From Quantum Optics to Non-Commutative Geometry : A Non-Commutative Version of the Hopf Bundle, Veronese Mapping and Spin Representation
Abstract
Description
In this paper we construct a non-commutative version of the Hopf bundle by making use of Jaynes-Commings model and so-called Quantum Diagonalization Method. The bundle has a kind of Dirac strings. However, they appear in only states containing the ground one (${\cal F}\times \{\ket{0}\} \cup \{\ket{0}\}\times {\cal F} \subset {\cal F}\times {\cal F}$) and don't appear in remaining excited states. This means that classical singularities are not universal in the process of non-commutativization.
Based on this construction we moreover give a non-commutative version of both the Veronese mapping which is the mapping from $\fukuso P^{1}$ to $\fukuso P^{n}$ with mapping degree $n$ and the spin representation of the group SU(2).
We also present some challenging problems concerning how classical (beautiful) properties can be extended to the non-commutative case.
Latex files, 28 pages. Minor changes (one page increased). To appear in the special issue of International Journal of Geometric Methods in Modern Physics
Latex files, 28 pages. Minor changes (one page increased). To appear in the special issue of International Journal of Geometric Methods in Modern Physics