Standard and NonStandard Quantum Models : A NonCommutative Version of the Classical System of SU(2) and SU(1,1) Arising from Quantum Optics
Abstract
This is a challenging paper including some review and new results. Since the noncommutative version of the classical system based on the compact group SU(2) has been constructed in (quantph/0502174) by making use of JaynesCommings model and socalled Quantum Diagonalization Method in (quantph/0502147), we construct a noncommutative version of the classical system based on the noncompact group SU(1,1) by modifying the compact case. In this model the Hamiltonian is not hermite but pseudo hermite, which causes a big difference between two models. For example, in the classical representation theory of SU(1,1), unitary representations are infinite dimensional from the starting point. Therefore, to develop a unitary theory of noncommutative system of SU(1,1) we need an infinite number of noncommutative systems, which means a kind of {\bf second noncommutativization}. This is a very hard and interesting problem. We develop a corresponding theory though it is not always enough, and present some challenging problems concerning how classical properties can be extended to the noncommutative case. This paper is arranged for the convenience of readers as the first subsection is based on the standard model (SU(2) system) and the next one is based on the nonstandard model (SU(1,1) system). This contrast may make the similarity and difference between the standard and nonstandard models clear.
 Publication:

arXiv eprints
 Pub Date:
 June 2005
 DOI:
 10.48550/arXiv.quantph/0506026
 arXiv:
 arXiv:quantph/0506026
 Bibcode:
 2005quant.ph..6026F
 Keywords:

 Quantum Physics;
 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 Latex files, 50 pages, minor changes. Instead of the paper quantph/0502174 this will be published in the special issue of International Journal of Geometric Methods in Modern Physics