Motion of vortex lines in nonlinear wave mechanics

We extend our previous analysis of the motion of vortex lines [I. Bialynicki-Birula, Z. Bialynicka-Birula and C. Sliwa, Phys. Rev. A 61, 032110 (2000)] from linear to a nonlinear Schroedinger equation with harmonic forces. We also argue that under certain conditions the influence of the contact nonlinearity on the motion of vortex lines is negligible. The present analysis adds new weight to our previous conjecture that the topological features of vortex dynamics are to a large extent universal.Comment: To appear in Phys. Rev. A, 4 page

Roots of the affine Cremona group

Let k[x_1,...,x_n] be the polynomial algebra in n variables and let A^n=Spec k[x_1,...,x_n]. In this note we show that the root vectors of the affine Cremona group Aut(A^n) with respect to the diagonal torus are exactly the locally nilpotent derivations x^a\times d/dx_i, where x^a is any monomial not depending on x_i. This answers a question due to Popov.Comment: 4 page

Electromagnetic vortex lines riding atop null solutions of the Maxwell equations

New method of introducing vortex lines of the electromagnetic field is outlined. The vortex lines arise when a complex Riemann-Silberstein vector $({\bm E} + i{\bm B})/\sqrt{2}$ is multiplied by a complex scalar function $\phi$. Such a multiplication may lead to new solutions of the Maxwell equations only when the electromagnetic field is null, i.e. when both relativistic invariants vanish. In general, zeroes of the $\phi$ function give rise to electromagnetic vortices. The description of these vortices benefits from the ideas of Penrose, Robinson and Trautman developed in general relativity.Comment: NATO Workshop on Singular Optics 2003 To appear in Journal of Optics

Collision entropy and optimal uncertainty

We propose an alternative measure of quantum uncertainty for pairs of arbitrary observables in the 2-dimensional case, in terms of collision entropies. We derive the optimal lower bound for this entropic uncertainty relation, which results in an analytic function of the overlap of the corresponding eigenbases. Besides, we obtain the minimum uncertainty states. We compare our relation with other formulations of the uncertainty principle.Comment: The manuscript has been accepted for publication as a Regular Article in Physical Review

Homogeneous components in the moduli space of sheaves and Virasoro characters

The moduli space $\mathcal M(r,n)$ of framed torsion free sheaves on the projective plane with rank $r$ and second Chern class equal to $n$ has the natural action of the $(r+2)$-dimensional torus. In this paper, we look at the fixed point set of different one-dimensional subtori in this torus. We prove that in the homogeneous case the generating series of the numbers of the irreducible components has a beautiful decomposition into an infinite product. In the case of odd $r$ these infinite products coincide with certain Virasoro characters. We also propose a conjecture in a general quasihomogeneous case.Comment: Published version, 19 page

Pinning and transport of cyclotron/Landau orbits by electromagnetic vortices

Electromagnetic waves with phase defects in the form of vortex lines combined with a constant magnetic field are shown to pin down cyclotron orbits (Landau orbits in the quantum mechanical setting) of charged particles at the location of the vortex. This effect manifests itself in classical theory as a trapping of trajectories and in quantum theory as a Gaussian shape of the localized wave functions. Analytic solutions of the Lorentz equation in the classical case and of the Schr\"odinger or Dirac equations in the quantum case are exhibited that give precise criteria for the localization of the orbits. There is a range of parameters where the localization is destroyed by the parametric resonance. Pinning of orbits allows for their controlled positioning -- they can be transported by the motion of the vortex lines.Comment: This version differs from the printed paper in having the full titles of all referenced pape

Quantum lattice gases and their invariants

The one particle sector of the simplest one dimensional quantum lattice gas automaton has been observed to simulate both the (relativistic) Dirac and (nonrelativistic) Schroedinger equations, in different continuum limits. By analyzing the discrete analogues of plane waves in this sector we find conserved quantities corresponding to energy and momentum. We show that the Klein paradox obtains so that in some regimes the model must be considered to be relativistic and the negative energy modes interpreted as positive energy modes of antiparticles. With a formally similar approach--the Bethe ansatz--we find the evolution eigenfunctions in the two particle sector of the quantum lattice gas automaton and conclude by discussing consequences of these calculations and their extension to more particles, additional velocities, and higher dimensions.Comment: 19 pages, plain TeX, 11 PostScript figures included with epsf.tex (ignore the under/overfull \vbox error messages

On the uncertainty relations and squeezed states for the quantum mechanics on a circle

The uncertainty relations for the position and momentum of a quantum particle on a circle are identified minimized by the corresponding coherent states. The sqeezed states in the case of the circular motion are introduced and discussed in the context of the uncertainty relations.Comment: 4 figure