frequency is (Gaussian units) and is the speed of the particle about the magnetic field, and B is the magnitude of the magnetic field. The speed formula comes from setting the classical kinetic energy equal to the spacing between Landau levels in the quantum treatment of a charged particle in a magnetic field. In this geometry, the interaction energy can be written where is the distance between the centers of the current loops and is a Bessel function of the first kind. In obtaining the interaction energy we made use of the integral Electric field due to a density perturbation The chemical potential near equilibrium, is given by where is the potential energy of an electron in an electric potential and and are the number of particles in the electron gas in the absence of and in the presence of an electrostatic potential, respectively. The density fluctuation is then where is the area of the material in the plane perpendicular to the magnetic field. Poisson's equation yields where The propagator is then and the interaction energy becomes where in the second equality (Gaussian units) we assume that the vortices had the same energy and the electron charge. In analogy with plasmons, the force carrier is the quantum version of the upper hybrid oscillation which is a longitudinal plasma wave that propagates perpendicular to the magnetic field. Currents with angular momentum Delta function currents Unlike classical currents, quantum current loops can have various values of the Larmor radius for a given energy. Landau levels, the energy states of a charged particle in the presence of a magnetic field, are multiply degenerate. The current loops correspond to angular momentum states of the charged particle that may have the same energy. Specifically, the charge density is peaked around radii of where is the angular momentum quantum number. When we recover the classical situation in which the electron orbits the magnetic field at the Larmor radius. If currents of two angular momentum and interact, and we assume the charge densities are delta functions at radius , then the interaction energy is The interaction energy for is given in Figure 1 for various values of . The energy for two different values is given in Figure 2. Quasiparticles For large values of angular momentum, the energy can have local minima at distances other than zero and infinity. It can be numerically verified that the minima