Yangian Dirac Harmonic Oscillator

C. N.Yang [1] and R. J. Baxter [2] separately established Quantum Yang-Baxter Equation (For short, QYBE) in 1960s. Since then the investigations on quantum integrable models have been greatly promoted. Worthy of mention especially is that the Yangian and quantum algebra was established by V. G. Drinfeld[7-9] in 1985 that offer a cogent mathematic method for the studies about the symmetry of quantum integrable models in physics. After several decades, people have made great progress both in different kinds of realizations of Yangian and in the investigations on quantum integrable models, giving new physical understanding and theoretical results. For example, the Yangian symmetry and RTT integrability of hydrogen atom had been reported at length [21]. But many pursuits were concentrated on non-relativistic area.In this paper, we mainly apply the theory of Yangian to relativistic area, studying Dirac oscillator by using Yangian. First, we find one kind of realization of Yangian and under this condition we investigate commutation relations between the generators of Yangian and the Hamiltonian. It can be concluded that Dirac oscillator doesn' t satisfy Yangian symmetry underthis condition, namely, from which we know that, where c is a constant and f(J) is the functionof J , we are able to construct the shift operators that are thefunctions of /(J) for different quantum states. The final results are accord with those shift operators obtained by0.L. delange [14]through algebraic method in 1990. Then we construct the corresponding transfer matrix to determine the RTT integrability of Dirac oscillator and gain the conserved quantities of the system according to quantum determinant.Second, we try to find another kind of realization of Yangian so that we can study the symmetry of this system based on a different point of view. We find that there is Yangian symmetry in Dirac oscillator, as a result, we are able to shift one degenerate state to anther in the same energy level. Then we construct the corresponding transfer matrix to determine the RTT integrability of Dirac oscillator.Thus it can be seen that we gained good results from Yangian. We research into its applications to Dirac oscillator, namely study Yangian symmetry of Dirac oscillator and find out the shift operators for different states through different realizations of Yangian.