Standard and Non-Standard Quantum Models : A Non-Commutative Version of the Classical System of SU(2) and SU(1,1) Arising from Quantum Optics

This is a challenging paper including some review and new results. Since the non-commutative version of the classical system based on the compact group SU(2) has been constructed in (quant-ph/0502174) by making use of Jaynes-Commings model and so-called Quantum Diagonalization Method in (quant-ph/0502147), we construct a non-commutative version of the classical system based on the non-compact 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 non-commutative system of SU(1,1) we need an infinite number of non-commutative systems, which means a kind of {\bf second non-commutativization}. 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 non-commutative 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 non-standard model (SU(1,1) system). This contrast may make the similarity and difference between the standard and non-standard models clear.