Quadratic algebras applied to noncommutative integration of the Klein-Gordon equation: Four-dimensional quadratic algebras containing three-dimensional nilpotent lie algebras

The study is continued on noncommutative integration of linear partial differential equations [1] in application to the exact integration of quantum-mechanical equations in a Riemann space. That method gives solutions to the Klein-Gordon equation when the set of noncommutative symmetry operations for that equation forms a quadratic algebra consisting of one second-order operator and of first-order operators forming a Lie algebra. The paper is a continuation of [2], where a single nontrivial example is used to demonstrate noncommutative integration of the Klein-Gordon equation in a Riemann space not permitting variable separation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 89–94, March, 1995.     

Noncommutative 5-dimensional subalgebras of a conformal algebra integrable in R1,3

The present study is concerned with the application and investigation of a new method of exact integration of systems of linear differential equations, the method of noncommutative integration. The method is based on the use of noncommutative subalgebras of symmetry for finding an exact solution. The investigation of 5-dimensional subalgebras of symmetry of the d'Alembert equation lead to the claim that there exists a class of subalgebras which generate exact solutions in explicit form but which it is not possible to obtain in explicit form by means of complete separation of the variables.Tomsk State University. Translated from Izvestiya Vysshikh Uchenbykh Zavedenii, Fizika, No. 6, pp. 115–119, June, 1995.     

Quadratic algebras and noncommutative integration of Klein-Gordon equations in non-steckel Riemann spaces

The method of noncommutative integration of linear partial differential equations is used to solve the Klein-Gordon equations in Riemann space, in the case when the set of noncommutating symmetry operators of this equation for a quadratic algebra consists of one second-order operator and several first-order operators. Solutions that do not permit variable separation are presented.Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 83–87, May, 1995.     

Integration of the d'Alembert equation by means of four-dimensional nonabelian symmetry subalgebras with a single second-order operator

In the present article a classification of four-dimensional symmetry subalgebras of a d'Alembert equation containing a single second-order operator and satisfying the condition of noncommutative integration is presented. Exact integration is performed by means of these subalgebras and by means of the method of complete separation of variables.Translated from Izvestiya Vysshaya Uchebnykh Zavedenii, Fizika, No. 8, pp. 48–51, August, 1995.     

Representations of Lie algebras and the problem of noncommutative integrability of linear differential equations

An algorithm is proposed for integrating linear partial differential equations with the help of a special set of noncommuting linear differential operators — an analogue of the method of noncommutative integration of finite-dimensional Hamiltonian systems. The algorithm allows one to construct a parametric family of solutions of an equation satisfying the requirement of completeness. The case is considered when the noncommutative set of operators form a Lie algebra. An essential element of the algorithm is the representation of this algebra by linear differential operators in the space of parameters. A connection is indicated of the given method with the method of separation of variables, and also with problems of the theory of representations of Lie algebras. Let us emphasize that on the whole the proposed algorithm differs from the method of separation of variables, in which sets of commuting symmetry operators are used.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 95–100, April, 1991.     

Use of noncommutative integration to construct the basis of wave-equation solutions

This study continues an earlier investigation of applications of the method of noncommutative integration of linear partial differential equations [A. V. Shapovalov and I. V. Shirokov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 4, 1995; No. 5, 33 (1991)], which was a generalization of the analogous method for Hamiltonian systems. The method of noncommutative integration uses nonabelian algebra to characterize the symmetry of the equation, which makes it possible to construct exact solutions going beyond the framework of the method of separation of variables. The condition of noncommutative integrability is used to select the algebras of waveequation symmetry needed for the given method in Minkowski space R_1,2. Nonequivalent noncommutative subalgebras of conformal algebra k_1,2 are used to construct the basis of solutions of the three-dimensional wave equation.Tomsk University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 54–60, May, 1995.     

