International Journal of Modern Physics A

I gave a rambling talk about gravity and its many mysteries at Chen-Ning Yang's 85th Birthday Celebration held in November 2007. I do not have any answers.

The commutator of enlarged vector fields was explicitly computed for integrable coupling systems associated with semidirect sums of Lie algebras. An algebraic structure of zero curvature representations is then established for such integrable coupling systems. As an application example of this algebraic structure, the commutation relations of Lax operators corresponding to the enlarged isospectral and nonisospectral AKNS flows are worked out, and thus a τ-symmetry algebra for the AKNS integrable couplings is engendered from this theory.

The algebra of observables of planar electrons subject to a constant background magnetic field B is given by , the product of two mutually commuting Moyal algebras. It describes the free Hamiltonian and the guiding center coordinates. We argue that itself furnishes a representation space for the actions of these two Moyal algebras, and suggest physical arguments for this choice of the representation space. We give the proper setup to couple the matter fields based on to electromagnetic fields which are described by the Abelian commutative gauge group , i.e. gauge fields based on . This enables us to give a manifestly gauge covariant formulation of integer quantum Hall effect (IQHE). Thus, we can view IQHE as an elementary example of interacting quantum topologies, where matter and gauge fields based on algebras with different θ′ appear. Two-particle wave functions in this approach are based on . We find that the full symmetry group in IQHE, which is the semidirect product acts on this tensor product using the twisted coproduct Δ_θ. Consequently, as we show, many particle sectors of each Landau level have twisted statistics. As an example, we find the twisted two particle Laughlin wave functions.

Extension of symplectic geometry on manifolds to the supersymmetric case is considered. In the even case it leads to the even symplectic geometry (or, equivalently, to the geometry on supermanifolds endowed with a nondegenerate Poisson bracket) or to the geometry on an even Fedosov supermanifolds. It is proven that in the odd case there are two different scalar structures (namely, an odd closed differential 2-form and the antibracket) which can be used for construction of two different types of odd geometries on supermanifolds.

The lowest order contribution of the amplitude of Dirac–Coulomb scattering in de Sitter space–time is calculated assuming that the initial and final states of the Dirac field are described by exact solutions of the free Dirac equation on de Sitter space–time with a given energy and helicity. We find that the total energy is conserved in the scattering process.

Through an ansatz specifying the azimuthal-angle dependence of the solution, the static field equation for vortex of the Faddeev model is converted to an algebraic ordinary differential equation. An approximate analytic expression of the vortex solution is explored so that the energy per unit vortex length becomes as small as possible. It is observed that the minimum energy of vortex is approximately proportional to the integer which specifies the solution.

We apply the S-matrix based finite temperature formalism to nonrelativistic Bose and Fermi gases in 1+1 and 2+1 dimensions. For the (2+1)-dimensional case in the constant scattering length approximation, the free energy is given in terms of Roger's dilogarithm in a way analagous to the thermodynamic Bethe ansatz for the relativistic (1+1)-dimensional case. The 1d fermionic case with a quasiperiodic two-body potential is closely connected with the Riemann hypothesis.

In two-dimensional noncommutative space for the case of both position–position and momentum–momentum noncommuting, the consistent deformed bosonic algebra at the nonperturbation level described by the deformed annihilation and creation operators is investigated. A general relation between noncommutative parameters is fixed from the consistency of the deformed Heisenberg–Weyl algebra with the deformed bosonic algebra. A Fock space is found, in which all calculations can be similarly developed as if in commutative space and all effects of spatial noncommutativity are simply represented by parameters.

The asymptotic iteration method (AIM) is applied to obtain highly accurate eigenvalues of the radial Schrödinger equation with the singular potential V(r) = r²+λ/r^α(α,λ>0) in arbitrary dimensions. Certain fundamental conditions for the application of AIM, such as a suitable asymptotic form for the wave function, and the termination condition for the iteration process, are discussed. Several suggestions are introduced to improve the rate of convergence and to stabilize the computation. AIM offers a simple, accurate, and efficient method for the treatment of singular potentials, such as V(r), valid for all ranges of coupling λ.

This paper is devoted to investigate the cold plasma wave properties outside the event horizon of the Schwarzschild planar analogue. The dispersion relations are obtained from the corresponding Fourier analyzed equations for nonrotating and rotating, nonmagnetized and magnetized backgrounds. These dispersion relations provide complex wave numbers. The wave numbers are shown in graphs to discuss the nature and behavior of waves and the properties of plasma lying in the vicinity of the Schwarzschild event horizon.

Detailed study of a neutron Laue diffraction in a noncentrosymmetric crystal was carried out using a prototype of the experimental setup to measure the neutron electric dipole moment (nEDM) by Laue diffraction method (LDM). A new kind of spin rotation effect was observed for the first time. We attribute this effect to a difference of the two Bloch waves amplitudes in the crystal, which is caused by the presence of the definite small crystal deformation. In a revised scheme of the experiment, this effect could be exploited for a purposeful manipulation of the Bloch waves.

In this paper, spinor and vector decomposition of SU(2) gauge potential are presented and their equivalence is constructed using a simple proposal. We also obtain the action of Skyrme–Faddeev model from the SU(2) massive gauge field theory which is proposed according to the gauge invariant principle. Then, the knot structure in Skyrme–Faddeev model is discussed in terms of the so-called ϕ-mapping topological current theory. The topological charge of the knot is characterized by the Hopf indices and the Brouwer degrees of ϕ-mapping, naturally. At last, we briefly discussed the topological invariant–Hopf invariant which describes the topology of these knots. It is shown that Hopf invariant is the total number of all the linking numbers and self-linking numbers of these knots.