4.3 Notes on Other Examples
65
[Y
a
j , Q
b
k ] = i δ ab jkm Q
a
m , [Y
a
j , P
b
k ] = i δ ab jkm P
a
m ,
(4.3.12)
[Y j , Q
a
k ] = i jkm Q
a
m , [Y j , P
a
k ] = i jkm P
a
m , [Y j , Y
a
k ] = i jkm Y
a
m .
(4.3.13)
Let us first consider the Lie algebra g 0 represented by generators Q
a
j , P
a
j and
Y j ( j = 1, 2, 3; a = 1, 2, . . . N ) of the representation U (G 0 ) of the corresponding
group G 0 , compare Proposition 4.2.2. We see that G 0 is a semidirect product of
of SU (2) with the Heisenberg group G 3N (with the notation from 4.2.1), G 0 =
SU (2) G 3N , where the Heisenberg group is a normal subgroup. Let us investigate
the orbits O ϕ := U(G 0 )ϕ ⊂ D G 0 in P(H) and the corresponding classical phase
spaces M ϕ . Since any O ϕ is a homogeneous space of G 0 , it can be generated from a
point ϕ satisfying (see 4.1.5)
T r(P ϕ X j ) = 0, for all j = 1, 2, . . . 6N .
(4.3.14)
The local structure of O ϕ is most easily seen in a neighbourhood of such ϕ. The
isotropy group of F ϕ ∈ g
∗
0 (see (3.1.5)) with respect to Ad
∗
(G 0 ) has the Lie algebra
generated by such C ∈ U (g 0 ), which are solutions of the system
T r(P ϕ [C, X j ]) = 0 ( j = 1, 2, . . . 6N ), T r(P ϕ [C, Y k ]) = 0 (k = 1, 2, 3).
(4.3.15)
The corank of the matrix of this homogeneous system is:
(i) equal to 3 iff T r(P ϕ Y k ) = 0 for all k = 1, 2, 3; in this situation there might
occur cases with dim O ϕ = 6N , 6N + 2, 6N + 3 corresponding to such ϕ, for
which Y ϕ = 0 for all Y ∈ U (so(3)), (resp. Y ϕ = 0 for just one linearly independent
Y ∈ U (so(3)), resp. Y ϕ = 0 for all Y = 0); as an example of the case of dim O ϕ =
6N + 3 we can take ϕ for N = 1 in Schrödinger realization of CCR:
ϕ(q) := ϕ(q 1 , q 2 , q 3 ) := c q 1 q 2 q 3 exp(−q
2
1 − q
2
2 − q
2
3 ), c :=
2
5
π
3
4
, (4.3.16)
corresponding to the value J = 3 of the total momentum. In all these cases of various
values of dim O ϕ the corresponding symplectic spaces M ϕ are homeomorphic to
T
∗
R
3N
= R
6N
.
(ii) equal to 1 in all other cases; now all the solutions C of (4.3.15) are proportional
to C ϕ of the form (4.3.4). If ϕ is an eigenvector of C ϕ , then dim O ϕ = dim M ϕ =
6N + 2. In the remaining case it is dim O ϕ = 6N + 3 and dim M ϕ = 6N + 2. If ϕ is
proportional to C ϕ ϕ, the orbit O ϕ is the fiber-bundle with base R
6N and typical fiber
S
2
; if ϕ is not an eigenvector of C ϕ , then the fiber on R
6N is the whole group SO(3).
In the both cases the phase space is T
∗
R
3N fibered by two dimensional spheres S
2
with the canonical symplectic form from P(H) being the sum of the canonical form
on T
∗
R
3N and that on S
2 described in (4.3.8).
65
[Y
a
j , Q
b
k ] = i δ ab jkm Q
a
m , [Y
a
j , P
b
k ] = i δ ab jkm P
a
m ,
(4.3.12)
[Y j , Q
a
k ] = i jkm Q
a
m , [Y j , P
a
k ] = i jkm P
a
m , [Y j , Y
a
k ] = i jkm Y
a
m .
(4.3.13)
Let us first consider the Lie algebra g 0 represented by generators Q
a
j , P
a
j and
Y j ( j = 1, 2, 3; a = 1, 2, . . . N ) of the representation U (G 0 ) of the corresponding
group G 0 , compare Proposition 4.2.2. We see that G 0 is a semidirect product of
of SU (2) with the Heisenberg group G 3N (with the notation from 4.2.1), G 0 =
SU (2) G 3N , where the Heisenberg group is a normal subgroup. Let us investigate
the orbits O ϕ := U(G 0 )ϕ ⊂ D G 0 in P(H) and the corresponding classical phase
spaces M ϕ . Since any O ϕ is a homogeneous space of G 0 , it can be generated from a
point ϕ satisfying (see 4.1.5)
T r(P ϕ X j ) = 0, for all j = 1, 2, . . . 6N .
(4.3.14)
The local structure of O ϕ is most easily seen in a neighbourhood of such ϕ. The
isotropy group of F ϕ ∈ g
∗
0 (see (3.1.5)) with respect to Ad
∗
(G 0 ) has the Lie algebra
generated by such C ∈ U (g 0 ), which are solutions of the system
T r(P ϕ [C, X j ]) = 0 ( j = 1, 2, . . . 6N ), T r(P ϕ [C, Y k ]) = 0 (k = 1, 2, 3).
(4.3.15)
The corank of the matrix of this homogeneous system is:
(i) equal to 3 iff T r(P ϕ Y k ) = 0 for all k = 1, 2, 3; in this situation there might
occur cases with dim O ϕ = 6N , 6N + 2, 6N + 3 corresponding to such ϕ, for
which Y ϕ = 0 for all Y ∈ U (so(3)), (resp. Y ϕ = 0 for just one linearly independent
Y ∈ U (so(3)), resp. Y ϕ = 0 for all Y = 0); as an example of the case of dim O ϕ =
6N + 3 we can take ϕ for N = 1 in Schrödinger realization of CCR:
ϕ(q) := ϕ(q 1 , q 2 , q 3 ) := c q 1 q 2 q 3 exp(−q
2
1 − q
2
2 − q
2
3 ), c :=
2
5
π
3
4
, (4.3.16)
corresponding to the value J = 3 of the total momentum. In all these cases of various
values of dim O ϕ the corresponding symplectic spaces M ϕ are homeomorphic to
T
∗
R
3N
= R
6N
.
(ii) equal to 1 in all other cases; now all the solutions C of (4.3.15) are proportional
to C ϕ of the form (4.3.4). If ϕ is an eigenvector of C ϕ , then dim O ϕ = dim M ϕ =
6N + 2. In the remaining case it is dim O ϕ = 6N + 3 and dim M ϕ = 6N + 2. If ϕ is
proportional to C ϕ ϕ, the orbit O ϕ is the fiber-bundle with base R
6N and typical fiber
S
2
; if ϕ is not an eigenvector of C ϕ , then the fiber on R
6N is the whole group SO(3).
In the both cases the phase space is T
∗
R
3N fibered by two dimensional spheres S
2
with the canonical symplectic form from P(H) being the sum of the canonical form
on T
∗
R
3N and that on S
2 described in (4.3.8).