Searcharxiv⌕ Search

arXiv subjects

A. N. Leznov

Publications and source records attributed to A. N. Leznov.

At least 37 records · Page 2Linked to original sources
arXiv2004

Theory of fields in quantized spaces

It is shown how the theory of the fields can be constructed in a consistent way in quantized spaces. All constructions are connected with unitary irreducible representations of real forms of six dimensional rotation algebras O(1,5), O(2,4), O(3,3) or equivalent real forms of four-dimensional unitary algebras SU^*(4),SU(2,2),SL(4). The most surprising fact is that that all operations in the construction of such field theories are connected with compact manifolds. The gauge invariant theory in quantized space is proposed.

hep-th↗
arXiv2003

Fine structure of the discrete transformation for multicomponent integrable systems

It is shown that in the case of multicomponent integrable systems connected with algebras $A_n$, the discrete transformation $T$ possesses the fine structure and can be represented in the form $T=\prod T_i^{l_i}$, where $T_i$ are n commuting basis discrete transformations, and $l_i$ are arbitrary natural numbers. All the calculations are conducted in detail for the case of a 3-wave interacting system.

hep-th↗
arXiv2003

Fine Structure of Matrix Darboux-Toda Integrable Mapping

We show here that matrix Darboux-Toda transformation can be written as a product of a number of mappings. Each of these mappings is a symmetry of the matrix nonlinear Shrodinger system of integro-differential equations. We thus introduce a completely new type of discrete transformations for this system. The discrete symmetry of the vector nonlinear Shrodinger system is a particular realization of these mappings.

hep-th↗
arXiv2003

Relativistic invariant Lie algebras for kinematical observables in quantum space-time

A deformation of the canonical algebra for kinematical observables of the quantum field theory in Minkowski space-time has been considered under the condition of Lorentz invariance. A relativistic invariant algebra obtained depends on additional fundamental constants M, L and H with the dimensions of mass, length and action, respectively. In some limiting cases the algebra goes over into the well-known Snyder or Yang algebras. In general case the algebra represents a class of Lie algebras, that consists of simple algebras and semidirect sums of simple algebras and integrable ones. Some algebras belonging to this class are noninvariant under T and C transformations. Possible applications of obtained algebras for descriptions of states of matter under extreme conditions are briefly discussed.

hep-th↗
arXiv2002

Some integrable models in quantized spaces

It is shown that in a quantized space determined by the $B_2\quad (O(5)=Sp(4))$ algebra with three dimensional parameters of the length $L^2$, momentum $(Mc)^2$, and action $S$, the spectrum of the Coulomb problem with conserving Runge-Lenz vector coincides with the spectrum found by Schrödinger for the space of constant curvature but with the values of the principal quantum number limited from the side of higher values. The same problem is solved for the spectrum of a harmonic oscillator.

hep-th↗
arXiv2000

Spherically symmetrical configurations of self-dual Yang-Mills and Einstein-Plebanski equations

Spherically symmetrical reductions of self-dual Yang-Mills and Einstein-Plebanski equations are constructed at the same manner. As in the first case we come back to known before solutions (under such kind of reduction but in some different form) if in the second one we obtain unknown before equation describing the spherically symmetrical configurations of Plebanski equation and some number of its particular solutions.

math-ph↗
arXiv2000

A Discourse on the Benney Equation

It is shown that the Benney system is equivalent to a one body classical dynamical problem in which the interaction potential is a functional of the action function. The general solution of this equation is responsible for the general solution of the Benney system. Some particular solutions of the selfconsistent Hamilton-Jacobi equation arising in this investigation are presented in explicit form.

math-ph↗
arXiv1999

The Complex Bateman Equation

The general solution to the Complex Bateman equation is constructed. It is given in implicit form in terms of a functional relationship for the unknown function. The known solution of the usual Bateman equation is recovered as a special case.

solv-int↗
arXiv1999

Integrable systems in spaces of arbitrary dimension

The $2n$ dimensional manifold with two mutually commutative operators of differentiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general solution of them is presented in explicit form.

math-ph↗
arXiv1999

The connection of Monge-Bateman equations with ordinary differential equations and their generalisation

It is shown that the Monge equation is equivalent to the ordinary differential equation $\ddot X=0$ of free motion. Equations of Monge type (with their general solutions) are connected with each ordinary differential equation of second order $\ddot X=F(\dot X,X; t)$, integrable by quadratures. The result is generalised to a system of equations of the second order, which is in one to one correspondence with the multidimensional Monge-Bateman system.

math-ph↗
arXiv1999

Representation of the quantum algebra $SU_q(2)$ in the basis with diagonal $"J_x"$ generator

Generators of the quantum $SU_q(2)$ algebra are obtained in the explicit form in the basis where the operator $\exp {J_z\over 2} J_x \exp {J_z\over 2}$ is diagonal. It is shown that the solution of this problem is related to the representation theory of the two-dimensional algebra $[s,r]=\tanh t (s^2-r^2+1)$. The relevance of such basis to some problems of quantum optics is discussed.

math-ph↗
arXiv1999

The invariant form of the generators of semisimple Lie and quantum algebras in their arbitrary finite-dimensional representation

An explicit form of the generators of quantum and ordinary semisimple algebras for an arbitrary finite-dimensional representation is found. The generators corresponding to the simple roots are obtained in terms of a solution of a system of matrix equations. The result is presented in the form of $N_l\times N_l$ matrices, where $N_l$ is the dimension of the corresponding representation, determined by the invariant Weyl formula.

math-ph↗
arXiv1999

New approach to representation theory of semisimple Lie algebras and quantum algebras

A method to construct in explicit form the generators of the simple roots of an arbitrary finite-dimensional representation of a quantum or standard semisimple algebra is found. The method is based on general results from the global theory of representations of semisimple groups. The rank two algebras $A_2$, $B_2=C_2$, $D_2$ and $G_2$ are considered as examples. The generators of the simple roots are presented as solutions of a system of finite difference equations and given in the form of $N_l\times N_l$ matrices, where $N_l$ is the dimension of the representation.

math-ph↗