4.3 Notes on Other Examples
63
Choose any nonzero ϕ ∈ H and form the orbit O ϕ := {U(g)ϕ : g ∈ SO(3)}. Let
us denote by Y ξ the generator of t → U (exp(tξ)) corresponding to an element ξ
of the Lie algebra g := so(3). We are interested in the Ad
∗
(SO(3))-action onto
F ϕ ∈ so(3)
∗
, where
F ϕ : ξ → F ϕ (ξ) := T r(P ϕ Y ξ ), ξ ∈ so(3).
(4.3.2)
Generators C ϕ := c
j
ϕ Y j of one-parameter subgroups of the isotropy group of F ϕ are
just all nonzero solutions of equations
T r(P ϕ [Y k , C ϕ ]) = 0, k = 1, 2, 3.
(4.3.3)
With y
k
:= y
k
(ϕ) := T r(P ϕ Y k ), the only linearly independent solution C ϕ of (4.3.3)
can be written:
C ϕ = y
k
(ϕ)Y k .
(4.3.4)
One could easily check that C ϕ = 0 in (4.3.4) for some ϕ, iff C ϕ = 0 for all
ϕ ∈ H, iff J = 0 (i.e. dim C H = 1), iff the matrix of the homogeneous equations
(4.3.3) is identically zero. In all other cases the rank of the matrix of the system
(4.3.3) equals to 2. For J = 0 the corresponding classical phase space degenerates
to a point: this corresponds to the traditional point of view according to of which
spin does not occur in classical mechanics.
For orbits O ϕ in representations with J = 0 we have two possibilities:
(i) The vector ϕ is an eigenvector of C ϕ and the orbit O ϕ is two-dimensional (any
generator Y ∈ U (so(3)) which is linearly independent of C ϕ cannot be a solution
of (4.3.3): T r(P ϕ [Y k , Y ]) = 0 for k = 1, 2, 3 implies Y = λC ϕ ; hence Y linearly
independent of C ϕ generate two-dimensional tangent space to O ϕ at ϕ).
(ii) If ϕ is not an eigenvector of C ϕ , then the generator C ϕ generates a onedimensional submanifold of O ϕ diffeomorphic to a circle S
1
(C ϕ generates the
isotropy subgroup of SO(3) at F ϕ , which is closed, hence compact). In this case
O ϕ is 3-dimensional.
Note that for J =
1
2
only the possibility (i) occurs, since H = C
2 and
dim R P(H) = 2.
It can be easily shown that in the both cases the corresponding classical phase
space M ϕ (in the case (i) identical with O ϕ ) is diffeomorphic to the sphere S
2 in
so(3)
∗ with coordinates
y
k
: F ϕ → y
k
(ϕ) := T r(P ϕ Y k ), k = 1, 2, 3; ϕ ∈ O ϕ .
(4.3.5)
Let t ∈ R and let τ ∈ R
3 be any unit vector:
k (τ
k
)
2
= 1. Let y(ϕ) ∈ R
3 be
given by coordinates y
k in (4.3.5) and τ · y :=
k τ
k y
k
. Using (4.3.1) we obtain:
63
Choose any nonzero ϕ ∈ H and form the orbit O ϕ := {U(g)ϕ : g ∈ SO(3)}. Let
us denote by Y ξ the generator of t → U (exp(tξ)) corresponding to an element ξ
of the Lie algebra g := so(3). We are interested in the Ad
∗
(SO(3))-action onto
F ϕ ∈ so(3)
∗
, where
F ϕ : ξ → F ϕ (ξ) := T r(P ϕ Y ξ ), ξ ∈ so(3).
(4.3.2)
Generators C ϕ := c
j
ϕ Y j of one-parameter subgroups of the isotropy group of F ϕ are
just all nonzero solutions of equations
T r(P ϕ [Y k , C ϕ ]) = 0, k = 1, 2, 3.
(4.3.3)
With y
k
:= y
k
(ϕ) := T r(P ϕ Y k ), the only linearly independent solution C ϕ of (4.3.3)
can be written:
C ϕ = y
k
(ϕ)Y k .
(4.3.4)
One could easily check that C ϕ = 0 in (4.3.4) for some ϕ, iff C ϕ = 0 for all
ϕ ∈ H, iff J = 0 (i.e. dim C H = 1), iff the matrix of the homogeneous equations
(4.3.3) is identically zero. In all other cases the rank of the matrix of the system
(4.3.3) equals to 2. For J = 0 the corresponding classical phase space degenerates
to a point: this corresponds to the traditional point of view according to of which
spin does not occur in classical mechanics.
For orbits O ϕ in representations with J = 0 we have two possibilities:
(i) The vector ϕ is an eigenvector of C ϕ and the orbit O ϕ is two-dimensional (any
generator Y ∈ U (so(3)) which is linearly independent of C ϕ cannot be a solution
of (4.3.3): T r(P ϕ [Y k , Y ]) = 0 for k = 1, 2, 3 implies Y = λC ϕ ; hence Y linearly
independent of C ϕ generate two-dimensional tangent space to O ϕ at ϕ).
(ii) If ϕ is not an eigenvector of C ϕ , then the generator C ϕ generates a onedimensional submanifold of O ϕ diffeomorphic to a circle S
1
(C ϕ generates the
isotropy subgroup of SO(3) at F ϕ , which is closed, hence compact). In this case
O ϕ is 3-dimensional.
Note that for J =
1
2
only the possibility (i) occurs, since H = C
2 and
dim R P(H) = 2.
It can be easily shown that in the both cases the corresponding classical phase
space M ϕ (in the case (i) identical with O ϕ ) is diffeomorphic to the sphere S
2 in
so(3)
∗ with coordinates
y
k
: F ϕ → y
k
(ϕ) := T r(P ϕ Y k ), k = 1, 2, 3; ϕ ∈ O ϕ .
(4.3.5)
Let t ∈ R and let τ ∈ R
3 be any unit vector:
k (τ
k
)
2
= 1. Let y(ϕ) ∈ R
3 be
given by coordinates y
k in (4.3.5) and τ · y :=
k τ
k y
k
. Using (4.3.1) we obtain: