Polarization symmetries of relativistic wave equations

Generators of the group invariance of polarization origin U(2s + 1) are constructed for relativistic wave equations (RWE), where s is the particle spin. Particular cases of the vector (mass) and Dirac fields are examined.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 74–77 November, 1982.     

The symmetries of wave equations on new lattices

This paper focuses on studying the symmetry of a practical wave equation on new lattices. It is a new step in that the new lattice equation is applied to reduce the discrete problem of motion of an elastic thin homogeneous bar. The equation of motion of the bar can be changed into a discrete wave equation. With the new lattice equation, the translational and scaling invariant, not only is the infinitesimal transformation given, but the symmetry and Lie algebras are also calculated. We also give a new form of invariant called the ratio invariant, which can reduce the process of the computing invariant with the characteristic equation.     

Some symmetries of the nonlinear heat and wave equations

The Lie-group formalism is applied to deduce the classical symmetries of the nonlinear heat equation, the diffusion-convection equation and the nonlinear wave equations. Some nonclassical symmetries are also presented.     

Local and other symmetries of the subsystem of maxwell rotation equations

A nonlocal representation of the Poincaré group is found, and, using a well-known technique, the conformal group and its generalization, relative to which both the complete system of free Maxwell equations and the subsystem of only the rotation equations, are also found. It is remarked that, by means of a simple and natural restatement of the equations for the free electromagnetic field, a universalized treatment for massless fields of arbitrary spin is possible. Uzhgorod State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 92–97, May, 1997.     

Lie symmetries of difference equations

The discrete heat equation is worked out to illustrate the search of symmetries of difference equations. Special attention it is paid to the Lie structure of these symmetries, as well as to their dependence on the derivative’s discretization. The case ofq-symmetries for discrete equations in aq-lattice is briefly considered at the end. Talk delivered by J. Negro at the DI-CRM Workshop held in Prague, 18–21 June 2000. This work has been partially supported by DGES of the Ministerio de Educación y Cultura of Spain under Projects PB98-0360 and the Junta de Castilla y León (Spain).     

Conformally covariant field equations II. First-order massless equations

The necessary and sufficient conditions for the finite as well as infinite component field equation

$\text{L}{}_{\mathrm{\mu }}\text{?}{}^{\mathrm{\mu }}\text{ψ(x) = 0}$

to be conformally covariant are derived and their consequences discussed. The derived results are specified for massless field equations treated in literature.     

Continuous symmetries of equations on lattices

A method is presented for calculating the Lie point symmetries of difference equations with one, or several, independent variables. The equations are given on a priori specified lattices. The Lie transformations act on the lattice, as well as on the equation. The transformations take solutions into solutions and can be used to perform symmetry reduction. Presented by P. Winternitz at the DI-CRM Workshop held in Prague, 18–21 June 2000. This article was written while S.T. and P.W. were visiting the Dipartimento di Fisica, Università di Roma Tre. They thank the Dipartimento, the INFN and the Agreement Università di Roma Tre — Université de Montréal for their support. The research of P.W. was partly supported by a research grant from NSERC of Canada.     

Continuous symmetries of three-dimensional diffusion equations

Lie transformation groups are given which leave the three-dimensional linear diffusion equation invariant, with and without chemical reactions. We show how similarity solutions and conserved currents can be obtained with the help of these groups. We apply these methods to nonlinear three-dimensional diffusion equations which can be exactly linearized by nonlinear transformations.     

Local characterization of massless radiation from point sources

We show that the energy-momentum tensor for the massless fields generated by a multipolar source, confined to a world-line, can be split into a radiation and a bound part, both of them having zero divergence off the source world-line. This allows us to compute in a simple way the energy of radiation emitted by the source. Scalar, vector, and tensor fields are studied. An exact expression for the radiated energy of a spinning electron is exhibited. By analizing a system of two electrons we show that radiation from extended sources cannot be described locally in the manner mentioned above.     

Symmetries of massless Dirac equations and supersymmetric quantum mechanics

It is shown how symmetries of Dirac equations can be used to obtain constants of motion for nonrelativistic supersymmetric quantum Hamiltonians. In particular, conserved supercharges are found for a spin-1/2 particle in the field of a dyon which yield under anticommutation the generalized Runge-Lenz symmetry of the system.This work is supported in part through funds provided by the Natural Sciences and Engineering Research Council (NSERC) of Canada and the Fonds FCAR of the Quebec Ministry of Education.     

On the new symmetries of Maxwell equations

The known symmetries of the Maxwell equations are summarized. Then new symmetries of these equations found by the authors are reviewed. These symmetries are generated by infinitesimal integrodifferential operators of the eight-dimensional Lie algebra. Their physical meaning is not clear.Invited talk presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.     

Lie symmetries of quadratic two-dimensional difference equations

We find all systems of first-order quadratic autonomous two-dimensional difference equations which have two linear Lie symmetries. Knowledge of these symmetries permits the systems to be integrated by a reduction procedure.     

Generalized symmetries of nonlinear partial differential equations

We propose a rigorous definition of the generalized infinitesimal symmetries using the notion of k-vector fields, and we derive an algorithm for their determination. We show that Bäcklund transformations between evolution equations are differential operators having prescribed symmetries.     

Generalized conditional symmetries of nonlinear differential-difference equations

Generalized conditional symmetry method for tackling nonlinear partial differential equations is extended to differential-difference equations. As the applications, some exact solutions to several nonlinear differential-difference equations are obtained.     

First order generalized symmetries of Maxwell's equations

The results of the calculation of first order generalized symmetries of Maxwell's equations in vacuum are summarized. The Lie algebra structure of the symmetries is also discussed.     

Non-abelian traveling wave solutions for massless QCD2

The field equations for quantum chromodynamics in 1 + 1 dimensions (QCD₂) with massless fermions are shown to admit classical non-abelian traveling wave solutions. In this case, the field equations reduce to the linear Frenet-Serret equations for a curve in the three-space corresponding to an SU(2) subalgebra of the SU(N) gauge group.     

New potential symmetries for some evolution equations

In this paper we derive new potential symmetries that seem not to be recorded in the literature. These potential symmetries are determined by considering a generalized potential system, rather than the natural potential system or a general integral variable. An inhomogeneous diffusion equation, a porous medium equation and the Fokker-Planck equation have been considered as application of this procedure.     

一类动力学方程的Mei对称性

研究一类动力学方程的Mei对称性的定义和判据，由Mei对称性通过Noether对称性可找到Noether守恒量.由Mei对称性通过Lie对称性可找到Hojman守恒量.同时，也可找到一类新型守恒量.