317 research outputs found
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
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
Geometric Quantization on the Super-Disc
In this article we discuss the geometric quantization on a certain type of
infinite dimensional super-disc. Such systems are quite natural when we analyze
coupled bosons and fermions. The large-N limit of a system like that
corresponds to a certain super-homogeneous space. First, we define an example
of a super-homogeneous manifold: a super-disc. We show that it has a natural
symplectic form, it can be used to introduce classical dynamics once a
Hamiltonian is chosen. Existence of moment maps provide a Poisson realization
of the underlying symmetry super-group. These are the natural operators to
quantize via methods of geometric quantization, and we show that this can be
done.Comment: 17 pages, Latex file. Subject: Mathematical physics, geometric
quantizatio
Yang-Mills Theory on a Cylinder Coupled to Point Particles
We study a model of quantum Yang-Mills theory with a finite number of gauge
invariant degrees of freedom. The gauge field has only a finite number of
degrees of freedom since we assume that space-time is a two dimensional
cylinder. We couple the gauge field to matter, modeled by either one or two
nonrelativistic point particles. These problems can be solved {\it without any
gauge fixing}, by generalizing the canonical quantization methods of
Ref.\[rajeev] to the case including matter. For this, we make use of the
geometry of the space of connections, which has the structure of a Principal
Fiber Bundle with an infinite dimensional fiber. We are able to reduce both
problems to finite dimensional, exactly solvable, quantum mechanics problems.
In the case of one particle, we find that the ground state energy will diverge
in the limit of infinite radius of space, consistent with confinement. In the
case of two particles, this does not happen if they can form a color singlet
bound state (`meson').Comment: 37 pages, UR-1327 ER-40685-77
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
Non-relativistic Lee Model on two Dimensional Riemannian Manifolds
This work is a continuation of our previous work (JMP, Vol. 48, 12, pp.
122103-1-122103-20, 2007), where we constructed the non-relativistic Lee model
in three dimensional Riemannian manifolds. Here we renormalize the two
dimensional version by using the same methods and the results are shortly given
since the calculations are basically the same as in the three dimensional
model. We also show that the ground state energy is bounded from below due to
the upper bound of the heat kernel for compact and Cartan-Hadamard manifolds.
In contrast to the construction of the model and the proof of the lower bound
of the ground state energy, the mean field approximation to the two dimensional
model is not similar to the one in three dimensions and it requires a deeper
analysis, which is the main result of this paper.Comment: 18 pages, no figure
Hadron Production in Neutrino-Nucleon Interactions at High Energies
The multi-particle production at high energy neutrino- nucleon collisions are
investigated through the analysis of the data of the experiment CERN-WA-025 at
neutrino energy less than 260GeV and the experiments FNAL-616 and FNAL-701 at
energy range 120-250 GeV. The general features of these experiments are used as
base to build a hypothetical model that views the reaction by a Feynman diagram
of two vertices. The first of which concerns the weak interaction between the
neutrino and the quark constituents of the nucleon. At the second vertex, a
strong color field is assumed to play the role of particle production, which
depend on the momentum transferred from the first vertex. The wave function of
the nucleon quarks are determined using the variation method and relevant
boundary conditions are applied to calculate the deep inelastic cross sections
of the virtual diagram.Comment: 6 pages PDF forma
Large N limit of SO(N) gauge theory of fermions and bosons
In this paper we study the large N_c limit of SO(N_c) gauge theory coupled to
a Majorana field and a real scalar field in 1+1 dimensions extending ideas of
Rajeev. We show that the phase space of the resulting classical theory of
bilinears, which are the mesonic operators of this theory, is OSp_1(H|H
)/U(H_+|H_+), where H|H refers to the underlying complex graded space of
combined one-particle states of fermions and bosons and H_+|H_+ corresponds to
the positive frequency subspace. In the begining to simplify our presentation
we discuss in detail the case with Majorana fermions only (the purely bosonic
case is treated in our earlier work). In the Majorana fermion case the phase
space is given by O_1(H)/U(H_+), where H refers to the complex one-particle
states and H_+ to its positive frequency subspace. The meson spectrum in the
linear approximation again obeys a variant of the 't Hooft equation. The linear
approximation to the boson/fermion coupled case brings an additonal bound state
equation for mesons, which consists of one fermion and one boson, again of the
same form as the well-known 't Hooft equation.Comment: 27 pages, no figure
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