Lie-algebraic Formulation Research Articles

Generalized Weierstrass–Enneper representations of Euclidean, spacelike, and timelike surfaces: a unified Lie-algebraic formulation

The complex-analytic approach to constructing minimal surfaces carried out by Weierstrass and Enneper has since been extended to create conformal parametrizations of minimal and more general surfaces, including Euclidean, spacelike, and timelike surfaces in three-dimensional Euclidean and Lorentz spaces. In this work, we present a Lie-algebraic formulation that unites these representations into one coherent framework and completes the formulas in the general timelike case. Integrable moving frames for surfaces are created inside of three-dimensional Lie algebras by applying a scaling, together with an isometric action of the appropriate Lie group, to a standard constant orthonormal frame. Using complex coordinates for Euclidean and spacelike surfaces and hyperbolic coordinates for timelike surfaces allows for simple characterizations of geometric properties of the surfaces in terms of the complex and hyperbolic structures. We give expressions in this framework of the first and second fundamental forms and of mean and Gaussian curvatures on Euclidean, spacelike, and timelike surfaces. The expression of the Gaussian curvature affirms in an efficient and elegant way that Gauss’ Theorem Egregium holds for surfaces of all three types. We also provide examples and illustrations of special surfaces and deformations of surfaces arising from the Weierstrass–Enneper representations.

Algebraic metrology: Nonoptimal but pretty good states and bounds

We investigate quantum metrology using a Lie algebraic approach for a class of Hamiltonians, including local and nearest-neighbor interaction Hamiltonians. Using this Lie algebraic formulation, we identify and construct highly symmetric states that admit Heisenberg scaling in precision for phase estimation in the absence of noise. For the nearest-neighbor Hamiltonian we also perform a numerical scaling analysis of the performance of pretty good states and derive upper bounds on the quantum Fisher information.

Geometric structure-preserving optimal control of a rigid body

In this paper, we study a discrete variational optimal control problem for a rigid body. The cost to be minimized is the external torque applied to move the rigid body from an initial condition to a pre-specified terminal condition. Instead of discretizing the equations of motion, we use the discrete equations obtained from the discrete Lagrange---d'Alembert principle, a process that better approximates the equations of motion. Within the discrete-time setting, these two approaches are not equivalent in general. The kinematics are discretized using a natural Lie-algebraic formulation that guarantees that the flow remains on the Lie group SO(3) and its algebra $\mathfrak{s}\mathfrak{o}(3)$ . We use the Lagrange method for constrained problems in the calculus of variations to derive the discrete-time necessary conditions. We give a numerical example for a three-dimensional rigid body maneuver.

SU(1,1) Coherent States for the Generalized Two-Mode Time-Dependent Quadratic Hamiltonian System

The SU(1,1) coherent states, so-called Barut-Girardello coherent state and Perelomov coherent state, for the generalized two-mode time-dependent quadratic Hamiltonian system are investigated through SU(1,1) Lie algebraic formulation. Two-mode Schrodinger cat states defined as an eigenstate of \(\hat{K}_{-}^{2}\) are also studied. We applied our development to two-mode Caldirola-Kanai oscillator which is a typical example of the time-dependent quadratic Hamiltonian system. The time evolution of the quadrature distribution for the probability density in the coherent states are analyzed for the two-mode Caldirola-Kanai oscillator by plotting relevant figures.

Dynamic Lie algebraic formulation of second-order optical nonlinearity for substituted benzenes

A dynamic Lie algebraic (DLA) formulation is applied to the study of nonlinear optical properties of the substituted benzenes. We have described the generation of the dynamic Lie algebra for the model Hamiltonian used in the present study. In terms of these elements of the dynamic algebra we express the evolution operator as a function of the group parameters, which can be determined by means of solving a system of coupled nonlinear differential equations. Thus, in terms of the density matrix operator formalism in statistical mechanics we obtain the statistical averages of the electric polarization and then derive readily an expression for the hyperpolarizability of the paradisubstituted benzenes. Comparisons with experimental observations and other quantum calculations of the hyperpolarizabilities for the paradisubstituted benzenes are made qualitatively and quantitatively. These results imply that the DLA method appears useful in describing the nonlinear optical phenomena in the substituted benzene molecules. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 93: 335–343, 2003

SO(10) UNIFICATION IN NONCOMMUTATIVE GEOMETRY REVISITED

We investigate the SO(10) unification model in a Lie-algebraic formulation of noncommutative geometry. The SO(10) symmetry is broken by a 45-Higgs and the Majorana mass term for the right neutrinos (126-Higgs) to the standard model structure group. We study the case where the fermion masses are as general as possible, which leads to two 10-multiplets, four 120-multiplets and two additional 126-multiplets of Higgs fields. This Higgs structure differs considerably from the two Higgs multiplets 16 ⊗ 16* and 16c ⊗ 16* used by Chamseddine and Fröhlich. We find the usual tree level predictions of noncommutative geometry: [Formula: see text], [Formula: see text] and g2=g3 as well as mH≤ mt.

SU(1,1) Lie algebraic approach to linear dissipative processes in quantum optics

An SU(1,1) Lie algebraic formulation is presented for investigating the linear dissipative processes in quantum optical systems. The Liouville space formulation, thermo field dynamics, and the disentanglement theorem of SU(1,1) Lie algebra play essential roles in this formulation. In the Liouville space, the time-evolution equation for the state vector of a system is solved algebraically by using the decomposition formulas of SU(1,1) Lie algebra and the thermal state condition of thermo field dynamics. The presented formulation is used for investigating a dissipative nonlinear oscillator, the quantum mechanical model of phase modulation, and the photon echo in the localized electron–phonon system. This algebraic formulation gives a systematic treatment for investigating the phenomena in quantum optical systems.