4.3 Notes on Other Examples
71
p
+∗
M
M
+ =
◦
+ .
(4.3.29)
The Proposition 3.2.10 is a special case of this assertion.
Note: In the above presented construction of the symplectic manifold (M
+
ϕ , ,
M
+ ),
we did not use any specific properties of the projector P + and of the group action
U (G N ). These properties enter in constructions of specific orbits.
4.3.7 We shall specify here the previous construction to the case of G N := N -
fold direct product of 2n + 1—dimensional Heisenberg group G with infinitedimensional unitary irreducible representations U in H. The linear space U N (g N ) is
spanned by elements
6
X ξ :=
N
j=1
X
j
ξ with any X
j
ξ ∈ U (g), ξ ∈ g N ,
(4.3.30)
where the index j has the following meaning: If ϕ ∈ H N has the form
ϕ := ϕ 1 ⊗ ϕ 2 ⊗ · · · ⊗ ϕ N ,
(4.3.31)
then the linear operator X
j on H N corresponds to an (equally denoted) operator on
H by:
X
j
ϕ := ϕ 1 ⊗ ϕ 2 ⊗ · · · ⊗ X
j
ϕ j ⊗ ϕ j+1 ⊗ · · · ⊗ ϕ N .
(4.3.32)
(No summation! In this subsection all sums are explicitly indicated.)
Let us work in the Schrödinger realization of CCR, i.e. H = L
2
(R
n
), H N =
L
2
(R
Nn
) and operators X
k
j (k = 1, 2, . . . N ; j = 1, 2, . . . 2n) acting on the k-th
copy of L
2
(R
n
) are chosen as in (4.1.5). Let ϕ ∈ H N be given by (4.3.31) with ϕ j ∈
L
2
(R
n
), supp ϕ j ∩ supp ϕ k = ∅ ( j = k) and such, that there is a neighbourhood
of unity e ∈ G so that for any g j ( j = 1, 2, . . . N ) in this neighbourhood also
U (g j )ϕ j and U (g k )ϕ k ( j = k) have disjoint supports. We assume, moreover, that ϕ
is a smooth function on R
Nn
. With these assumptions, we obtain from (4.3.28) in
a neighbourhood of the point s
ϕ
± (ϕ) on Z
m
ϕ± (the following result shows that the
mappings P P
ϕ
± have at ϕ the maximal rank):
ϕ
(±) (v η , v ξ ) = i
N
j=1
(ϕ j , [X
j
η , X
j
ξ ]ϕ j ),
(4.3.33)
where we assumed for all the j : :ϕ j = 1, and X η , X ξ in (4.3.28) are of the form
(4.3.30). The expression (4.3.33) shows, that Z ϕ± = M
±
ϕ is a 2N n-dimensional sym6 For g N =
N
j=1 g ( j) , g ( j) are copies of g, one has ξ :=
N
j=1 ξ j with ξ j ∈ g ( j) , X
j
ξ := X ξ j ∈
U (g).
71
p
+∗
M
M
+ =
◦
+ .
(4.3.29)
The Proposition 3.2.10 is a special case of this assertion.
Note: In the above presented construction of the symplectic manifold (M
+
ϕ , ,
M
+ ),
we did not use any specific properties of the projector P + and of the group action
U (G N ). These properties enter in constructions of specific orbits.
4.3.7 We shall specify here the previous construction to the case of G N := N -
fold direct product of 2n + 1—dimensional Heisenberg group G with infinitedimensional unitary irreducible representations U in H. The linear space U N (g N ) is
spanned by elements
6
X ξ :=
N
j=1
X
j
ξ with any X
j
ξ ∈ U (g), ξ ∈ g N ,
(4.3.30)
where the index j has the following meaning: If ϕ ∈ H N has the form
ϕ := ϕ 1 ⊗ ϕ 2 ⊗ · · · ⊗ ϕ N ,
(4.3.31)
then the linear operator X
j on H N corresponds to an (equally denoted) operator on
H by:
X
j
ϕ := ϕ 1 ⊗ ϕ 2 ⊗ · · · ⊗ X
j
ϕ j ⊗ ϕ j+1 ⊗ · · · ⊗ ϕ N .
(4.3.32)
(No summation! In this subsection all sums are explicitly indicated.)
Let us work in the Schrödinger realization of CCR, i.e. H = L
2
(R
n
), H N =
L
2
(R
Nn
) and operators X
k
j (k = 1, 2, . . . N ; j = 1, 2, . . . 2n) acting on the k-th
copy of L
2
(R
n
) are chosen as in (4.1.5). Let ϕ ∈ H N be given by (4.3.31) with ϕ j ∈
L
2
(R
n
), supp ϕ j ∩ supp ϕ k = ∅ ( j = k) and such, that there is a neighbourhood
of unity e ∈ G so that for any g j ( j = 1, 2, . . . N ) in this neighbourhood also
U (g j )ϕ j and U (g k )ϕ k ( j = k) have disjoint supports. We assume, moreover, that ϕ
is a smooth function on R
Nn
. With these assumptions, we obtain from (4.3.28) in
a neighbourhood of the point s
ϕ
± (ϕ) on Z
m
ϕ± (the following result shows that the
mappings P P
ϕ
± have at ϕ the maximal rank):
ϕ
(±) (v η , v ξ ) = i
N
j=1
(ϕ j , [X
j
η , X
j
ξ ]ϕ j ),
(4.3.33)
where we assumed for all the j : :ϕ j = 1, and X η , X ξ in (4.3.28) are of the form
(4.3.30). The expression (4.3.33) shows, that Z ϕ± = M
±
ϕ is a 2N n-dimensional sym6 For g N =
N
j=1 g ( j) , g ( j) are copies of g, one has ξ :=
N
j=1 ξ j with ξ j ∈ g ( j) , X
j
ξ := X ξ j ∈
U (g).