## Journal of Geometry and Symmetry in Physics

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

#### Abstract

We use the mathematical structure of group algebras and $H^{+}$-algebras for describing certain problems concerning the quantum dynamics of systems of angular momenta, including also the spin systems. The underlying groups are ${\rm SU}(2)$ and its quotient ${\rm SO}(3,\mathbb{R})$. The proposed scheme is applied in two different contexts. Firstly, the purely group-algebraic framework is applied to the system of angular momenta of arbitrary origin, e.g., orbital and spin angular momenta of electrons and nucleons, systems of quantized angular momenta of rotating extended objects like molecules. Secondly, the other promising area of applications is Schrödinger quantum mechanics of rigid body with its often rather unexpected and very interesting features. Even within this Schrödinger framework the algebras of operators related to group algebras are a very useful tool. We investigate some problems of composed systems and the quasiclassical limit obtained as the asymptotics of “large” quantum numbers, i.e., “quickly oscillating” wave functions on groups. They are related in an interesting way to geometry of the coadjoint orbits of ${\rm SU}(2)$.

#### Article information

Source
J. Geom. Symmetry Phys., Volume 21 (2011), 61-94.

Dates
First available in Project Euclid: 25 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.jgsp/1495677828

Digital Object Identifier
doi:10.7546/jgsp-21-2011-61-94

Mathematical Reviews number (MathSciNet)
MR2856236

Zentralblatt MATH identifier
1238.81133

#### Citation

Slawianowski, Jan J.; Kovalchuk, Vasyl; Martens, Agnieszka; Golubowska, Barbara; Rozko, Ewa E. Quasiclassical and Quantum Systems of Angular Momentum. Part I. Group Algebras as a Framework for Quantum-Mechanical Models with Symmetries. J. Geom. Symmetry Phys. 21 (2011), 61--94. doi:10.7546/jgsp-21-2011-61-94. https://projecteuclid.org/euclid.jgsp/1495677828