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.     

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.     

Quadratic algebras in the method of noncommutative integration of the wave equation

In this paper, we continue an investigation of applications of the method of noncommutative integration of linear differential equations in partial derivatives (A. V. Shapovalov and I. V. Shirokov, Izv. Vyssh. Uchebn. Zaved., Fizika., No. 4, 116 (1991); ibid., No. 5, 100 (1991)). We demonstrate the application of quadratic algebras (allowing for second-order operators) to the problem of constructing an exact basis for solutions of the wave equation in unseparated variables. For a nontrivial example, we have integrated the three-dimensional wave equation using a nonabelian quadratic algebra.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 18–23, August, 1995.     

Noncqmmutative integration of Klein-Gordon and Dirac equations in Riemannian spaces with a group of motions

By the method of non-coimmutative integration of linear differential equations proposed by the authors [Izv. Vyssh. Uchebn. Zaved., Fiz., No. 4, 95 (1991)] the Klein-Gordon and Dirac equations are integrated in four-dimensional Riemannian spaces, not admitting separation of variables.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 33–38, May, 1991.     

Klein-Gordon oscillators in noncommutative phase space

We study the Klein-Gordon oscillators in non-commutative (NC) phase space. We find that the Klein-Gordon oscillators in NC space and NC phase-space have a similar behaviour to the dynamics of a particle in commutative space moving in a uniform magnetic field. By solving the Klein-Gordon equation in NC phase space, we obtain the energy levels of the Klein-Gordon oscillators, where the additional terms related to the space-space and momentum-momentum non-commutativity are given explicitly.     

The ⋆-value equation and Wigner distributions in noncommutative Heisenberg algebras

We consider the quantum mechanical equivalence of the Seiberg-Witten map in the context of the Weyl-Wigner-Groenewold-Moyal phase-space formalism in order to construct a quantum mechanics over noncommutative Heisenberg algebras. The formalism is then applied to the exactly soluble Landau and harmonic oscillator problems in the 2-dimensional noncommutative phase-space plane, in order to derive their correct energy spectra and corresponding Wigner distributions. We compare our results with others that have previously appeared in the literature.Dedicated to Mike Ryan on his sixtieth birthday, who as a scientist always understood that it is nice to be good, but that it is better to be nice.     

Klein-Gordon oscillators in noncommutative phase space

We study the Klein-Gordon oscillators in non-commutative (NC) phase space.We find that the Klein-Gordon oscillators in NC space and NC phase-space have a similar behaviour to the dynamics of a particle in commutative space moving in a uniform magnetic field.By solving the Klein-Gordon equation in NC phase space,we obtain the energy levels of the Klein-Gordon oscillators,where the additional terms related to the space-space and momentum-momentum non-commutativity are given explicitly.     

Riemannian manifolds in noncommutative geometry

We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spin^c${}^c$ manifolds; and conversely, in the presence of a spin^c${}^c$ structure. We also show how to obtain an analogue of Kasparov’s fundamental class for a Riemannian manifold, and the associated notion of Poincaré duality. Along the way we clarify the bimodule and first-order conditions for spectral triples.     

On the spaces of positive and negative frequency solutions of the Klein-Gordon equation in curved space-times

In a space-time $\text{(V}{}_{\mathrm{n}}\text{×}\text{R}\text{;g)}$ with V_n closed (n ≠ 2) satisfying certain global conditions, we can write the Klein-Gordon equation, relative to a suitable class of atlases, in the evolution form du/dt = T^-1(t)u, on Sobolev spaces K^l(V_n) = H^l(V_n) × H^l?1(V_n), where the spectrum of T^-1(t) is imaginary. Following papers by T. Kato and J. Kisyński we prove the existence of the evolution operator for this equation. The space $\text{K}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(V}{}_{\mathrm{n}}\text{)}$ has a natural strongly-symplectic structure ω. We determine the explicit form of complex-structure-positive operators of this structure. We prove that any two such operators, say J₁, J₂, are symplectically equivalent, (i.e. there is a symplectic transformation S such that J₂ = SJ₁S^-1). Spaces of positive and negative frequency solutions are then unique modulo symplectic equivalence. Each operator J determines a regular kernel on space-time which satisfies the properties of the kernel postulated by A. Lichnérowich in his program of quantization of fields in curved space-times. We carry out explicit calculations in the case of Robertson-Walker space-times. If an additional condition is satisfied by the given space-time, a unique complex-structure-positive operator can be selected in a natural way. This condition is satisfied by globally stationary space-times.     

Integration of Dirac equation in Riemannian spaces with five-dimensional group of motions

In the present article a classification of Riemannian spaces with five-dimensional group of motion is described from the point of view of a solution of the Dirac equation. A class of spaces is identified in which the Dirac equation does not admit a complete separation of variables, and exact solutions of the Dirac equation are obtained in these spaces by means of the method of noncommutative integration. Omsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 24–28, August, 1997.     

Schwarzschild black hole in noncommutative spaces

We study the effects of noncommutative spaces on the horizon, the area spectrum and Hawking temperature of a Schwarzschild black hole. The results show deviations from the usual horizon, area spectrum and the Hawking temperature. The deviations depend on the parameter of space/space noncommutativity.     

Classification of quadratic symmetry algebras of the Schrödinger equation

Noncommuntative quadratic symmetry algebras of a certain class for the Schrödinger equation are classified. For each such algebra, the permissible potential is found. The application of noncommuntative integration of partial differential equations by means of quadratic algebras is demonstrated for a nontrivial example. The solution obtained forms the basis for the representation of quadratic algebras.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 11–17, August, 1995.     

Trajectory length constraint and the Klein-Gordon equation

It is shown that the relativistic constraint of the length of possible trajectories in the method of integrals over the trajectories results in the Klein-Gordon equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 63–65, December, 1984.     

The reduction of the Laplace equation in certain Riemannian spaces and the resulting Type II hidden symmetries

We prove a general theorem which allows the determination of Lie symmetries of the Laplace equation in a general Riemannian space using the conformal group of the space. Algebraic computing is not necessary. We apply the theorem in the study of the reduction of the Laplace equation in certain classes of Riemannian spaces which admit a gradient Killing vector, a gradient Homothetic vector and a special Conformal Killing vector. In each reduction we identify the source of Type II hidden symmetries. We find that in general the Type II hidden symmetries of the Laplace equation are directly related to the transition of the CKVs from the space where the original equation is defined to the space where the reduced equation resides. In particular we consider the reduction of the Laplace equation (i.e., the wave equation) in the Minkowski space and obtain the results of all previous studies in a straightforward manner. We consider the reduction of Laplace equation in spaces which admit Lie point symmetries generated from a non-gradient HV and a proper CKV and we show that the reduction with these vectors does not produce Type II hidden symmetries. We apply the results to general relativity and consider the reduction of Laplace equation in locally rotational symmetric space times (LRS) and in algebraically special vacuum solutions of Einstein’s equations which admit a homothetic algebra acting simply transitively. In each case we determine the Type II hidden symmetries.     

Photon velocity from the Klein-Gordon equation

An approximate mathematical relationship for the velocity of a photon as a non-zero rest-mass quantum particle is derived from the field-free Klein-Gordon equation in the framework of the de Broglie-Bohm theory of quantum mechanics.     

The Klein-Gordon equation and scattering theory

Using our previously developed theory, we prove the main conclusions of scattering theory for the Klein-Gordon equation under hypotheses weaker than presently known. We consider more general equations and obtain stronger results.