Abstract
Different phenomena can be described by different theoretical schemes. These schemes should be, however, mutually consistent. It is shown in the book that classical mechanics can be found as a subtheory of quantum mechanics. The introductory Chap. 1 sketches main theoretical tools by which this goal can be reached.
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Notes
- 1.
Consider here macroscopic quantal effects (e.g. superconductivity, superfluidity) vanishing for \(\hbar \rightarrow \)0.
- 2.
Some more specific hints on this possible classical stochastic evolutions from quantal time development could be found perhaps in [29].
- 3.
The macroscopic quantal effects like superfluidity and superconductivity are additional effects observed in these ‘classical subsystems’ of the large quantal systems.
- 4.
where the quantal interpretation of classical quantities (i.e. expectation values of generators of U(G) in corresponding states) was different from the classical interpretation (i.e. sharp values of corresponding classical generators).
- 5.
The center \(\varvec{\mathcal{Z}(\mathfrak {A})}\) of a \(C^*\)-algebra \(\mathfrak {A}\) is the commutative \(C^*\)-subalgebra of \(\mathfrak {A}\) consisting of all elements of \(\mathfrak {A}\), each commuting with all elements of \(\mathfrak {A}\): \(\mathcal{Z}(\mathfrak {A}):=\{z\in \mathfrak {A}:z\!\cdot \!x-x\!\cdot \!z=0, \forall x\in \mathfrak {A}\}\).
- 6.
Ideas of this kind could, perhaps, reconcile the basic idea of Niels Bohr [26, 27] on fundamental role of a “classical background” in formulations of QM with the postulate that QM is the basic theory.
- 7.
The concepts of “system”, and “physical system” are taken here to be as intuitively clear.
- 8.
This point was important also in the discussion about (im-)possibility of deducing the linearity of QM-time evolutions from mere quantal kinematics together with the so called “No-Signaling Condition”, cf. [46].
- 9.
Let us remember here that no unbounded symmetric linear operator A acting on a Hilbert space \(\mathcal{H}\) can be defined on the whole space \(\mathcal{H}:\ D(A)\subsetneqq \mathcal{H}\).
- 10.
This is so called “passive symmetry transformation”, contrasted to the “active” one, when the ‘physical system’ is moved in the fixed environment; these two ways of understanding of transformations applied to a system are mathematically equivalent.
- 11.
Our formalism is built for the nonrelativistic situations. If the space V was the Minkowski space and our considerations were Einstein-Lorentz–relativistic, the condition for the commutativity in (1.4.2) would be the space–like separation instead of the disjointness of the domains \(u,v\subset V\).
- 12.
A densely defined linear mapping \(\delta :D(\delta )\subset \mathfrak {A}\rightarrow \mathfrak {A}\) is a derivation on \(\mathfrak {A}\) if it satisfies the Leibniz rule : \(\delta (xy)=\delta (x)y+x\delta (y)\ \forall \ x,y\in D(\delta )\subset \mathfrak {A}\).
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Bóna, P. (2020). Introduction. In: Classical Systems in Quantum Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-030-45070-0_1
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DOI: https://doi.org/10.1007/978-3-030-45070-0_1
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