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From Quantum Optics to Non-Commutative Geometry : A Non-Commutative Version of the Hopf Bundle, Veronese Mapping and Spin Representation | | Kazuyuki Fujii
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26 Feb 2005 | | Subject: | Quantum Physics; Mathematical Physics | quant-ph hep-th math-ph math.MP | | Affiliation: | Yokohama City University | | Abstract: | 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 (${cal F} imes {ket{0}cup {ket{0}} imes {cal F} subset {cal F} imes {cal F}$) 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. | | Source: | arXiv, quant-ph/0502174 | | Services: | Forum | Review | PDF | Favorites | |
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