Physical Interpretation of Angular Projection Operators

The fuzzy disc is a disc-shaped region in a noncommutative plane,
and is a fuzzy approximation of a commutative disc. In our paper http://arxiv.org/abs/1206.6602 , we showed 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 studied properties of angular projection operators, which correspond to the decomposition of the fuzzy disc radially.

There are soliton solutions in a (2+1)-dimensional field theory with noncommutative space coordinates. We demonstrated it by explicitly constructing such solutions with a help of the angular projection operators. As a physical interpretation of them, we focused on the fact that they have the same mass (tension) as the D0-branes. Actually, the GMS solitons, which correspond to a concentric cutting of the fuzzy disc, can be regarded as the D0-brane.

Along this line, we had been seeking a clearer interpretation of the angular noncommutative solitons. Now my collaborator and I have just realized the similarity between the angular noncommutative solitons and a physical quantity in optics or laser physics.

As we have already mentioned in our previous paper, both of them are expressed by the Laguarre-Gaussian function, though the origins are completely different in each case. We are now trying to give a proof that through a key word, "the Laguarre-Gaussian beam", the angular noncommutative solitons can be regarded as a motion of the D0-brane.