62
4 Examples of Classical Mechanical Projections
4.2.11 The quantal and classical evolutions corresponding to the generator A, cf.
4.2.3, (resp. to the Hamiltonian function f A ) coincide in our examples in the sense
of 3.3.9 + 3.3.10, independently of the dimension (= 2n or 2n+l) of the orbit O ϕ . The
time-evolved quantal states remain all the time on the orbit O ϕ . We might be interested
also in time evolution of other quantities than (the expectations of) X j in the quantal
interpretation. According to 4.1.7(i), in the case of dim O ϕ = 2n = dim M ϕ , any
‘spreading of the wave packet’ does not occur. The situation is different, however,
on 2n+l - dimensional orbits. For various ϕ j ( j = 1, 2) corresponding to distinct
quantal states in the same leaf [ϕ] ∈ M ϕ we have in general (cf. 4.1.4 for notation)
T r(P
ϕ 1
x (X j − x j )(X k − x k )) = T r(P
ϕ 2
x (X j − x j )(X k − x k )).
(4.2.24)
This is the case of e.g., free particle motions. This fact makes a certain difference between classical and quantal interpretations of the ‘extended phase spaces’
O ϕ (dim O ϕ = 2n + 1). This will be briefly discussed later on, in 4.3.5.
4.3 Notes on Other Examples
4.3.1 By the method developed in our Chap. 3, we can construct from an arbitrary
continuous unitary representation U (G) of a Lie group G ‘classical phase spaces’,
which are diffeomorphic (and even symplectomorphic) to orbits of Ad
∗
(G). It can
be shown, [174, 15.2], that any symplectic homogenous space of any connected Lie
group G is a covering symplectic space of either an orbit of Ad*(G), or an orbit
of Ad
∗
(G 1 ), where G 1 is a central extension of G by R—see also 1.3.7. On the
other hand, unitary continuous representations of G can be constructed from orbits
of Ad
∗
(G), [174]. Considerations in Sects. 1.2 and 1.3 show reasons for modeling
state spaces of CM-systems as homogeneous symplectic spaces of some Lie groups,
at least for ‘basic’ or ‘elementary’ physical systems. In this section we shall outline
further examples of obtaining CM-systems from unitary group representations which
suggest, that all generally accepted models of ‘elementary’ finite dimensional CMsystems could be obtained in this way.
4.3.2 Classical spin from SO(3): Let U be a (projective) irreducible representation
of the compact Lie group SO(3)—the connected component of the 3-dimensional
orthogonal group O(3) of orthogonal transformations of a 3-dimensional Euclidean
space E 3 . The representation space H = C
2J +1
(J =
n
2
, n ∈ Z + ) is finite dimensional. Generators Y k (k = 1, 2, 3) of U corresponding to rotations around orthogonal axes in E 3 satisfy the commutation relations (with jkm = − k jm = − jmk ,
123 = 1):
[Y k , Y m ] = i km j Y j .
(4.3.1)
4 Examples of Classical Mechanical Projections
4.2.11 The quantal and classical evolutions corresponding to the generator A, cf.
4.2.3, (resp. to the Hamiltonian function f A ) coincide in our examples in the sense
of 3.3.9 + 3.3.10, independently of the dimension (= 2n or 2n+l) of the orbit O ϕ . The
time-evolved quantal states remain all the time on the orbit O ϕ . We might be interested
also in time evolution of other quantities than (the expectations of) X j in the quantal
interpretation. According to 4.1.7(i), in the case of dim O ϕ = 2n = dim M ϕ , any
‘spreading of the wave packet’ does not occur. The situation is different, however,
on 2n+l - dimensional orbits. For various ϕ j ( j = 1, 2) corresponding to distinct
quantal states in the same leaf [ϕ] ∈ M ϕ we have in general (cf. 4.1.4 for notation)
T r(P
ϕ 1
x (X j − x j )(X k − x k )) = T r(P
ϕ 2
x (X j − x j )(X k − x k )).
(4.2.24)
This is the case of e.g., free particle motions. This fact makes a certain difference between classical and quantal interpretations of the ‘extended phase spaces’
O ϕ (dim O ϕ = 2n + 1). This will be briefly discussed later on, in 4.3.5.
4.3 Notes on Other Examples
4.3.1 By the method developed in our Chap. 3, we can construct from an arbitrary
continuous unitary representation U (G) of a Lie group G ‘classical phase spaces’,
which are diffeomorphic (and even symplectomorphic) to orbits of Ad
∗
(G). It can
be shown, [174, 15.2], that any symplectic homogenous space of any connected Lie
group G is a covering symplectic space of either an orbit of Ad*(G), or an orbit
of Ad
∗
(G 1 ), where G 1 is a central extension of G by R—see also 1.3.7. On the
other hand, unitary continuous representations of G can be constructed from orbits
of Ad
∗
(G), [174]. Considerations in Sects. 1.2 and 1.3 show reasons for modeling
state spaces of CM-systems as homogeneous symplectic spaces of some Lie groups,
at least for ‘basic’ or ‘elementary’ physical systems. In this section we shall outline
further examples of obtaining CM-systems from unitary group representations which
suggest, that all generally accepted models of ‘elementary’ finite dimensional CMsystems could be obtained in this way.
4.3.2 Classical spin from SO(3): Let U be a (projective) irreducible representation
of the compact Lie group SO(3)—the connected component of the 3-dimensional
orthogonal group O(3) of orthogonal transformations of a 3-dimensional Euclidean
space E 3 . The representation space H = C
2J +1
(J =
n
2
, n ∈ Z + ) is finite dimensional. Generators Y k (k = 1, 2, 3) of U corresponding to rotations around orthogonal axes in E 3 satisfy the commutation relations (with jkm = − k jm = − jmk ,
123 = 1):
[Y k , Y m ] = i km j Y j .
(4.3.1)