First-order Symmetry Operators Research Articles

Algebra of Symmetry Operators for Klein-Gordon-Fock Equation

All external electromagnetic fields in which the Klein-Gordon-Fock equation admits the first-order symmetry operators are found, provided that in the space-time V4 a group of motion G3 acts simply transitively on a non-null subspace of transitivity V3. It is shown that in the case of a Riemannian space Vn, in which the group Gr acts simply transitively, the algebra of symmetry operators of the n-dimensional Klein-Gordon-Fock equation in an external admissible electromagnetic field coincides with the algebra of operators of the group Gr.

The first-order symmetry operator on gravitational perturbations in the 5D Myers–Perry spacetime with equal angular momenta

Abstract It has been revealed that the first-order symmetry operator for the linearized Einstein equation on a vacuum spacetime can be constructed from a Killing–Yano 3-form. This might be used to construct all or part of the solutions to the field equation. In this paper, we perform a mode decomposition of a metric perturbation on the Schwarzschild spacetime and the Myers–Perry spacetime with equal angular momenta in 5 dimensions, and investigate the action of the symmetry operator on specific modes concretely. We show that, on such spacetimes, there is no transition between the modes of a metric perturbation by the action of the symmetry operator, and it ends up being the linear combination of the infinitesimal transformations of isometry.

Symmetry algebra of the time-dependent Schrödinger equation in constant and uniform fields

The problems of constructing the symmetry operators of the time-dependent Schrödinger equation in the constant and uniform electric and magnetic fields with the use of the obtained operators in the method of non-commutative integration of linear differential equations are considered. A system of defining equations for the first-order symmetry operators is written. The paper investigates the structure of the symmetry algebras. It is shown that in the case of electric field the resulting algebra is the deformation of the Schrödinger algebra sch(1, 3).

Алгебры операторов симметрии уравнения Клейна - Гордона - Фока для групп движений, действующих транзитивно на двумерных подпространствах пространственно-временного многообразия

All external electromagnetic fields are found in which the Klein-Gordon-Fock equation for a charged test particle admits first-order symmetry operators provided that the groups G 3, r £ 3, of motions act transitively on the two-dimensional subspace V 2.

Twistor spinors and extended conformal superalgebras

We show that the first-order symmetry operators of twistor spinors can be constructed from conformal Killing–Yano forms in conformally-flat backgrounds. We express the conditions on conformal Killing–Yano forms to obtain mutually commuting symmetry operators of twistor spinors. Conformal superalgebras which consist of conformal Killing vectors and twistor spinors and play important roles in supersymmetric field theories in conformal backgrounds are extended to more general superalgebras by using the graded Lie algebra structure of conformal Killing–Yano forms and the symmetry operators of twistor spinors. The even part of the extended conformal superalgebra corresponds to conformal Killing–Yano forms and the odd part consists of twistor spinors.

On symmetry operators for the Maxwell equation on the Kerr-NUT-(A)dS spacetime

We focus on the method recently proposed by Lunin and Frolov–Krtouš–Kubizňák to solve the Maxwell equation on the Kerr-NUT-(A)dS spacetime by separation of variables. In their method, it is crucial that the background spacetime has hidden symmetries because they generate commuting symmetry operators with which the separation of variables can be achieved. In this work we reproduce these commuting symmetry operators in a covariant fashion. We first review the procedure known as the Eisenhart–Duval lift to construct commuting symmetry operators for given equations of motion. Then we apply this procedure to the Lunin–Frolov–Krtouš–Kubizňák (LFKK) equation. It is shown that the commuting symmetry operators obtained for the LFKK equation coincide with the ones previously obtained by Frolov–Krtouš–Kubizňák, up to first-order symmetry operators corresponding to Killing vector fields. We also address the Teukolsky equation on the Kerr-NUT-(A)dS spacetime and its symmetry operator is constructed.

