60
4 Examples of Classical Mechanical Projections
4.2.5 Let C := C ϕ have the form (4.2.9) and let (4.2.10) be fulfilled. Then C
satisfies the system of linear equations:
T r(P ϕ [C, X j ]) = 0, j = 1, 2, . . . 2n,
(4.2.13)
T r(P ϕ [C, A]) = 0,
(4.2.14)
where (4.2.14) follows from (4.2.13). The equations (4.2.13) have unique solution
C of the form (4.2.9) for any ϕ ∈ D G , even if the relation (4.2.10) is not fulfilled:
c
j
ϕ = h
jk T r(P ϕ X k ).
(4.2.15)
The corresponding operator C ϕ represents the generator of the isotropy subgroup
K ϕ ⊂ G at the point F ϕ ∈ g
∗ in the Ad
∗
(G)-representation; here F ϕ (ξ) := T r(P ϕ X ξ )
for ξ ∈ g, compare 3.2.3 and 3.2.6. From (4.2.15) and (4.2.9), we have immediately:
4.2.6 Lemma. If ϕ ∈ O ϕ is chosen such that T r (P ϕ X j ) = 0 for all j =
1, 2, . . . 2n, then C ϕ = −A.
4.2.7 Proposition. The orbit O ϕ of U(G) is 2n-dimensional iff it contains an eigenprojector P ϕ of A, i.e. iff for some ϕ ∈ O ϕ it is
T r(P ϕ A
2
) =
T r(P ϕ A)
2 .
(4.2.16)
Proof. In any orbit lying in D G there is a point ϕ satisfying the conditions of the
Lemma 4.2.6, compare 4.1.5. The assertion is an immediate consequence of the
Lemmas 4.2.6 and 4.2.4.
4.2.8 Corollaries. (i) Let ϕ 0 ∈ U (G)ϕ satisfy (4.1.15). If ϕ is an eigenvector of
A, then also ϕ 0 is an eigenvector of A, if ϕ ∈ D G .
(ii) If ϕ 0 ∈ D G max satisfies (4.1.15), then those relations are satisfied by all the
vectors
ϕ
A
t := exp(−it A)ϕ 0
(4.2.17)
for all t ∈ R and all the choices of A; (4.2.8).
4.2.9 Proposition. For any choice of A in 4.2.3 there is in P(H) a 2n+1 - dimensional orbit of the corresponding representation U(G) (defined in 4.2.3), which is
an infinitely differentiable immersed submanifold of P(H).
Proof. Remember that any A is an unbounded selfadjoint operator. Let U π := P + −
P − be the ‘parity operator’ defined by
3
U
∗
π = U
−1
π = U π , U π X j U π = −X j ( j = 1, 2, . . . 2n),
(4.2.18)
3 In H ≡ L 2 (R n , d n x), it is defined as [U π ψ](x) := ψ(−x), ∀ψ ∈ H, x ∈ R n .
4 Examples of Classical Mechanical Projections
4.2.5 Let C := C ϕ have the form (4.2.9) and let (4.2.10) be fulfilled. Then C
satisfies the system of linear equations:
T r(P ϕ [C, X j ]) = 0, j = 1, 2, . . . 2n,
(4.2.13)
T r(P ϕ [C, A]) = 0,
(4.2.14)
where (4.2.14) follows from (4.2.13). The equations (4.2.13) have unique solution
C of the form (4.2.9) for any ϕ ∈ D G , even if the relation (4.2.10) is not fulfilled:
c
j
ϕ = h
jk T r(P ϕ X k ).
(4.2.15)
The corresponding operator C ϕ represents the generator of the isotropy subgroup
K ϕ ⊂ G at the point F ϕ ∈ g
∗ in the Ad
∗
(G)-representation; here F ϕ (ξ) := T r(P ϕ X ξ )
for ξ ∈ g, compare 3.2.3 and 3.2.6. From (4.2.15) and (4.2.9), we have immediately:
4.2.6 Lemma. If ϕ ∈ O ϕ is chosen such that T r (P ϕ X j ) = 0 for all j =
1, 2, . . . 2n, then C ϕ = −A.
4.2.7 Proposition. The orbit O ϕ of U(G) is 2n-dimensional iff it contains an eigenprojector P ϕ of A, i.e. iff for some ϕ ∈ O ϕ it is
T r(P ϕ A
2
) =
T r(P ϕ A)
2 .
(4.2.16)
Proof. In any orbit lying in D G there is a point ϕ satisfying the conditions of the
Lemma 4.2.6, compare 4.1.5. The assertion is an immediate consequence of the
Lemmas 4.2.6 and 4.2.4.
4.2.8 Corollaries. (i) Let ϕ 0 ∈ U (G)ϕ satisfy (4.1.15). If ϕ is an eigenvector of
A, then also ϕ 0 is an eigenvector of A, if ϕ ∈ D G .
(ii) If ϕ 0 ∈ D G max satisfies (4.1.15), then those relations are satisfied by all the
vectors
ϕ
A
t := exp(−it A)ϕ 0
(4.2.17)
for all t ∈ R and all the choices of A; (4.2.8).
4.2.9 Proposition. For any choice of A in 4.2.3 there is in P(H) a 2n+1 - dimensional orbit of the corresponding representation U(G) (defined in 4.2.3), which is
an infinitely differentiable immersed submanifold of P(H).
Proof. Remember that any A is an unbounded selfadjoint operator. Let U π := P + −
P − be the ‘parity operator’ defined by
3
U
∗
π = U
−1
π = U π , U π X j U π = −X j ( j = 1, 2, . . . 2n),
(4.2.18)
3 In H ≡ L 2 (R n , d n x), it is defined as [U π ψ](x) := ψ(−x), ∀ψ ∈ H, x ∈ R n .