Classification of F algebras and noncommutative integration of the Klein-Gordon equation in Riemannian spaces

The method of noncommutative integration of linear differential equations [A. V. Shapovalov and I. V. Shirokov, Izv. Vyssh. Uchebn. Zaved. Fiz., No. 4, 116; No. 5, 100 (1991)] is used to integrate the Klein-Gordon equation in Riemannian spaces. The situation is investigated where the set of noncommuting symmetry operators of the Klein-Gordon equation consists of first-order operators and one second-order operator and forms a so-called F algebra, which generalizes the concept of a Lie algebra. The F algebra is a quadratic algebra in the given situation. A classification of four- and five-dimensional F algebras is given. The integration of the Klein-Gordon equation in a Riemannian space, which does not admit separation of variables, is demonstrated in a nontrivial example.V. V. Kuibyshev State University, Tomsk. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 45–50, January, 1993.     

On the Algebra of Symmetry of the Dirac Equation in Flat Space and De Sitter Space

The structure of the quadratic algebras of spinor symmetry operators for the Dirac equation is studied in a four-dimensional flat space and in the de Sitter space of arbitrary signature. The algebras are shown to be standard equivalent. Linear noncommutative subalgebras meeting the conditions of the noncommutative integrability theorem are found in these algebras.     

Algebraic properties of the Dirac equation

The first-order symmetry operators of the Dirac equation are classified according to their tensor properties under transformations of the homogeneous Lorentz group; a minimal system of generators for the ring of symmetry operators of the free Dirac equation is obtained, and the physical meaning of the spin operators is considered; fields are found which admit symmetry operators of first order.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 84–89, February, 1972.The author is grateful to V. N. Shapovalov for discussions and valuable suggestions.     

Some Problems of the Existence of Symmetry Spinor Operators for the Dirac Equation

Conditions necessary for the existence of a class of fields that can be used to construct the spinor symmetry operators for the Dirac equation in Riemannian space are specified in the present paper. The metrics of spaces with four-dimensional groups of motions in which these fields exist are indicated. A class of spaces is identified in which the Dirac equation admits no separation of variables within the framework of the definition adopted, but the algebra of symmetry of the Dirac equation satisfies the conditions of theorems of the noncommutative intergrability.     

On Symmetry of the Generalized Breit Equation

Abstract

In this paper we find the complete set of symmetry operators for the two-particle Breit equation in the class of first-order differential operators with matrix coefficients. A new integral of motion is obtained.     

Fredholm Determinants and Pole-free Solutions to the Noncommutative Painlevé II Equation

We extend the formalism of integrable operators à la Its-Izergin-Korepin-Slavnov to matrix-valued convolution operators on a semi–infinite interval and to matrix integral operators with a kernel of the form \fracE₁^T(l) E₂(m)l+m{\frac{E_1^T(\lambda) E_2(\mu)}{\lambda+\mu}}, thus proving that their resolvent operators can be expressed in terms of solutions of some specific Riemann-Hilbert problems. We also describe some applications, mainly to a noncommutative version of Painlevé II (recently introduced by Retakh and Rubtsov) and a related noncommutative equation of Painlevé type. We construct a particular family of solutions of the noncommutative Painlevé II that are pole-free (for real values of the variables) and hence analogous to the Hastings-McLeod solution of (commutative) Painlevé II. Such a solution plays the same role as its commutative counterpart relative to the Tracy–Widom theorem, but for the computation of the Fredholm determinant of a matrix version of the Airy kernel.     

Algebras of Symmetry Operators of the Klein–Gordon–Fock Equation for Groups Acting Transitively on Two-Dimensional Subspaces of a Space-Time Manifold

Russian Physics Journal - All external electromagnetic fields are found in which the Klein–Gordon–Fock equation for a charged test particle admits first-order symmetry operators...     

Separating the variables in a massless Dirac equation in Minkowski space

Exact integration of the Dirac equation is a classical topic in mathematical physics, which has been researched for several decades. A basic method is complete segregation of the variables. Such separation can be attained in a Dirac equation containing an external electromagnetic field in Minkowski space by means of complete sets of first-order symmetry matrix operators. The purpose of this paper is to solve an analogous case for a free massless Dirac equation. That task has a special feature because external fields are absent and the massless equation is reduced to a D'Alambert equation by squaring. Nevertheless, interest attaches to states defined by the first-order symmetry-operator matrices that cannot be obtained by setting the mass to zero in systems containing a mass Dirac equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 105–110, January, 1995.     

Complete sets of symmetry operators containing a second-order operator and the problem of separation of variables in the wave equation

For the wave equation in Minkowski space, a space is defined of nontrivial local second-order differential symmetry operators. The algebraic conditions, which, in accordance with the general theorems on the separation of variables, must be satisfied by the commutative subalgebras, including two first-order operators and second-order operator, are formulated in coordinate-free form. On the basis of these subalgebras, there are obtained all the complete sets of symmetry operators of types (2.0), (2.1). Sets are presented which do not have analogues in papers of other authors.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 115–119, April, 1991.     

Solving the Noncommutative Batalin–Vilkovisky Equation

Given an odd symmetry acting on an associative algebra, I show that the summation over arbitrary ribbon graphs gives the construction of the solutions to the noncommutative Batalin–Vilkovisky equation, introduced in (Barannikov in IMRN, rnm075, 2007), and to the equivariant version of this equation. This generalizes the known construction of A _∞-algebra via summation over ribbon trees. I give also the generalizations to other types of algebras and graph complexes, including the stable ribbon graph complex. These solutions to the noncommutative Batalin–Vilkovisky equation and to its equivariant counterpart, provide naturally the supersymmetric matrix action functionals, which are the gl(N)-equivariantly closed differential forms on the matrix spaces, as in (Barannikov in Comptes Rendus Mathematique vol 348, pp. 359–362.     

Quantum projection. Integration of the open Tod quantum chain

A quantum projection method is developed on the basis of noncommutative integration of linear differential equations and the results of M. A. Ol’shanetskii and A. M. Perelomov on the integration of classical Hamiltonian systems (projection method). The method proposed makes it possible to obtain in explicit form solutions of the quantum equations whose classical analogs can be integrated by projection. Then the semisimplicity property of the symmetry algebra of the original equation is no longer a factor. The solution basis of a Schrödinger equation with the potential of an open three-particle Tod chain is constructed as a nontrivial example.     

Classification of the Dirac equation with an external SU(3) gauge field admitting a first-order symmetry operator of special type

A classification is performed of massless gauge fields admitting one first-order symmetry operator of special type for the Dirac equation in Minkowski space. The gauge group is chosen to be SU(3). The factors multiplying the derivatives of the symmetry operator do not contain generators of the gauge group, which allows us to classify the fields according to symmetry operators of the free Dirac equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 22–21, June, 1989.     

Maxwell–Chern–Simons Topologically Massive Gauge Fields in the First-Order Formalism

We find the canonical and Belinfante energy-momentum tensors and their nonzero traces. We note that the dilatation symmetry is broken and the divergence of the dilatation current is proportional to the topological mass of the gauge field. It was demonstrated that the gauge field possesses the ‘scale dimensionality’ d=1/2. Maxwell–Chern–Simons topologically massive gauge field theory in 2+1 dimensions is formulated in the first-order formalism. It is shown that 6×6-matrices of the relativistic wave equation obey the Duffin–Kemmer–Petiau algebra. The Hermitianizing matrix of the relativistic wave equation is given. The projection operators extracting solutions of field equations for states with definite energy-momentum and spin are obtained. The 5×5-matrix Schr?dinger form of the equation is derived after the exclusion of non-dynamical components, and the quantum-mechanical Hamiltonian is obtained. Projection operators extracting physical states in the Schr?dinger picture are found.