Angles in Fuzzy Disc and Angular Noncommutative Solitons

The fuzzy disc, introduced by the authors of (1), is a disc-shaped region in a noncommutative plane, and is a fuzzy approximation of a commutative disc. In this paper we show that one can introduce a concept of angles to the fuzzy disc, by using the phase operator and phase states known in quantum optics. We gave a description of the fuzzy disc in terms of operators and their commutation relations, and studied properties of angular projection operators. A similar construction for the fuzzy annulus is also given. As an application, we constructed fan-shaped soliton solutions of a scalar eld theory on the fuzzy disc. We also applied this concept to the theory of noncommutative gravity we proposed in (2). In addition, possible connections to some systems in physics are suggested.