Killing tensors and the separation of the Hamilton-Jacobi equation

This paper investigates the relationship between Killing Tensors and separable systems for the geodesic Hamilton-Jacobi equation in Riemannian and Lorentzian manifolds: locally, a separable system consists of the vector and covector associated with a separable coordinate. It is shown that there are only two types of separable system, those associated with local symmetry groups and those which can be obtained by a simple transformation from orthogonal systems. Some sufficient conditions for existence are given and some global problems are enumerated. The results are illustrated with a demonstration that the existence of separable systems in a certain class of {2, 2} space-times is a consequence of the algebraic properties of the Weyl tensor.     

Integration of the dirac equation,which does not presume complete separation of variables,in Stäckel spaces

The method of noncommutative integration of linear differential equations is used to construct an exact solution of the Dirac equation, which does not presume complete separation of variables, in Stäckel spaces. The Dirac equation in an external electromagnetic field is integrated by this method, using one example. The Stäckel space under consideration does not enable one to solve this equation exactly within the framework of the theory of separation of variables.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 31–37, January, 1996.     

Mayer expansions and the Hamilton-Jacobi equation

We review the derivation of Wilson's differential equation in (infinitely) many variables, which describes the infinitesimal change in an effective potential of a statistical mechanical model or quantum field theory when an infinitesimal integration out is performed. We show that this equation can be solved for short times by a very elementary method when the initial data are bounded and analytic. The resulting series solutions are generalizations of the Mayer expansion in statistical mechanics. The differential equation approach gives a remarkable identity for connected parts and precise estimates which include criteria for convergence of iterated Mayer expansions. Applications include the Yukawa gas in two dimensions past the=4 threshold and another derivation of some earlier results of Göpfert and Mack.     

Stäckel spaces in electrovacuum,permitting complete separation of variables in the diagonalized dirac equation. Type (1.1)

Institute of High-Current Electronics, Siberian Branch of the Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 31–34, June, 1992.     

Probability densities in configuration space and the Hamilton-Jacobi equation

The characteristic solutions of the Hamilton-Jacobi equation give the energies of conservative physical systems as functions of position and time. It is shown that these expressions are useful in the formation of probability densities in configuration space for canonical ensembles. Applications are given and discussed.     

On the Hamilton-Jacobi equation for second-class constrained systems

We discuss a general procedure for arriving at the Hamilton-Jacobi equation of second-class constrained systems, and illustrate it in terms of a number of examples by explicitely obtaining the respective Hamilton principal function, and verifying that it leads to the correct solution to the Euler-Lagrange 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.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 79–84, May, 1990.     

Geometric phase within the complex quantum Hamilton-Jacobi formalism

We derive a geometric phase using the quantum kinematic approach within the complex quantum Hamilton-Jacobi formalism. The single valuedness of the wave function implies that the geometric phase along an arbitrary path in the complex plane must be equal to an integer multiple of 2π. The nonzero geometric phase indicates that we travel along the path through the branch cut of the phase function from one Riemann sheet to another.     

Conformally flat stäckel spaces and the problem of separation of variables in the Laplace-Beltrami equation

A full proof of the statement that all orthogonal systems of coordinates admitting separation of variables in the Laplace-Beltrami equation are cyclidal coordinates is presentedInstitute of High Current Electrotechniques, Siberian Branch, Russian Academy of Sciences, Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 97–103, October, 1992.     

Separation of variables in the Dirac equation

In this paper we investigate the possibility of separating variables in the Dirac equation by using complete sets, containing second-order differential operators and integral operators having a definite structure.Translated from Izvestiya VUZ. Fizika, No. 6, pp. 94–100, June, 1973.     

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.     

Back-Reaction of Black Hole Radiation from Hamilton-Jacobi Method

In the frame of Hamilton-Jacobi method, the back-reactions of the radiating particles together with the total entropy change of the whole system are investigated. The emission probability from this process is found to be equivalent to the null geodesic method. However its physical picture is more clear: the negative energy one of a virtual particle pair is absorbed by the black hole, resulting in the temperature, electric potential and angular velocity increase; then the black hole amount of heat, electric charge and angular momentum can spontaneously transfer to the positive energy particle; when obtaining enough energy, it can escape away to infinity, visible to distant observers. And this method can be applied to any sort of horizons and particles without a specific choice of (regular-across-the-horizon) coordinates.     

Solution of the Dirac equation by separation of variables in spherical coordinates for a large class of non-central electromagnetic potentials

A systematic and intuitive approach for the separation of variables of the three-dimensional Dirac equation in spherical coordinates is presented. Using this approach, we consider coupling of the Dirac spinor to electromagnetic four-vector potential that satisfies the Lorentz gauge. The space components of the potential have angular (non-central) dependence such that the Dirac equation becomes separable in all coordinates. We obtain exact solutions for a class of three-parameter static electromagnetic potential whose time component is the Coulomb potential. The relativistic energy spectrum and corresponding spinor wave functions are obtained. The Aharonov–Bohm and magnetic monopole potentials are included in these solutions.