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Quasiclassical and quantum systems of angular momentum. I. Group algebras as a framework for quantum-mechanical models with symmetries. (English) Zbl 1238.81133

The purpose of this work (being the first part of three) is to provide a solid formal framework for quantum-mechanical models and their symmestries. To this extent, an approach based on the theory of \(H^{+}\) algebras (i.e., a Banach algebra the underlying space of which is a Hilbert space) introduced by W. Ambrose [Trans. Am. Math. Soc. 57, 364–386 (1945; Zbl 0060.26906)] and group algebras is considered for the description of quantum dynamics with underlying groups \(SU(2)\) and \(SO(3)\). The use of group algebras offers itself two different but related ways. The first one uses the structure of topological groups (locally compact), and the possibility of the pointwise product of functions is used in addition to the convolution product of group algebras. Analyzing these products the well known Clebsch-Gordan coefficients are recovered, which further provides the physical interpretation of composing angular momenta [A. R. Edmonds, Angular momentum in quantum mechanics. Investigations in Physics, No. 4. Princeton, New Jersey: Princeton University Press (1957; Zbl 0079.42204)]. The second scheme is based on the Schrödinger framework of Quantum Mechanics. In this context, the representations of the group algebra are used to analyze composed systems and the quasiclassical limits. A nice interpretation in terms of the coadjoint orbits of the group \(SU(2)\) is obtained.

MSC:

81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81R15 Operator algebra methods applied to problems in quantum theory
22E70 Applications of Lie groups to the sciences; explicit representations
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