Oscillator Quantum Algebra and Deformed su(2) Algebra

A difference operator realization of quantum deformed oscillator algebra Hq(1) with a Casimir operator freedom is introduced. We show that thisHq(1) have a nonlinear mapping to the deformed quantum su(2) which was introduced by Fujikawa et al. We also examine the cyclic representation obtained by this difference operator realization and the possibility to analyze a Bloch electron problem by Hq(1). Quantum deformed algebra[1] was firstly introduced to study the inverse scattering problems and the integral systems, which have rich structures, Yang-Baxter equations[2]. They are going to be standard techniques of theoretical physics. Wiegmann and Zabrodin found that a system of the Bloch electron on a two dimensional square lattice [5][6][7] can be expressed in a linear combination of generators of the algebra Uq(sl2). This corresponds to the fact that Uq(sl2) gives a foundation to obtain an exact solution of the Bethe Ansatz equation. Bidenharn[3] and Macfarlane[4] introduced q-deformed oscillator algebraHq(1) to construct Uq(sl2) in the manner of Schwinger’s construction of conventional su(2). Recently a new method to construct a representation of Hq(1), which is manifestly free of negative norm, was proposed[12]. This Hq(1) enjoys Hopf structure[14]. This algebra was used to analyze the phase operator problem [9][10] of the photon with the notion of index [13]. Fujikawa et al[8] constructed a new deformation of su(2) algebra with a “Schwinger term” ∗JSPS fellow, E-mail address : [email protected]