B. Sakita

The quantum collective field method and its application to the planar limit

Abstract We formulate a general method of collective fields in quantum theory, which represents a direct generalization of the Bohm-Pines treatment of plasma oscillations. The present method provides a complete procedure for reformulating a given quantum system in terms of a most general (overcomplete) set of commuting operators. We point out and exemplify how this formalism offers a new powerful method for studying the large- N limit. For illustration we discuss the collective motions of N identical harmonic oscillators. As a much more important application, we show how, based on the present formalism, one solves the planar limit of a non-trivial SU( N ) symmetric quantum theory.

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.

Generalizations of dual models

Generalizations of the dual model are discussed in the context of functional integral formulation of the theory. We notice that all the symmetries of dual amplitudes are led from the symmetries of the Lagrangian used for the measure of functional integration. Irreducible fields under conformal transformations are classified by their dimension and conformal spin. Since the factorizability of dual amplitudes requires a local Langrarian, we construct the general form of local Langrarian density such that the action integral is conformal invariant. We found that if one restricts the form of the Langrangian to bilinear form of the fields, the kinetic term of the Langrarian is possible to construct only for conformal spin 0 and 12 field. The former yields the usual dual model, while the latter yields the Bardakci-Halpern model and the Neveu-Schwarz one. Quantization of conformal spin 12 field is discussed in detail. The generating functional of this field is constructed in terms of the functional integral technique and used to construct the general N-particle amplitudes.

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.

Chern-Simons matrix model: Coherent states and relation to Laughlin wave functions

Received 11 June 2001; published 4 December 2001) Using a coherent state representation we derive many-body probability distributions and wave functions for the Chern-Simons matrix model proposed by Polychronakos and compare them to the Laughlin ones. We analyze two different coherent state representations, corresponding to different choices for electron coordinate bases. In both cases we find that the resulting probability distributions do not quite agree with the Laughlin ones. There is agreement on the long distance behavior, but the short distance behavior is different.

Baryon number violation at high temperature in the standard model

Abstract We explore a suggestion that baryon number violation is unsuppressed in the standard model at high energies and temperatures. We find that baryon number violation is negligible in high energy scattering. On the other hand, in agreement with previous authors, we find that it is indeed unsuppressed at high temperatures. We give simple, qualitative arguments valid even in systems with many degrees of freedom. In particular, we show that there is no extra entropy suppression for production of sphaleron or soliton-antisoliton configurations. Above the weak interaction phase transition, a simple scaling argument gives the rate up to an unknown numerical factor. We also show that passage over the barrier is associated with fermion production. We illustrate the general considerations with a simple model which, for this problem, has features identical to those of the standard model. In this model it is possible to perform many computations explicitly, and a semiclassical analysis may be performed even at high temperature. In addition, we explain why certain criticisms which have been leveled at these ideas are not correct.

Orthogonal basis for the energy eigenfunctions of the Chern-Simons matrix model

We study the spectrum of the Chern-Simons matrix model and identify an orthogonal set of states. The connection to the spectrum of the Calogero model is discussed.

WEAK INTERACTIONS ARE WEAK AT HIGH-ENERGIES

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.

W∞ gauge transformations and the electromagnetic interactions of electrons in the lowest Landau level

Abstract We construct a W ∞ gauge field theory of electrons in the lowest Landau level. For this purpose we introduce an external gauge potential A such that its W ∞ gauge transformations cancel against the gauge transformation of the electron field. We then show that the electromagnetic interactions of electrons in the lowest Landau level are obtained through a non-linear realization of A in terms of the U (1) gauge potential A μ . As applications we derive the effective Lagrangians for circular droplets and for the ν = 1 quantum Hall system.

Real-time approach to instanton phenomena: (I). Multidimensional potentials with degenerate absolute minima

Abstract The tunneling problem of degenerate classical ground states is discussed by using the WKB wave function for systems with many degrees of freedom. The method is elaborated for a simple multi-dimensional potential system with two absolute minima. The WKB wave function is constructed for the forbidden region of configuration space by using imaginary time (Euclidean) solutions of classical equations of motion. This imaginary time is used as a geometrical parameter of phase space and is not related to the true time of the problem, which is eliminated by studying stationary-state wave functions. The matching of the WKB wave function to the harmonic oscillator wave function in the allowed region is discussed to obtain the ground state energy. The result is compared with the imaginary time path integral computation. The agreement between the two results follows from scattering theory applied to the small fluctuation equation.