Zernike System from Free Motion on 3-Sphere
171
In Sect. 2 we present the well-known Lie algebra realization of the orthogonal
groups on spheres. We are using the algebra so(4) and the 3-sphere as its
homogeneous space, so in Sect. 3 we introduce two coordinate systems, where the
Laplace–Beltrami operator appears in different differential forms. The eigenfunctions there already relate through Clebsch–Gordan coefficients, and continue to do
so when the 3-sphere is projected on a two-sphere in Sect. 4, resulting in a restricted
set of Clebsch–Gordan’s as interbasis coefficients between solutions of the Zernike
system in the two coordinate systems, shown in Sect. 5. In the concluding Sect. 6
we add some remarks on the significance of free motions on conics that project to
remarkable physical systems.
2 Realization of SO(4) on the 3-Sphere
Lie algebras, when exponentiated to the group, can act faithfully and transitively,
on any of its homogeneous coset spaces. Corresponding to cosets by the group
identity {1}, the action is on the group itself, a manifold of dimension
1
2 N(N−1).
One may have spaces of cosets by SO(2), SO(3), etc. up to cosets by SO(N −1) [8].
The last is a privileged space because SO(N )/SO(N −1) = S
N −1 is the (N−1)dimensional manifold of a sphere. In this space one can realize the generators as
K i,j = s i ∂ j − s j ∂ i (∂ j := ∂/∂s j ), that generate rotations of S
N −1 ; the s i are the
Cartesian coordinates restricted to the sphere by
N
i=1 s 2
i = 1. Thus we realize
so(4) as the generators of rotations of the 3-dimensional manifold of S
3 .
While the Lie algebra so(3) has one well-known invariant J 2 :=
3
i,j =1 K 2
i,j
with eigenvalues j (j + 1) where j ∈ Z
+
0 , the Lie algebra so(4) has two seconddegree invariant Casimir operators. The first is the sum of all squares, i.e., the
Laplace–Beltrami operator on the 3-sphere, Δ
(3)
LB =
4
i,j =1 K 2
i,j with spectrum
J (J +2), J ∈ Z
+
0 on S
3 ; the second invariant,
3
i,j,k=1 ε i,j,k K i,j K k,4 = 0 vanishes
in the coset space of the sphere. This implies that
J
(1) 2
=
1
4
3
i,j,k=1
(K j,k ± K i,4 )
2
= J
(2) 2
⇒ j
(1)
= j
(2)
=: j, (7)
Δ
(3)
LB = 2J
(1) 2 + 2J
(2) 2 spectrum J (J + 2)
4J
(1) 2 = 4J
(2) 2 spectrum 4 j (j + 1)
⇒ J = 2j ∈ Z
+
0 .
(8)
Finally, in (5) the so(3) Casimir invariant
1
2
3
i,j =1 K 2
i,j has the spectrum + 1),
∈ Z
+
0 , with the range of the branching rule 0 J .
171
In Sect. 2 we present the well-known Lie algebra realization of the orthogonal
groups on spheres. We are using the algebra so(4) and the 3-sphere as its
homogeneous space, so in Sect. 3 we introduce two coordinate systems, where the
Laplace–Beltrami operator appears in different differential forms. The eigenfunctions there already relate through Clebsch–Gordan coefficients, and continue to do
so when the 3-sphere is projected on a two-sphere in Sect. 4, resulting in a restricted
set of Clebsch–Gordan’s as interbasis coefficients between solutions of the Zernike
system in the two coordinate systems, shown in Sect. 5. In the concluding Sect. 6
we add some remarks on the significance of free motions on conics that project to
remarkable physical systems.
2 Realization of SO(4) on the 3-Sphere
Lie algebras, when exponentiated to the group, can act faithfully and transitively,
on any of its homogeneous coset spaces. Corresponding to cosets by the group
identity {1}, the action is on the group itself, a manifold of dimension
1
2 N(N−1).
One may have spaces of cosets by SO(2), SO(3), etc. up to cosets by SO(N −1) [8].
The last is a privileged space because SO(N )/SO(N −1) = S
N −1 is the (N−1)dimensional manifold of a sphere. In this space one can realize the generators as
K i,j = s i ∂ j − s j ∂ i (∂ j := ∂/∂s j ), that generate rotations of S
N −1 ; the s i are the
Cartesian coordinates restricted to the sphere by
N
i=1 s 2
i = 1. Thus we realize
so(4) as the generators of rotations of the 3-dimensional manifold of S
3 .
While the Lie algebra so(3) has one well-known invariant J 2 :=
3
i,j =1 K 2
i,j
with eigenvalues j (j + 1) where j ∈ Z
+
0 , the Lie algebra so(4) has two seconddegree invariant Casimir operators. The first is the sum of all squares, i.e., the
Laplace–Beltrami operator on the 3-sphere, Δ
(3)
LB =
4
i,j =1 K 2
i,j with spectrum
J (J +2), J ∈ Z
+
0 on S
3 ; the second invariant,
3
i,j,k=1 ε i,j,k K i,j K k,4 = 0 vanishes
in the coset space of the sphere. This implies that
J
(1) 2
=
1
4
3
i,j,k=1
(K j,k ± K i,4 )
2
= J
(2) 2
⇒ j
(1)
= j
(2)
=: j, (7)
Δ
(3)
LB = 2J
(1) 2 + 2J
(2) 2 spectrum J (J + 2)
4J
(1) 2 = 4J
(2) 2 spectrum 4 j (j + 1)
⇒ J = 2j ∈ Z
+
0 .
(8)
Finally, in (5) the so(3) Casimir invariant
1
2
3
i,j =1 K 2
i,j has the spectrum + 1),
∈ Z
+
0 , with the range of the branching rule 0 J .