Poisson Algebra of Wilson Loops and Derivations of Free Algebras

We describe a finite analogue of the Poisson algebra of Wilson loops in Yang-Mills theory. It is shown that this algebra arises in an apparently completely different context; as a Lie algebra of vector fields on a non-commutative space. This suggests that non-commutative geometry plays a fundamental role in the manifestly gauge invariant formulation of Yang-Mills theory. We also construct the deformation of the loop algebra induced by quantization, in the large N_c limit.Comment: 20 pages, no special macros necessar

Parton Model from Bi-local Solitonic Picture of the Baryon in two-dimensions

We study a previously introduced bi-local gauge invariant reformulation of two dimensional QCD, called 2d HadronDynamics. The baryon arises as a topological soliton in HadronDynamics. We derive an interacting parton model from the soliton model, thus reconciling these two seemingly different points of view. The valence quark model is obtained as a variational approximation to HadronDynamics. A succession of better approximations to the soliton picture are obtained. The next simplest case corresponds to a system of interacting valence, `sea' and anti-quarks. We also obtain this `embellished' parton model directly from the valence quark system through a unitary transformation. Using the solitonic point of view, we estimate the quark and anti-quark distributions of 2d QCD. Possible applications to Deep Inelastic Structure Functions are pointed out.Comment: 12 page

Geometric Quantization and Two Dimensional QCD

In this article, we will discuss geometric quantization of 2d QCD with fermionic and bosonic matter fields. We identify the respective large-N_c phase spaces as the infinite dimensional Grassmannian and the infinite dimensional Disc. The Hamiltonians are quadratic functions, and the resulting equations of motion for these classical systems are nonlinear. In a previous publication, the first author has shown that the linearization of the equations of motion for the Grassmannian gave the `t Hooft equation. We will see that the linearization in the bosonic case leads to the scalar analog of the `t Hooft equation found by Tomaras.Comment: 46 pages, Latex, no figure

Poisson Algebra of Wilson Loops in Four-Dimensional Yang-Mills Theory

We formulate the canonical structure of Yang--Mills theory in terms of Poisson brackets of gauge invariant observables analogous to Wilson loops. This algebra is non--trivial and tractable in a light--cone formulation. For U(N) gauge theories the result is a Lie algebra while for SU(N) gauge theories it is a quadratic algebra. We also study the identities satsfied by the gauge invariant observables. We suggest that the phase space of a Yang--Mills theory is a co--adjoint orbit of our Poisson algebra; some partial results in this direction are obtained.Comment: 32 Pages, 7 figures upon reques

A comment on generalized Schwinger effect

A spatially homogeneous, time-dependent, electric field can produce charged particle pairs from the vacuum. When the electric field is constant, the mean number of pairs which are produced depends on the electric field and the coupling constant in a non-analytic manner, showing that this result cannot be obtained from the standard perturbation theory of quantum electrodynamics. When the electric field varies with time and vanishes asymptotically, the result may depend on the coupling constant either analytically or non-analytically. We investigate the nature of this dependence in detail. We show that the dependence of particle production on coupling constant is non-analytic for a class of time-dependent electric fields which vanish asymptotically when a specific condition is satisfied. We also demonstrate that for another class of electric fields, which vary rapidly, the dependence of particle production on coupling constant is analytic.Comment: published versio

Symmetry Algebras of Large-N Matrix Models for Open Strings

We have discovered that the gauge invariant observables of matrix models invariant under U($N$) form a Lie algebra, in the planar large-N limit. These models include Quantum Chromodynamics and the M(atrix)-Theory of strings. We study here the gauge invariant states corresponding to open strings (`mesons'). We find that the algebra is an extension of a remarkable new Lie algebra ${\cal V}_{\Lambda}$ by a product of more well-known algebras such as $gl_{\infty}$ and the Cuntz algebra. ${\cal V}_{\Lambda}$ appears to be a generalization of the Lie algebra of vector fields on the circle to non-commutative geometry. We also use a representation of our Lie algebra to establish an isomorphism between certain matrix models (those that preserve `gluon number') and open quantum spin chains. Using known results on quantum spin chains, we are able to identify some exactly solvable matrix models. Finally, the Hamiltonian of a dimensionally reduced QCD model is expressed explicitly as an element of our Lie algebra.Comment: 44 pages, 8 eps figures, 3 tables, LaTeX2.09; this is the published versio

Multiparticle Schrodinger operators with point interactions in the plane

We study a system of N bosons in the plane interacting with delta function potentials. After a coupling constant renormalization we show that the Hamiltonian defines a self-adjoint operator and obtain a lower bound for the energy. The same results hold if one includes a regular inter-particle potential.Comment: 17 pages, Late