From Quantum Optics to Non–Commutative Geometry : A Non–Commutative Version of the Hopf Bundle, Veronese Mapping and Spin Representation

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 (F × {|0〉} ∪ {|0〉} × F ⊂ F × 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 CP 1 to CPn 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. ∗E-mail address : [email protected] 1