Narrow Results

The dynamical (super)symmetries for various monopole systems are reviewed. For a Dirac monopole, non-smooth Runge–Lenz vector can exist; there is, however, a spectrum-generating conformal o(2,1) dynamical symmetry that extends into osp(1/1) or osp(1/2) for spin 1/2 particles. Self-dual 't Hooft–Polyakov-type monopoles admit an su(2/2) dynamical supersymmetry algebra, which allows us to reduce the fluctuation equation to the spin 0 case. For large r, the system reduces to a Dirac monopole plus a suitable inverse-square potential considered before by McIntosh and Cisneros, and by Zwanziger in the spin 0 case, and to the "dyon" of D'Hoker and Vinet for spin 1/2. The asymptotic system admits a Kepler-type dynamical symmetry as well as a "helicity-supersymmetry" analogous to the one Biedenharn found in the relativistic Kepler problem. Similar results hold for the Kaluza–Klein monopole of Gross–Perry–Sorkin. For the magnetic vortex, the N = 2 supersymmetry of the Pauli Hamiltonian in a static magnetic field in the plane combines with the o(2) × o(2,1) bosonic symmetry into an o(2) × osp(1/2) dynamical superalgebra.

In the context of Friedmann–Robertson–Walker (FRW) spacetime with zero spatial curvature, we consider a multi-scalar tensor cosmology model under the pretext of obtaining quadratic conservation laws. We propose two new interaction potentials of the scalar field. Integral to this task is the existence of dynamical Noether symmetries which are Lie–Bäcklund transformations of the physical system. Finally, analytical solutions of the field are found corresponding to each new model. In one of the models, we find that the scale factor mimics $\mathrm{\Lambda }$-cosmology in a special case.

Several dynamical symmetries of the Dirac Hamiltonian are reviewed and the conditions under which such symmetries hold are considered. These include relativistic spin and orbital angular momentum symmetries, SO(4)×SU_σ(2) symmetry for generalized relativistic hydrogen atom that includes an extra Lorentz scalar potential, SU(3)×SU_σ(2) symmetry for the relativistic simple harmonic oscillator. The energy spectrum in each case is calculated from group-theoretic considerations.

We consider a multiscalar tensor cosmology model described by Friedmann–Robertson–Walker (FRW) spacetime with zero spatial curvature. Three specific scalar interaction potentials that characterize the model are analyzed under a set of coordinate transformations. By implication, we solve for the wave function of the universe, reduce the dimension of the underlying Hamiltonian system and consequently, establish analytical solutions of the multiscalar model’s field equations.

The elementary (one-particle) $U(d_1+d_2)\supset U(d_1)\otimes U(d_2)$ isoscalar factors, involving two-rowed initial and final irreducible representations (irreps), are obtained. Using the latter, further some of the multiplicity-free $U(d_1+d_2)\supset U(d_1)\otimes U(d_2)$ isoscalar factors, involving the $U(d_1+d_2)$ couplings of the type $[E_1,E_2,\dot{0}]\times [p,\dot{0},-q]\to [E_1^{\prime },E_2^{\prime },\dot{0}]$, are obtained using the building-up procedure. The present results extend the ones obtained earlier for the one-rowed initial and final irreps.

In this paper I start from the early history of application of group theory to nuclear physics, then I discuss the seniority scheme in nuclear shell model under strong pairing interaction. Then I go to the Interacting Boson Model (IBM) and its applications. As the microscopic foundation, I describe nucleon pair approximation and its application to complex nuclei. Finally I discuss the microscopic method to derive the IBM Hamiltonian in strongly deformed nuclei from the mean-field theory.