Chapter 5
Macroscopic Limits
5.1 Multiple Systems
5.1.1 We shall construct in this chapter classical subsystems of a large quantal system. We shall assume here that the large system consists of infinite number of copies
of a finite subsystem of the type dealt with in preceding chapters. The infinite “macroscopic” system is obtained as an inductive limit of a net of systems consisting of an
increasing number of copies of the mentioned finite systems. The symmetry group
G of a single finite subsystem is then also a symmetry group of the large system.
An essential formal difference with respect to the systems discussed in preceding
sections is that the action of G on the large system is not described by a continuous unitary representation, hence we cannot introduce generators corresponding to
one-parameter subgroups of G as operators in some Hilbert space.
5.1.2 To make the following considerations more intuitive, let us come back for a
while to finite systems consisting of N equal subsystems. Let the unitary representation V N (G) and its generators X
N
ξ := X ξ (ξ ∈ g) be defined as in 4.3.7 and 4.3.8,
esp. in (4.3.30). Then
[X
N
ξ , X
N
η ] = i X
N
[ξ,η] (ξ, η ∈ g)
(5.1.1)
and the restriction to the orbit O
N
ϕ := V N (G)ϕ (ϕ ∈ H N ) of the canonical symplectic form
N on P(H N ) is determined by
N
ϕ (σ ξ , σ η ) = i T r(P ϕ [X
N
ξ , X
N
η ]), (ξ, η ∈ g).
(5.1.2)
Here σ ξ is the vector field on P(H N ) corresponding to the unitary flow
(t; ϕ) → exp(−it X
N
ξ )ϕ, ϕ ∈ H N , t ∈ R.
(5.1.3)
For N → ∞, the operators X
N
ξ diverge and V N (G) does not converge to any
continuous unitary representation—compare the next subsection. Let
© Springer Nature Switzerland AG 2020, corrected publication 2020
P. Bóna, Classical Systems in Quantum Mechanics,
https://doi.org/10.1007/978-3-030-45070-0_5
75