4.3 Notes on Other Examples
73
X ξ ∈ V N (g) commute with projectors P ± . Hence V N leaves the subspaces H
+
N and
H
−
N invariant, and we can obtain the classical projections of this ‘macroscopic’ (for
large N ) subsystem in the standard way, (Sect. 3.2); the obtained classical phase
spaces are orbits of Ad
∗
(G) with their canonical symplectic structure—there is no
difference in the kinds of statistics, from the point of view of kinematics.
In trying to extend our constructions to systems consisting of infinite number
N → ∞ of equal (or identical) subsystems, we meet the problems of divergence of
‘global (or collective) observables’ X
N
ξ := X ξ and of discontinuity of the resulting
representation V ∞ of G. We give a formalization of this ‘large N limit’ in the next
Sect. 5.1, and in the Sect. 5.2 we outline a possible generalization of obtaining classical subsystems of collective observables from infinite quantal systems. We shall
not take any care of statistics of subsystems, what could be motivated by results of
the last two subsections: the statistics seems to have no essential influence upon the
classical phase spaces of systems of identical particles.
73
X ξ ∈ V N (g) commute with projectors P ± . Hence V N leaves the subspaces H
+
N and
H
−
N invariant, and we can obtain the classical projections of this ‘macroscopic’ (for
large N ) subsystem in the standard way, (Sect. 3.2); the obtained classical phase
spaces are orbits of Ad
∗
(G) with their canonical symplectic structure—there is no
difference in the kinds of statistics, from the point of view of kinematics.
In trying to extend our constructions to systems consisting of infinite number
N → ∞ of equal (or identical) subsystems, we meet the problems of divergence of
‘global (or collective) observables’ X
N
ξ := X ξ and of discontinuity of the resulting
representation V ∞ of G. We give a formalization of this ‘large N limit’ in the next
Sect. 5.1, and in the Sect. 5.2 we outline a possible generalization of obtaining classical subsystems of collective observables from infinite quantal systems. We shall
not take any care of statistics of subsystems, what could be motivated by results of
the last two subsections: the statistics seems to have no essential influence upon the
classical phase spaces of systems of identical particles.