Symmetry of the Time-Dependent Schrödinger Equation in Electromagnetic Fields Invariant Under Three-Dimensional Е(3) Subgroups

In this paper, the classification of electromagnetic fields invariant under three-dimensional E(3) subgroups is given. For every electromagnetic field from this classification, all first-order symmetry operators commuting with the operator of the time-dependent Schrodinger equation are calculated.

Symmetry operators and separation of variables in the (2 + 1)-dimensional Dirac equation with external electromagnetic field

We obtain and analyze equations determining first-order differential symmetry operators with matrix coefficients for the Dirac equation with an external electromagnetic potential in a [Formula: see text]-dimensional Riemann (curved) spacetime. Nonequivalent complete sets of mutually commuting symmetry operators are classified in a [Formula: see text]-dimensional Minkowski (flat) space. For each of the sets, we carry out a complete separation of variables in the Dirac equation and find a corresponding electromagnetic potential permitting separation of variables.

On the Lie Symmetry Algebras of the Stationary Schrödinger and Pauli Equations

A general method for constructing first-order symmetry operators for the stationary Schrodinger and Pauli equations is proposed. It is proven that the Lie algebra of these symmetry operators is a one-dimensional extension of some subalgebra of an e(3) algebra. We also assemble a classification of stationary electromagnetic fields for which the Schrodinger (or Pauli) equation admits a Lie algebra of first-order symmetry operators.

$\mathbb{Z}_2\times \mathbb{Z}_2$-graded Lie symmetries of the Lévy-Leblond equations

The first-order differential L\'evy-Leblond equations (LLE's) are the non-relativistic analogs of the Dirac equation, being square roots of ($1+d$)-dimensional Schr\"odinger or heat equations. Just like the Dirac equation, the LLE's possess a natural supersymmetry. In previous works it was shown that non supersymmetric PDE's (notably, the Schr\"odinger equations for free particles or in the presence of a harmonic potential), admit a natural ${\mathbb Z}_2$-graded Lie symmetry. In this paper we show that, for a certain class of supersymmetric PDE's, a natural ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie symmetry appears. In particular, we exhaustively investigate the symmetries of the $(1+1)$-dimensional L\'evy-Leblond Equations, both in the free case and for the harmonic potential. In the free case a ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie superalgebra, realized by first and second-order differential symmetry operators, is found. In the presence of a non-vanishing quadratic potential, the Schr\"odinger invariance is maintained, while the ${\mathbb Z}_2$- and ${\mathbb Z}_2\times{\mathbb Z}_2$- graded extensions are no longer allowed. The construction of the ${\mathbb Z}_2\times {\mathbb Z}_2$-graded Lie symmetry of the ($1+2$)-dimensional free heat LLE introduces a new feature, explaining the existence of first-order differential symmetry operators not entering the super Schr\"odinger algebra.

Noncommutative Integration and Symmetry Algebra of the Dirac Equation on the Lie Groups

The algebra of first-order symmetry operators of the Dirac equation on four-dimensional Lie groups with right-invariant metric is investigated. It is shown that the algebra of symmetry operators is in general not a Lie algebra. Noncommutative reduction mediated by spin symmetry operators is investigated. For the Dirac equation on the Lie group SO(2,1) a parametric family of particular solutions obtained by the method of noncommutative integration over a subalgebra containing a spin symmetry operator is constructed.

Symmetry operators of Killing spinors and superalgebras in AdS5

We construct the first-order symmetry operators of Killing spinor equation in terms of odd Killing-Yano forms. By modifying the Schouten-Nijenhuis bracket of Killing-Yano forms, we show that the symmetry operators of Killing spinors close into an algebra in AdS5 spacetime. Since the symmetry operator algebra of Killing spinors corresponds to a Jacobi identity in extended Killing superalgebras, we investigate the possible extensions of Killing superalgebras to include higher-degree Killing-Yano forms. We found that there is a superalgebra extension but no Lie superalgebra extension of the Killing superalgebra constructed out of Killing spinors and odd Killing-Yano forms in AdS5 background.

