Quasiclassical and Quantum Systems of Angular Momentum. Part I. Group Algebras as a Framework for Quantum-Mechanical Models with Symmetries

摘要

We use the mathematical structure of group algebras and $H^{+}$-algebras fordescribing certain problems concerning the quantum dynamics of systems ofangular momenta, including also the spin systems. The underlying groups are${\rm SU}(2)$ and its quotient ${\rm SO}(3,\mathbb{R})$. The scheme developedis applied in two different contexts. Firstly, the purely group-algebraicframework is applied to the system of angular momenta of arbitrary origin,e.g., orbital angular momenta of electrons and nucleons, systems of quantizedangular momenta of rotating extended objects like molecules. The otherpromising area of applications is Schr\"odinger quantum mechanics of rigid bodywith its often rather unexpected and very interesting features. Even withinthis Schr\"odinger framework the algebras of operators related to groupalgebras are a very useful tool. We investigate some problems of composedsystems and the quasiclassical limit obtained as the asymptotics of "large"quantum numbers, i.e., "quickly oscillating" functions on groups. They arerelated in an interesting way to geometry of the coadjoint orbits of ${\rmSU}(2)$.