Abstract
We shall construct in this chapter classical subsystems of a large quantum mechanical 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. Also more general algebraic descriptions are provided based on special classifications of classical quantities of general large quantum mechanical systems.
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Notes
- 1.
i.e. all the vectors \(\Phi ^b\) for which \(b_j\ne a_j\) for finite number of indices \(j\in \Pi \) only.
- 2.
We shall use sometimes projectors instead of the corresponding subspaces.
- 3.
Where \(\Psi \in \mathcal{H}_{\Pi }\) such that there is an \(F\in \mathfrak {g}^*\) satisfying: \(E_{\mathfrak {g}}^\#(F)\Psi = \Psi \).
- 4.
A face S of a compact convex set K is defined to be a subset of K with the property that if \(\omega =\sum _{i=1}^n\lambda _i\omega _i\) is a convex combination of elements \(\omega _i\in K\) such that \(\omega \in S\) then \(\omega _i\in S,\ \forall \ i= 1,2,\ldots n\).
- 5.
The relation (5.1.154) has been proved in the assumption that any projector \(p\in \mathfrak {M}_G^\Pi \) is of the form \(p= E_{\mathfrak {g}}^\Pi (B)\) for some \(B\subset {\mathfrak {g}}^{*},\) if \(p(I-p_G)=0\).
- 6.
\(a_N\) forms a zero-dimensional orbit of \(Ad^*(G):\ a_N([\xi ,\eta ])\equiv 0\).
- 7.
The present author was informed about some important set-theoretical concepts connected with this Proposition by the late colleague Ivan Korec (1943–1998).
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Bóna, P. (2020). Macroscopic Limits. In: Classical Systems in Quantum Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-030-45070-0_5
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DOI: https://doi.org/10.1007/978-3-030-45070-0_5
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