On q-analogues of the quantum harmonic oscillator and the quantum group \(SU(2)_ q\). (English) Zbl 0722.17009

Macfarlane, A. J.

J. Phys. A, Math. Gen. 22, No. 21, 4581-4588 (1989).

From the introduction: “Here we wish to generalize to \(SU(2)_ q\), the simplest quantum group, the q-deformation of the Lie algebra of SU(2), Schwinger’s approach [J. Schwinger, On angular momentum’, in ‘Quantum Theory of Angular Momentum (ed. by L. C. Biedenharn and H. van Dam), Academic Press, 1965, p. 229-279] to the quantum theory of angular momentum. To achieve this, a q-deformation of the quantum harmonic oscillator formalism has to be developed. Much of the present paper is devoted to this task. Then the algebra of \(SU(2)_ q\), and its representations can be realized in terms of the variables of two independent q-deformed harmonic oscillators. Our results involve explicit coordinate of wavefunctions representations as well as abstract Hilbert space versions.”

Reviewer: V.Pestov (Novosibirsk)

MSC:

17B37	Quantum groups (quantized enveloping algebras) and related deformations
81R50	Quantum groups and related algebraic methods applied to problems in quantum theory
53D50	Geometric quantization

Keywords:

quantum group; Schwinger’s approach; q-deformation; quantum harmonic oscillator; wavefunctions representations