Realization of the three-dimensional quantum Euclidean space by differential operators

S. Schraml; J. Wess

doi:doi:10.1007/s100520000472

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2019 Impact factor 4.389

Particles and Fields

Eur. Phys. J. C 17, 353-358
DOI 10.1007/s100520000472

Realization of the three-dimensional quantum Euclidean space by differential operators

S. Schraml - J. Wess

Max-Planck-Institut für Physik, Föhringer Ring 6, 80805 München, Germany - Sektion Physik, Universität München, Theresienstrasse 37, 80333 München, Germany

Received: 27 June 2000 / Published online: 9 August 2000 - © Springer-Verlag 2000

Abstract
The three-dimensional quantum Euclidean space is an example of a non-commutative space that is obtained from Euclidean space by q-deformation. Simultaneously, angular momentum is deformed to so_q(3), it acts on the q-Euclidean space that becomes a so_q(3)-module algebra this way. In this paper it is shown, that this algebra can be realized by differential operators acting on $C^{\infty}$ functions on $\mathbb{R} ^3$ . On a factorspace of $C^{\infty}(\mathbb{R} ^3)$ a scalar product can be defined that leads to a Hilbert space, such that the action of the differential operators is defined on a dense set in this Hilbert space and algebraically self-adjoint becomes self-adjoint for the linear operator in the Hilbert space. The self-adjoint coordinates have discrete eigenvalues, the spectrum can be considered as a q-lattice.