed submanifold of P(H).
Proof. Bijectivity of u : G/K
◦
→ O z and differentiability in a neighbourhood of m
imply, that the tangent mapping T m u : T m (G/K
◦
) → T u(m) P(H) is an isomorphism
onto a finite dimensional subspace of the tangent space of P(H) at u(m). Since each
finite dimensional real subspace of a Banach space is complementable, the restriction
of u to a neighbourhood is an immersion. Hence, there is a neighbourhood N m of m
satisfying the first statement, compare [74, p. 549]. The rest is a consequence of the
invariance of PD G and PA G as well as of the inverse mapping theorem, see also
[51].
3.1.3 We shall assume in the following that z ∈ PA G , for the orbit O z which we
shall consider. Many of the following considerations are valid, however, also for
orbits passing through z ∈ PD G . Let σ ξ (ξ ∈ g) be the (densely defined) vector field
on P(H) corresponding to the generator X ξ according to 2.3.9 and 2.3.5. According
to the definition of O z , for any x ∈ O z , the vectors σ ξ (x) (ξ ∈ g) are well defined,
they span T x O z and depend analytically on x ∈ O z . (Note: Here and in the following,
we use without comments the topology on O z inherited from G/K
◦ via the mapping
u introduced in 3.1.1) Let K
◦
x be the stability subgroup of G at the point x ∈ O z , and
let its Lie algebra be k
◦
x . Then the Lie algebra g of G is the direct sum
g = m
◦
x ⊕ k
◦
x
(3.1.3)
of two vector spaces (the choice of m
◦
x ⊂ g is nonunique). If {ξ j ∈ g : j = 1, 2, . . . n :=
dim O z } is a basis of m
◦
x , then σ ξ j span tangent spaces to O z in any point y lying in
some neighbourhood of x in O z . Then integral curves of σ ξ j ( j = 1, 2, . . . n) can be
used to introduce a natural coordinate system on O z in a neighbourhood of x (see
[152, Ch.II. Lemma 4.1]). In these coordinates, the point