Algebra of Symmetry Operators and Integration of the Klein–Gordon Equation in an External Electromagnetic Field

The structure of the algebra of first-order differential symmetry operators of the Klein-Gordon equation in an external electromagnetic field is investigated from positions of the theory of cohomologies of Lie algebras. A method for integrating the given equation is proposed, based on noncommutative reduction, and, as a consequence, the corresponding integrability condition is obtained. An example of an integrable Klein- Gordon equation in a spacetime with a Steckel metric is considered, which does not admit separation of variables when an electromagnetic field is switched on. tensor of which is invariant with respect to the group of motions of the spacetime metric. The goal of this work is to describe the structure of the algebra of first-order symmetry operators of the equation under consideration from positions of the theory of cohomologies of Lie algebras, and also to propose a method of integration using the indicated symmetry algebra. An integrability condition for the Klein-Gordon equation in an electromagnetic field is derived, expressed in terms of the dimensionality of the Lie algebra of the group of motions of the metric and the so-called cohomology index of this algebra, this index depending on the class of cohomologies defined by the electromagnetic field. Note that in contrast to the well-known method of separation of variables, the integration method proposed in this work is based on the idea of noncommutative reduction (1), which makes more effective use of the algebra of first-order differential symmetry operators. In a number of cases, this eliminates the necessity of investigating hidden symmetries of the problem, associated with Killing tensors of second rank (and higher), which must often be done in the case of separation of variables (2, 3). To wrap things up, this paper considers an example of the Klein-Gordon equation in spacetime with a four- dimensional group of motions of the metric, which is integrable with the help of the approach proposed here. An interesting qualitative aspect of the example is the possibility of separation of variables in this equation in the absence of an external electromagnetic field and the impossibility of doing so in its presence.

Electron in higher-dimensional weakly charged rotating black hole spacetimes

We demonstrate separability of the Dirac equation in weakly charged rotating black hole spacetimes in all dimensions. The electromagnetic field of the black hole is described by a test field approximation, with the vector potential proportional to the primary Killing vector field. It is shown that the demonstrated separability can be intrinsically characterized by the existence of a complete set of mutually commuting first-order symmetry operators generated from the principal Killing-Yano tensor. The presented results generalize the results on integrability of charged particle motion and separability of charged scalar field studied in V. P. Frolov and P. Krtous [Phys. Rev. D 83, 024016 (2011)].

Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets

In this paper we derive the most general first-order symmetry operator commuting with the Dirac operator in all dimensions and signatures. Such an operator splits into Clifford even and Clifford odd parts which are given in terms of odd Killing-Yano and even closed conformal Killing-Yano inhomogeneous forms respectively. We study commutators of these symmetry operators and give necessary and sufficient conditions under which they remain of the first-order. In this specific setting we can introduce a Killing-Yano bracket, a bilinear operation acting on odd Killing-Yano and even closed conformal Killing-Yano forms, and demonstrate that it is closely related to the Schouten-Nijenhuis bracket. An important non-trivial example of vanishing Killing-Yano brackets is given by Dirac symmetry operators generated from the principal conformal Killing-Yano tensor [hep-th/0612029]. We show that among these operators one can find a complete subset of mutually commuting operators. These operators underlie separability of the Dirac equation in Kerr-NUT-(A)dS spacetimes in all dimensions [arXiv:0711.0078].

Hidden symmetry in the presence of fluxes

We derive the most general first-order symmetry operator for the Dirac equation coupled to arbitrary fluxes. Such an operator is given in terms of an inhomogeneous form ω which is a solution to a coupled system of first-order partial differential equations which we call the generalized conformal Killing–Yano system. Except trivial fluxes, solutions of this system are subject to additional constraints. We discuss various special cases of physical interest. In particular, we demonstrate that in the case of a Dirac operator coupled to the skew symmetric torsion and U ( 1 ) field, the system of generalized conformal Killing–Yano equations decouples into the homogeneous conformal Killing–Yano equations with torsion introduced in D. Kubiznak et al. (2009) [8] and the symmetry operator is essentially the one derived in T. Houri et al. (2010) [9]. We also discuss the Dirac field coupled to a scalar potential and in the presence of 5-form and 7-form fluxes.

