First-order Symmetry Operators Research Articles

Approximate Lie symmetries of the Navier-Stokes equations

In the framework of the theory of approximate transformation groups proposed by Baikov, Gaziziv and Ibragimov [1], the first-order approximate symmetry operator is calculated for the Navier-Stokes equations. The symmetries of the coupled system obtained by expanding the dependent variables of the Navier-Stokes equations in the perturbation series with respect to a small parameter (viscosity) are used to derive approximate symmetries in the sense by Baikov et al.

On the solution of the Dirac equation in de Sitter space

It is shown that the maximal number of first-order symmetry operators for the Dirac equation (including spin symmetries), both in arbitrary signature flat space and in de Sitter space, is equal. The isomorphic representation of 11-dimensional nonlinear symmetry algebra (W-algebra) of first-order operators for the Dirac operator in flat space and de Sitter space is considered. The algebra is an extension of the Lie algebra of the group of pseudo-orthogonal rotations and this extension is unique. We have found all linear Lie subalgebras in the nonlinear algebra that satisfy the conditions of the noncommutative integration theorem. Using one subalgebra we have integrated the Dirac equation in the generalized spherical system of coordinates and have constructed the complete class of exact solutions. The solution is found by a method that differs from the variable separation method and is new in the literature. The massive particle spectrum, models of particle into antiparticle transmutation, the disappearance of particles and the quantization conditions of the motion are discussed.One can use the results of the paper to pose the boundary problem for the Dirac equation in de Sitter space if the interval is used in the boundary condition. As an example, we consider a model of asymptotically flat space that is glued from the de Sitter space and flat space. We interpret the model as a gravitational well or barrier.

Riemann functions and the group E(1,1)

Historically Lie algebras of first-order symmetry operators have proven to be a useful method for finding equivalence classes of Riemann functions. Here this idea is extended to higher order symmetries. The approach is to seek self-adjoint linear hyperbolic partial differential equations that separate variables in more than one coordinate system under the action of the group E(1,1). The equations derived admit no nontrivial first-order operators and can only be obtained from second-order symmetry operators. Using this symmetry structure, a new equivalence class of Riemann functions can then be found.

First-order Dirac symmetry operators

We show how, for all dimensions and signatures, the most general first-order linear symmetry operator for the massive Dirac equation is given in terms of Killing–Yano tensors. In the massless case the Killing–Yano condition is relaxed to the conformal Killing–Yano generalization.

About a Symmetry of the Dirac Massless Equation in a Riemannian Space

First-order differential symmetry operators of the Dirac massless equation are investigated in the present paper.

Noncommutative solutions of the d'Alembert equation

All the subalgebras of first-order symmetry operators for the d'Alembert equation, generating the bases of solutions in the method of noncommutative integration of linear differential equations, which cannot be constructed in the method of separation of variables, are found. These bases themselves are then given in explicit form. The complete systems of solutions of the d'Alembert equation, determined by noncommutative sets of first-order symmetry operators, are thereby classified.

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.

Separation of variables in the Dirac equation in Stackel spaces. II. External gauge fields

For pt.I see ibid., vol.7, p.517, (1990). This is the second paper in a series of research started by Bagrov et al. The Abelian complete sets of matrix first-order differential symmetry operators for the Dirac equation are used for the separation of variables in the Dirac equation in Stackel spaces with external gauge fields. The permissible space metrics and gauge field potentials are not limited by field equations.

Commutative subalgebras of three first-order symmetry operators and separation of variables in the wave equation

The problem of complex separation of variables in the wave equation is considered in four-dimensional Minkowskii space-time. In contrast to the known series of researches by Kalnins and Miller (see Ref. Zh., Fiz., 2B9 (1978); 1B208 and 1B209 (1979), e.g.), underlying this research is a theorem on the necessary and sufficient conditions of total separation of variables in the non-parabolic V. N. Shapovalov equation (Differents. Uravn.,16, No. 10, 1864–1874 (1980)). Nonequivalent complete sets of three differential first-order symmetry operators are constructed, appropriate coordinate systems are found, and complete separation of variables is performed in the wave equation.

Separation of variables in the Dirac equation in Stackel spaces

The subspaces of Riemannian space of signature (+---) that admit separation of the Dirac equation have been found in the case of Riemannian space admitting the separation of the Hamilton-Jacobi equation. For the separation of variables in the Hamilton-Jacobi equation it is necessary for the complete set of Killing vectors and tensors to be of a special kind. Every complete set defines its own type of metric of Riemannian space which is called Stackel space. The Dirac equation does not permit the separation of variables in general cases of Stackel space. The main idea of the paper is in the construction, in Stackel space, of a complete set of another kind. This complete set consists of three matrix first-order differential symmetry operators of the Dirac equation. The operators are pairwise commuting and linearly independent. The complete set structure is in agreement with the structure of the Killing vectors and tensors of Stackel space. The separation of variables in the Dirac equation has been carried out in the explicit form in Stackel spaces which admit complete sets of symmetry operators. These operators have been used essentially in the process of separation that differs from Chandrasekhar method.

Yang-Mills fields permitted by abelian complete sets of first-order symmetry operators of the Dirac equation

Nonequivalent complete sets of first-order symmetry operators of the Dirac free equation determine the Yang-Mills field, permitting complete variable separation in the Dirac equations with an external Yang-Mills field. Typical representatives of the classes of permissible fields are considered.

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.

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.