Boson realization of nonlinear SO(3) algebra

Fan Hong-Yi

Guy Burdet, M. Perrin

Eugene R. Marshalek

Two new closely related boson realizations of the Heisenberg–Weyl algebra are derived that are directly equivalent to the introduction of spherical coordinates for the isotropic three-dimensional harmonic oscillator. This provides another way to obtain matrix elements of all physical operators connecting angular-momentum eigenstates, which are represented here by simple products of boson operators acting on the vacuum. Two of these bosons carry the angular-momentum quantum numbers while the third is a novel ‘‘radial boson.’’ In this approach, one begins by finding the nonunitary Dyson and then the unitarized Holstein–Primakoff realizations for the spectrum-generating SU(1,1)×SO(3) algebra of the oscillator. The corresponding mappings of the Heisenberg–Weyl generators are then obtained from their tensor properties under SU(1,1)×SO(3). The Dyson map of the Heisenberg–Weyl generators is not uniquely determined in this way, but the Holstein–Primakoff map is nevertheless unique. The possibility exists for generalization to isotropic oscillators of higher dimension, which are of interest in connection with nuclear collective motion.

Zhang Li, Gao Yibo, Yanmin Li

When a boson interacts with another to form a composite system with SO(3) dynamic symmetry, it is shown that there exists the q-deformed bosonic excitation satisfying the q-deformed Heisenberg commutation relation in the quasi-classical limit that the angular momentum j for SO(3) is large, but not infinite. In second quantization this quasi-excitation is associated with the boson realization of SO(3) Lie algebra. Physically, the phenomena of q-deformed excitation can happen in many models of quantum dynamics, such as super emission from a system of many identical two-level atoms, the spin wave in Heisenberg chain, the high spin precession and the coherent output of Bose–Einstein atoms in a trap. Especially, in these models, the deformation parameter q is well defined intrinsically by a conservative quantity, such as the total atomic number and the angular momentum.

M. Wagner

P. Gulshani

Various definitions of the coherent states of the angular momentum are shown to be special cases of the oscillator-like coherent states of the groups SU(2) and SO(3) obtained by Mikhailov on the basis of a generalized Schwinger boson realization of the angular momentum algebra. This realization is then generalized to that of the angular momentum algebra of an asymmetric top by means of a transformation from the Euler angles to the Cayley–Klein parameters. The oscillator-like coherent states of an asymmetric top, analogous to those of Mikhailov, are then constructed. It is, then, shown that Janssen's and Mostowski's definitions of the coherent states of a top are special cases of these.