Abstract
Specific examples of the “classical projections”, introduced in a general form in Chap. 3, are obtained here by choosing specific symmetry groups of physical systems, i.e. the Heisenberg \(2N+1\) dimensional group and some of its extensions, the rotation group SO(3) for the description of “classical spin”; briefly mentioned are also the Galilean group and the Poincaré group. The transition to classical description of systems of identical particles is also described.
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Notes
- 1.
It is left to the reader’s assessment, whether the forthcoming reformulation could be helpful for better understanding of the “classical limit \( \hbar \rightarrow 0\)” of the dynamics.
- 2.
This fact was kindly announced to the author by Prof. Klaus Hepp (in 1985).
- 3.
In \(\mathcal{H}\equiv L^2({{\mathbb R}}^n,\mathrm{d}^n x)\), it is defined as \([U_\pi \psi ](x):=\psi (-x),\ \forall \psi \in \mathcal{H},\ x\in {{\mathbb R}}^n\).
- 4.
This relation between spin and statistics can be obtained as a consequence of mathematical axiomatics of relativistic quantum field theory, cf. e.g. [301].
- 5.
This vector space is, as could be seen from the formula, the image of the tangent space \(T_{{{\varphi }}'}\widetilde{O}_{{{\varphi }} +}\) by the tangent map of the mapping \(PP_+^{{\varphi }}\).
- 6.
For \(\mathfrak {g}_N=\bigoplus _{j=1}^N \mathfrak {g}^{(j)},\ \mathfrak {g}^{(j)}\) are copies of \(\mathfrak {g}\), one has \(\xi :=\sum _{j=1}^N \xi _j\) with \(\xi _j\in \mathfrak {g}^{(j)},\ X^j_\xi :=X_{\xi _j}\in U(\mathfrak {g})\).
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Bóna, P. (2020). Examples of Classical Mechanical Projections. In: Classical Systems in Quantum Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-030-45070-0_4
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DOI: https://doi.org/10.1007/978-3-030-45070-0_4
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