4.2 Extension of CCR by a Quadratic Generator
61
and P + (resp.P − ) are corresponding orthogonal projectors,
P + + P − = I H .
(4.2.19)
Choose a dense invariant linear subset D G of H consisting of infinitely differentiable
vectors of U (G) such, that (as usually)
U π D G ⊂ D G , hence P ± D G ⊂ D G .
(4.2.20)
This condition implies, that P + D G (resp. P − D G ) is dense in H + := P + H (resp. in
H − := P − H). For any ϕ ∈ D
+
G ∪ D
−
G (with D
±
G := P ± D G ), the assumption of 4.2.6
is fulfilled due to (4.2.18). If ϕ is not an eigenvector of A, then the orbit O ϕ is
2n+1-dimensional. Assume, that Aϕ = λϕ. Let ϕ ∈ D
+
G , for definiteness. Since A
is U π -invariant: [A, U π ] = 0, its spectral measure E A commutes with projectors P ± .
Denote for any Borel set B ⊂ R
E
±
A (B) := P ± E A (B), hence E A = E
+
A + E
−
A ,
(4.2.21)
and E
+
A is the spectral measure of the restriction of A to the U (G)-invariant (infinite
dimensional) subspace H + of H. Due to unboundedness of A, we can assume that the
subspace (P + − E
+
A ({λ}))H of H + is nonempty; here E
+
A ({λ}) is the eigenprojector
of P + A corresponding to the eigenvalue λ. Choose a nonzero vector
ϕ
∈ (P + − E
+
A ({λ}))H
(4.2.22)
and assume the normalization ϕ = =ϕ
= 1. Let χ :=
1
√
2
(ϕ
+ ϕ). Since D
+
G is
dense in H + , we can find for arbitrarily small δ > 0 a vector ϕ 0 :
ϕ 0 ∈ D
+
G : :ϕ 0 − χ
2
< δ, ϕ 0 = 1.
(4.2.23)
With δ < 2 −
√
2, the vector ϕ 0 cannot be an eigenvector of A and, moreover, it
satisfies (4.1.15). Hence the corresponding orbit O ϕ 0 is 2n + 1-dimensional. The
manifold structure was proved in 3.1.2.
4.2.10 Let O ϕ (with ϕ ∈ D G ) be a 2n+1-dimensional orbit of U(G) and let
◦ be
the restriction of the standard symplectic form on P(H) onto O ϕ , compare 3.2.2.
According to the previous results (Sects. 3.2 and 4.1),
◦ is a closed two-form of
the maximal rank 2n, hence it is a contact two-form on O ϕ (see, e.g. [1, Chap. 5.1.]).
The equations (4.2.13) determine the characteristic line-bundle of
◦ in terms of
operators C = C ϕ corresponding to generators of stability groups of F ϕ ∈ g
∗ (see
4.2.5) with respect to Ad
∗
(G). The characteristic line bundle of
◦ is integrable,
determining a regular foliation of O ϕ . The factorization of O ϕ with respect to this
foliation is the symplectic manifold M ϕ (symplectomorphic to the classical phase
space T
∗
R
n ) as it was constructed in Sect. 3.2 (for definition of the cotangent bundle
T
∗
(M) of a general manifold M see e.g. [37, A.3.6 Definitions (v)]).
61
and P + (resp.P − ) are corresponding orthogonal projectors,
P + + P − = I H .
(4.2.19)
Choose a dense invariant linear subset D G of H consisting of infinitely differentiable
vectors of U (G) such, that (as usually)
U π D G ⊂ D G , hence P ± D G ⊂ D G .
(4.2.20)
This condition implies, that P + D G (resp. P − D G ) is dense in H + := P + H (resp. in
H − := P − H). For any ϕ ∈ D
+
G ∪ D
−
G (with D
±
G := P ± D G ), the assumption of 4.2.6
is fulfilled due to (4.2.18). If ϕ is not an eigenvector of A, then the orbit O ϕ is
2n+1-dimensional. Assume, that Aϕ = λϕ. Let ϕ ∈ D
+
G , for definiteness. Since A
is U π -invariant: [A, U π ] = 0, its spectral measure E A commutes with projectors P ± .
Denote for any Borel set B ⊂ R
E
±
A (B) := P ± E A (B), hence E A = E
+
A + E
−
A ,
(4.2.21)
and E
+
A is the spectral measure of the restriction of A to the U (G)-invariant (infinite
dimensional) subspace H + of H. Due to unboundedness of A, we can assume that the
subspace (P + − E
+
A ({λ}))H of H + is nonempty; here E
+
A ({λ}) is the eigenprojector
of P + A corresponding to the eigenvalue λ. Choose a nonzero vector
ϕ
∈ (P + − E
+
A ({λ}))H
(4.2.22)
and assume the normalization ϕ = =ϕ
= 1. Let χ :=
1
√
2
(ϕ
+ ϕ). Since D
+
G is
dense in H + , we can find for arbitrarily small δ > 0 a vector ϕ 0 :
ϕ 0 ∈ D
+
G : :ϕ 0 − χ
2
< δ, ϕ 0 = 1.
(4.2.23)
With δ < 2 −
√
2, the vector ϕ 0 cannot be an eigenvector of A and, moreover, it
satisfies (4.1.15). Hence the corresponding orbit O ϕ 0 is 2n + 1-dimensional. The
manifold structure was proved in 3.1.2.
4.2.10 Let O ϕ (with ϕ ∈ D G ) be a 2n+1-dimensional orbit of U(G) and let
◦ be
the restriction of the standard symplectic form on P(H) onto O ϕ , compare 3.2.2.
According to the previous results (Sects. 3.2 and 4.1),
◦ is a closed two-form of
the maximal rank 2n, hence it is a contact two-form on O ϕ (see, e.g. [1, Chap. 5.1.]).
The equations (4.2.13) determine the characteristic line-bundle of
◦ in terms of
operators C = C ϕ corresponding to generators of stability groups of F ϕ ∈ g
∗ (see
4.2.5) with respect to Ad
∗
(G). The characteristic line bundle of
◦ is integrable,
determining a regular foliation of O ϕ . The factorization of O ϕ with respect to this
foliation is the symplectic manifold M ϕ (symplectomorphic to the classical phase
space T
∗
R
n ) as it was constructed in Sect. 3.2 (for definition of the cotangent bundle
T
∗
(M) of a general manifold M see e.g. [37, A.3.6 Definitions (v)]).