Quantized Knizhnik-Zamolodchikov equations,quantum Yang-Baxter equation,and difference equations forq-hypergeometric functions

Thes₂ quantized Knizhnik-Zamolodchikov equations are solved inq-hypergeometric functions. New difference equations are derived for generalq-hypergeometric functions. The equations are given in terms of quantum Yang-Baxter matrices and have the form similar to quantum Knizhnik-Zamolodchikov equations for quantum affine algebras introduced by Frenkel and Reshetikhin.This work was supported by NSF grant DMS-9203929.     

Hypergeometric solutions of Knizhnik-Zamolodchikov equations

Explicit integral formulas are presented for the solutions of Knizhnik-Zamolodchikov equations associated with an arbitrary Kac-Moody Lie algebra.     

Quantum Knizhnik-Zamolodchikov equations and affine root systems

Quantum (difference) Knizhnik-Zamolodchikov equations [S1, FR] are generalized for theR-matrices from [Ch1] with the arguments in arbitrary root systems (and their formal counterparts). In particular, QKZ equations with certain boundary conditions are introducted. The self-consistency of the equations from [FR] and the cross-derivative integrability conditions for ther-matrix KZ equations from [Ch2] are obtained as corollaries. A difference counterpart of the quantum many-body problem connected with Macdonald's operators is defined as an application.Partially supported by NSF PYI Award DMS-9057144     

Discretization of Liouville type nonautonomous equations preserving integrals

The problem of constructing semi-discrete integrable analogues of the Liouville type integrable PDE is discussed. We call the semi-discrete equation a discretization of the Liouville type PDE if these two equations have a common integral. For the Liouville type integrable equations from the well-known Goursat list for which the integrals of minimal order are of the order less than or equal to two we presented a list of corresponding semi-discrete versions. The list contains new examples of non-autonomous Darboux integrable chains.     

Form factors,deformed Knizhnik-Zamolodchikov equations and finite-gap integration

We study the limit of asymptotically free massive integrable models in which the algebra of nonlocal charges turns into affine algebra. The form factors of fields in that limit are described by KZ equations on level 0. We show the limit to be connected with finite-gap integration of classical integrable equations.     

Elliptic quantum many-body problem and double affine Knizhnik-Zamolodchikov equation

The elliptic-matrix quantum Olshanetsky-Perelomov problem is introduced for arbitrary root systems by means of an elliptic version of the Dunkl operators. Its equivalence with the double affine generalization of the Knizhnik-Zamolodchikov equation (in the induced representations) is established.Partially supported by NSF grant number DMS-9301114 and UNC Research Counsel grant     

Quantum algebra structure of certain Jackson integrals

Theq-difference system satisfied by Jackson integrals with a configuration ofA-type root system is studied. We explicitly construct some linear combination of Jackson integrals, which satisfies the quantum Knizhnik-Zamolodchikov equation for the 2-point correlation function ofq-vertex operators, introduced by Frenkel and Reshetikhin, for the quantum affine algebra . The expression of integrands for then-point case is conjectured, and a set of linear relations for the corresponding Jackson integrals is proved.     

Stochastic differential equations in quantum statistical mechanics. Observables and multiple wiener integrals

The physical and mathematical framework for quantum mechanical stochastic differential equations is discussed as the quantization ofc-number equations that typically describe Brownian motion in polynomial potentials.     

Shift operators for the quantum Calogero-Sutherland problems via Knizhnik-Zamolodchikov equation

We give a natural interpretation of the shift operators for Calogero-Sutherland quantum problem via KZ equation using Matsuo-Cherednik mappings. The explicit formulas for the inversions of these mappings and versions of shift operators for KZ equations are also found. As an application we show that the shift operator can be described via a factorization problem for an appropriate quantum integral (discriminant) of the Calogero system.     

Langevin equations and stochastic integrals

The ambiguity of stochastic integrals involved in Langevin equations is removed by the postulate of invariance with respect to nonlinear transformations of the coordinates. The Stratonovich sense of the integrals, which is imposed thereby, is also strongly suggested by stability considerations requiring small changes of the solutions whenever the perturbations are changed by a small amount. The associated Fokker-Planck equation must include the spurious drift which arises from the transition from the Stratonovich to the Itô sense of the Langevin equations and describes one aspect of the systematic motion due to nonconstant fluctuations.     

Jackson-type integrals,bethe vectors,and solutions to a difference analog of the Knizhnik-Zamolodchikov system

It is shown how to find solutions to the quantum Knizhnik-Zamolodchikov system using the Bethe ansatz technique.This work is supported by Alfred Sloan Foundation and by NSF Grant DMS-9015821.     

Functional equations for path integrals

We consider the density matrices that arise in the statistical mechanics of the electron-phonon systems. In the path integral representation the phonon coordinates can be eliminated. This leads to an action that depends on pairs of points on a path, that depends explicitly on time differences, and that contains the phonon occupation numbers. The integral is reduced to a standard form by scaling to the thermal length. We use the technique of integration by parts and add specially chosen generating functionals to the action. We set down functional derivative equations for the source-dependent density matrix and for the mass operator. This allows us to develop a series of approximations for the operator in terms of exact propagators. The crudest approximation is a coherent potential approximation applicable at a general temperature.     

Functional equations for Feynman integrals

New types of equations for Feynman integrals are found. It is shown that the latter satisfy functional equations that relate integrals with different kinematics. A regular method for obtaining such relations is proposed. A derivation of the functional equations for one-loop two-, three-, and four-point functions with arbitrary masses and external momenta is given. It is demonstrated that the functional equations can be used to analytically continue Feynman integrals to various kinematical domains.     

Nonlinear relativistic and quantum equations with a common type of solution

Generalizations of the three main equations of quantum physics, namely, the Schr?dinger, Klein-Gordon, and Dirac equations, are proposed. Nonlinear terms, characterized by exponents depending on an index q, are considered in such a way that the standard linear equations are recovered in the limit q→1. Interestingly, these equations present a common, solitonlike, traveling solution, which is written in terms of the q-exponential function that naturally emerges within nonextensive statistical mechanics. In all cases, the well-known Einstein energy-momentum relation is preserved for arbitrary values of q.     

Spectral families of quantum stochastic integrals

We develop a theory of spectral integration for quantum stochastic integrals of certain families of processes driven by creation, conservation and annihilation processes in Fock space. These give a non-commutative generalisation of classical stochastic integrals driven by Poisson random measures. A stochastic calculus for these processes is developed and used to obtain unitary operator valued solutions of stochastic differential equations. As an application we construct stochastic flows on operator algebras driven by Lévy processes with finite Lévy measure.     

On path integrals and relativistic quantum mechanics

A proposal for formulation of relativistic quantum mechanics in terms of path integrals is presented.We are deeply indebted to Dr. M.Petrá for many stimulating discussions.     

Closed path integrals and the quantum action

We suggest a closed form expression for the path integral of quantum transition amplitudes. We introduce a quantum action with parameters different from the classical action. We present numerical results for the harmonic oscillator with weak perturbation, the quartic potential, and the double well potential. The quantum action is relevant for quantum chaos and quantum instantons.