66
4 Examples of Classical Mechanical Projections
Let us take now all the operators Q
a
j , P
a
j , Y
a
j (a = 1, 2, . . . N ; j = 1, 2, 3) as
generators of the considered representation U (G) (now G is semi direct product
of the Heisenberg group G 3N and of the direct product of N copies of the group
SU (2)). The orbits and corresponding phase spaces arising from the action of this
group G on P(H) with H = L
2
(R
3N
) = L
2
(R
3
) ⊗ L
2
(R
3
) ⊗ . . . L
2
(R
3
) (N-tuple
tensor product) can be constructed as N-tuple direct product manifolds; each of the
multipled manifolds can be obtained by the above described procedure with N = 1.
Examples of classical systems obtained in this subsection include systems of
several nonrelativistic spinning particles. Here the ‘classical spin’ was obtained from
quantal orbital momentum.
4.3.4 The groups which are, perhaps, physically most important ones, are Galilean
and Poincaré groups. Because of relative complexity of any complete exposition of
these important examples, we shall restrict our present exposition to several notes
and remarks. For more detailed nice exposition see e.g. in [321].
(i) The Galilean group.
This group realizes the nonrelativistic (better: Galilean relativistic) conception of
relative positions and motions of mechanical systems (particles, bodies etc.). It is a
ten parameter Lie group, the parameters of which can be chosen to describe time and
space translations (4 parameters), space rotations (3 parameters) and transition to
uniformly moving systems (3 coordinates of a velocity). Any unitary (vector) representation of this group cannot be, however, interpreted in terms of really observed
physical systems, see e.g. [321, Sect. XII.8]. Physically interpreted projective representations correspond to multipliers m τ of the Galilean group characterized by a
real parameter τ —the mass of the system. Let us denote by G the central extension (cf. [174, 321], resp. also [37, Note 3.3.6]) of (the covering group of) the
Galilean group by R corresponding to a multiplier m τ with τ = 0 (all such groups
are mutually isomorphic). Orbits of Ad
∗
(G) (described e.g. in [5]) are just one particle phase spaces obtained in our subsection 4.3.3. Unitary representations of G, in
which the central subgroup R acts by a multiplication by constants, correspond to
physically interesting projective representations of the Galilean group. Irreducible
representations of G describe one-particle systems. The projected orbits O ϕ of these
representations are either seven or nine or ten dimensional (this is a consequence of
4.3.3, 4.2.7 and absolute continuity of the spectrum of the time-evolution generator
P
2
1 + P
2
2 + P
2
3 of U (G)). In the cases dim O ϕ = 7 or 9 the manifolds O ϕ with the
two-form
◦ (cf. 3.2.2) are just contact manifolds of the extended phase spaces,
dim O ϕ = dim M ϕ + 1.
(ii) The Poincaré group.
Let now G be the ten-parameter covering group of the Poincaré group. Physical
interpretation of the parameters is the same as that of the corresponding parameters of the Galilean group. In the present case of G, however, the conception of
Galilean relativity is replaced by the conception of Einstein relativity of mechanical
motions. Since the second cohomology group of G is now trivial, we have to deal with
unitary (vector) representations of G only. The orbits of the coadjoint action of G
4 Examples of Classical Mechanical Projections
Let us take now all the operators Q
a
j , P
a
j , Y
a
j (a = 1, 2, . . . N ; j = 1, 2, 3) as
generators of the considered representation U (G) (now G is semi direct product
of the Heisenberg group G 3N and of the direct product of N copies of the group
SU (2)). The orbits and corresponding phase spaces arising from the action of this
group G on P(H) with H = L
2
(R
3N
) = L
2
(R
3
) ⊗ L
2
(R
3
) ⊗ . . . L
2
(R
3
) (N-tuple
tensor product) can be constructed as N-tuple direct product manifolds; each of the
multipled manifolds can be obtained by the above described procedure with N = 1.
Examples of classical systems obtained in this subsection include systems of
several nonrelativistic spinning particles. Here the ‘classical spin’ was obtained from
quantal orbital momentum.
4.3.4 The groups which are, perhaps, physically most important ones, are Galilean
and Poincaré groups. Because of relative complexity of any complete exposition of
these important examples, we shall restrict our present exposition to several notes
and remarks. For more detailed nice exposition see e.g. in [321].
(i) The Galilean group.
This group realizes the nonrelativistic (better: Galilean relativistic) conception of
relative positions and motions of mechanical systems (particles, bodies etc.). It is a
ten parameter Lie group, the parameters of which can be chosen to describe time and
space translations (4 parameters), space rotations (3 parameters) and transition to
uniformly moving systems (3 coordinates of a velocity). Any unitary (vector) representation of this group cannot be, however, interpreted in terms of really observed
physical systems, see e.g. [321, Sect. XII.8]. Physically interpreted projective representations correspond to multipliers m τ of the Galilean group characterized by a
real parameter τ —the mass of the system. Let us denote by G the central extension (cf. [174, 321], resp. also [37, Note 3.3.6]) of (the covering group of) the
Galilean group by R corresponding to a multiplier m τ with τ = 0 (all such groups
are mutually isomorphic). Orbits of Ad
∗
(G) (described e.g. in [5]) are just one particle phase spaces obtained in our subsection 4.3.3. Unitary representations of G, in
which the central subgroup R acts by a multiplication by constants, correspond to
physically interesting projective representations of the Galilean group. Irreducible
representations of G describe one-particle systems. The projected orbits O ϕ of these
representations are either seven or nine or ten dimensional (this is a consequence of
4.3.3, 4.2.7 and absolute continuity of the spectrum of the time-evolution generator
P
2
1 + P
2
2 + P
2
3 of U (G)). In the cases dim O ϕ = 7 or 9 the manifolds O ϕ with the
two-form
◦ (cf. 3.2.2) are just contact manifolds of the extended phase spaces,
dim O ϕ = dim M ϕ + 1.
(ii) The Poincaré group.
Let now G be the ten-parameter covering group of the Poincaré group. Physical
interpretation of the parameters is the same as that of the corresponding parameters of the Galilean group. In the present case of G, however, the conception of
Galilean relativity is replaced by the conception of Einstein relativity of mechanical
motions. Since the second cohomology group of G is now trivial, we have to deal with
unitary (vector) representations of G only. The orbits of the coadjoint action of G