Erratum: Separability of a modified Dirac equation in a five-dimensional rotating, charged black hole in string theory [Phys. Rev. D80, 044037 (2009)

The aim of this paper is to investigate the separability of a spin-1/2 spinor field in a five-dimensional rotating, charged black hole constructed by Cvetic and Youm in string theory, in the case when three U(1) charges are set equal. This black hole solution represents a natural generalization of the famous four-dimensional Kerr-Newman solution to five dimensions with the inclusion of a Chern-Simons term to the Maxwell equation. It is shown that the usual Dirac equation can not be separated by variables in this general spacetime with two independent angular momenta. However if one supplements an additional counterterm into the usual Dirac operator, then the modified Dirac equation for the spin-1/2 spinor particles is separable in this rotating, charged Einstein-Maxwell-Chern-Simons black hole background geometry. A first-order symmetry operator that commutes with the modified Dirac operator has exactly the same form as that previously found in the uncharged Myers-Perry black hole case. It is expressed in terms of a rank-three totally antisymmetric tensor and its covariant derivative. This tensor obeys a generalized Killing-Yano equation and its square is a second-order symmetric Stackel-Killing tensor admitted by the five-dimensional rotating, charged black hole spacetime.

Separability of a modified Dirac equation in a five-dimensional rotating, charged black hole in string theory

The aim of this paper is to investigate the separability of a spin-1/2 spinor field in a five-dimensional rotating, charged black hole constructed by Cvetic and Youm in string theory, in the case when three U(1) charges are set equal. This black hole solution represents a natural generalization of the famous four-dimensional Kerr-Newman solution to five dimensions with the inclusion of a Chern-Simons term to the Maxwell equation. It is shown that the usual Dirac equation cannot be separated by variables in this general spacetime with two independent angular momenta. However if one supplements an additional counterterm into the usual Dirac operator, then the modified Dirac equation for the spin-1/2 spinor particles is separable in this rotating, charged Einstein-Maxwell-Chern-Simons black hole background geometry. A first-order symmetry operator that commutes with the modified Dirac operator has exactly the same form as that previously found in the uncharged Myers-Perry black hole case. It is expressed in terms of a rank-three totally antisymmetric tensor and its covariant derivative. This tensor obeys a generalized Killing-Yano equation and its square is a second-order symmetric Staeckel-Killing tensor admitted by the five-dimensional rotating, charged black hole spacetime.

First-order symmetries of the Dirac equation in a curved background: a unified dynamical symmetry condition**Dedicated to Professor T Dereli on the occasion of his 60th birthday.

It has been shown that, for all dimensions and signatures, the most general first-order linear symmetry operators for the Dirac equation including interaction with Maxwell field in a curved background are given in terms of Killing–Yano (KY) forms. As a general gauge invariant condition it is found that among all KY forms of the underlying (pseudo) Riemannian manifold, only those which Clifford commute with the Maxwell field take part in the symmetry operator. It is also proved that associated with each KY form taking part in the symmetry operator, one can define a quadratic function of velocities which is a geodesic invariant as well as a constant of motion for the classical trajectory. Some geometrical and physical implications of the existence of KY forms are also elucidated.

Symmetry operators and the separability of massive Klein–Gordon and Dirac equations in the general five-dimensional Kerr–(anti-)de Sitter black hole background

It is shown that the Dirac equation is separable by variables in a five-dimensional rotating Kerr–(anti-)de Sitter black hole with two independent angular momenta. A first-order symmetry operator that commutes with the Dirac operator is constructed in terms of a rank-3 Killing–Yano tensor whose square is a second-order symmetric Stäckel–Killing tensor admitted by the five-dimensional Kerr–(anti-)de Sitter spacetime. We highlight the construction procedure of such a symmetry operator. In addition, the first law of black hole thermodynamics has been extended to the case that the cosmological constant can be viewed as a thermodynamical variable.