Dimitra Karabali

FERMIONS IN THE LOWEST LANDAU LEVEL: Bosonization, W∞ Algebra, Droplets, Chiral Bosons

Abstract We present field theoretical descriptions of massless (2+1) dimensional nonrelativistic fermions in an external magnetic field, in terms of a fermionic and bosonic second quantized language. An infinite dimensional algebra, W∞ appears as the algebra of unitary transformations which preserve the lowest Landau level condition and the particle number. In the droplet approximation it reduces to the algebra of area-preserving diffeomorphisms, which is responsible for the existence of a universal chiral boson lagrangian independent of the electrostatic potential. We argue that the bosonic droplet approximation is the strong magnetic field limit of the fermionic theory. The relation to the c = 1 string model is discussed.

One-dimensional fermions as two-dimensional droplets via Chern-Simons theory☆

Abstract Based on the observation that particle motion in one dimension maps to two-dimentional motion of a charged particle in a uniform magnetic field, constrained in the lowest Landau level, we formulate a system of one-dimensional nonrelativistic fermions by using a Chern-Simons field theory in 2 + 1 dimensions. Using a hydrodynamical formulation we obtain a two-dimensional droplet picture of one-dimensional fermions. The dynamics involved is that of the boundary between a uniform density of particles and vortices. We use the sharp boundary approximation. In order to test our approach we apply it to a system of fermions in a harmonic oscillator potential. In the case of well separated boundaries we derive the one-dimensional collective field hamiltonian. Symmetries of the theory are also discussed as properties of curves in two dimensions.

Fuzzy spaces, the M(atrix) model and the quantum Hall effect

This is a short review of recent work on fuzzy spaces in their relation to the M(atrix) theory and the quantum Hall effect. We give an introduction to fuzzy spaces and how the limit of large matrices is obtained. The complex projective spaces

WEAK INTERACTIONS ARE WEAK AT HIGH-ENERGIES

{bf CP}^k

Gauge invariant variables and the Yang–Mills–Chern–Simons theory

, and to a lesser extent spheres, are considered. Quantum Hall effect and the behavior of edge excitations of a droplet of fermions on these spaces and their relation to fuzzy spaces are also discussed.

Relativistic Particle and Relativistic Fluids: Magnetic Moment and Spin-Orbit Interactions

Abstract We use a Schrodinger picture version of the instanton calculus to analyze the suggestion that nonperturbative instanton effects, such as baryon number violation in the electroweak interaction, can become large in high-energy scattering. We confirm that the euclidean instanton results of Ringwald, Espinosa and others are correct, for models with instantons of definite size, at low energy, and present a physical explanation of the energy dependence of amplitudes found by these authors. However, we show that their dilute instanton gas approximation breaks down at higher energy. Our techniques permit us to extend the energy regine in which a reliable calculation can be done. We find that the basic e 1/α suppression of the total cross section for instanton-induced processes persists at all energies, even ∼ M /α.

Diffractive Effects and General Boundary Conditions in Casimir Energy

Abstract A Hamiltonian analysis of Yang–Mills (YM) theory in (2+1) dimensions with a level k Chern–Simons term is carried out using a gauge invariant matrix parametrization of the potentials. The gauge boson states are constructed and the contribution of the dynamical mass gap to the gauge boson mass is obtained. Long distance properties of vacuum expectation values are related to a Euclidean two-dimensional YM theory coupled to k flavors of Dirac fermions in the fundamental representation. We also discuss the expectation value of the Wilson loop operator and give a comparison with previous results.

Exact operator Hamiltonians and interactions in the droplet bosonization method

We consider relativistic charged particle dynamics and relativistic magnetohydrodynamics using symplectic structures and actions given in terms of co-adjoint orbits of the Poincare group. The particle case is meant to clarify some points such as how minimal coupling (as defined in text) leads to a gyromagnetic ratio of 2, and to set the stage for fluid dynamics. The general group-theoretic framework is further explained and is then used to set up Abelian magnetohydrodynamics including spin effects. An interesting new physical effect is precession of spin density induced by gradients in pressure and energy density. The Euler equation (the dynamics of the velocity field) is also modified by gradients of the spin density.

Boundary Conditions as Dynamical Fields

The effect of edges and apertures on the Casimir energy of an arrangement of plates and boundaries can be calculated in terms of an effective nonlocal lower-dimensional field theory that lives on the boundary. This formalism has been developed in a number of previous papers and applied to specific examples with Dirichlet boundary conditions. Here we generalize the formalism to arbitrary boundary conditions. As a specific example, the geometry of a flat plate and a half-plate placed parallel to it is considered for a number of different boundary conditions and the area-dependent and edge dependent contributions to the Casimir energy are evaluated. While our results agree with known results for those special cases (such as the Dirichlet and Neumann limits) for which other methods of calculation have been used, our formalism is suitable for general boundary conditions, especially for the diffractive effects.

Casimir effect: Edges and diffraction

We derive the exact form of the bosonized Hamiltonian for a many-body fermion system in one spatial dimension with arbitrary dispersion relations, using the droplet bosonization method. For a single-particle Hamiltonian polynomial in the momentum, the bosonized Hamiltonian is a polynomial of one degree higher in the bosonic ‘boundary’ field and includes subleading lower-order and derivative terms. This generalizes the known results for massless relativistic and nonrelativistic fermions (quadratic and cubic bosonic Hamiltonians, respectively). We also consider two-body interactions and demonstrate that they lead to interesting collective behavior and phase transitions in the Fermi sea.