166 research outputs found
Integrable Systems and Factorization Problems
The present lectures were prepared for the Faro International Summer School
on Factorization and Integrable Systems in September 2000. They were intended
for participants with the background in Analysis and Operator Theory but
without special knowledge of Geometry and Lie Groups. In order to make the main
ideas reasonably clear, I tried to use only matrix algebras such as
and its natural subalgebras; Lie groups used are either GL(n)
and its subgroups, or loop groups consisting of matrix-valued functions on the
circle (possibly admitting an extension to parts of the Riemann sphere). I hope
this makes the environment sufficiently easy to live in for an analyst. The
main goal is to explain how the factorization problems (typically, the matrix
Riemann problem) generate the entire small world of Integrable Systems along
with the geometry of the phase space, Hamiltonian structure, Lax
representations, integrals of motion and explicit solutions. The key tool will
be the \emph{% classical r-matrix} (an object whose other guise is the
well-known Hilbert transform). I do not give technical details, unless they may
be exposed in a few lines; on the other hand, all motivations are given in full
scale whenever possible.Comment: LaTeX 2.09, 69 pages. Introductory lectures on Integrable systems,
Classical r-matrices and Factorization problem
Classical and Quantum Nonultralocal Systems on the Lattice
We classify nonultralocal Poisson brackets for 1-dimensional lattice systems
and describe the corresponding regularizations of the Poisson bracket relations
for the monodromy matrix . A nonultralocal quantum algebras on the lattices for
these systems are constructed.For some class of such algebras an
ultralocalization procedure is proposed.The technique of the modified
Bethe-Anzatz for these algebras is developed.This technique is applied to the
nonlinear sigma model problem.Comment: 33 pp. Latex. The file is resubmitted since it was spoiled during
transmissio
Drinfeld-Sokolov reduction for difference operators and deformations of W-algebras. II. General Semisimple Case
The paper is the sequel to q-alg/9704011. We extend the Drinfeld-Sokolov
reduction procedure to q-difference operators associated with arbitrary
semisimple Lie algebras. This leads to a new elliptic deformation of the Lie
bialgebra structure on the associated loop algebra. The related classical
r-matrix is explicitly described in terms of the Coxeter transformation. We
also present a cross-section theorem for q-gauge transformations which
generalizes a theorem due to R.Steinberg.Comment: 19 pp., AMS-LaTeX. The paper replaces a temporarily withdrawn text;
the first part (written by E. Frenkel, N. Reshetikhin, and M. A.
Semenov-Tian-Shansky) is available as q-alg/970401
Classification of All Poisson-Lie Structures on an Infinite-Dimensional Jet Group
A local classification of all Poisson-Lie structures on an
infinite-dimensional group of formal power series is given. All
Lie bialgebra structures on the Lie algebra {\Cal G}_{\infty} of
are also classified.Comment: 11 pages, AmSTeX fil
Path Integral Quantization of the Symplectic Leaves of the SU(2)* Poisson-Lie Group
The Feynman path integral is used to quantize the symplectic leaves of the
Poisson-Lie group SU(2)*. In this way we obtain the unitary representations of
U_q(su(2)). This is achieved by finding explicit Darboux coordinates and then
using a phase space path integral. I discuss the *-structure of SU(2)* and give
a detailed description of its leaves using various parametrizations and also
compare the results with the path integral quantization of spin.Comment: 24 pages, LaTeX, no figures, full postscript available from
http://phyweb.lbl.gov/theorygroup/papers/40890.p
Differential technique for the covariant orbital angular momentum operators
The orbital angular momentum operator expansion turns to be a powerful tool
to construct the fully covariant partial wave amplitudes of hadron decay
reactions and hadron photo- and electroproduction processes. In this paper we
consider a useful development of the orbital angular momentum operator
expansion method. We present the differential technique allowing the direct
calculation of convolutions of two orbital angular momentum operators with an
arbitrary number of open Lorentz indices. This differential technique greatly
simplifies calculations when the reaction subject to the partial wave analysis
involves high spin particles in the initial and/or final states. We also
present a useful generalization of the orbital angular momentum operators.Comment: 14 page
Pion and photon beam initiated backward charmonium or lepton pair production
Hard exclusive reactions initiated by pion or photon beams within the
near-backward kinematical regime specified by the small Mandelstam variable
can be studied to access pion-to-nucleon and photon-to-nucleon Transition
Distribution Amplitudes (TDAs). Checking the validity of collinear factorized
description of pion and photon induced reactions in terms of TDAs allows to
test the universality of TDAs between the space-like and time-like regimes that
is the indispensable feature of the QCD collinear factorization approach. In
this short review we consider the exclusive pion- and photo-production off
nucleon of a highly virtual lepton pair (or heavy quarkonium) in the
near-backward region. We first employ a simplistic cross channel nucleon
exchange model of pion-to-nucleon TDAs to estimate the magnitude of the
corresponding cross sections for the kinematical conditions of J-PARC. We then
illustrate the flexibility of our approach by building a two parameter model
for the photon-to-nucleon TDAs based on preliminary results for near threshold
photoproduction at JLab and provide our estimates for near-backward
photoproduction and Timelike Compton Scattering cross sections for the
kinematical conditions of JLab and of future EIC and EicC.Comment: 22 pages, 9 figures; The paper is extended by adding in Sec. 7 a
discussion on the near-backward charmonium photoproduction employing the
photon-to-nucleon TDA model driven by the recent GlueX data on the
photoproduction in arXiv:2304.0384
- âŠ