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 the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces

The eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces of arbitrary dimension are computed by separating variables in geodesic polar coordinates. These eigenfunctions are then used to derive the heat kernel of the iterated Dirac operator on these spaces. They are then studied as cross sections of homogeneous vector bundles, and a group-theoretic derivation of the spinor spherical functions and heat kernel is given based on Harish-Chandra's formula for the radial part of the Casimir operator.     

On the geometry of Lorentz orbit spaces

The orbit space of the Lorentz group acting on the product ofn real, or complex, Minkowski spaces is stratified into subspaces isomorphic to certain products of Grassmann manifolds and varieties of Gram matrices. The Lorentz orbits (of nonzero dimension) are completely classified by the Stiefel manifolds of standard orthogonal bases for the linear subspaces of the Minkowski space. Several representations of the spaces ofn-point Lorentz invariant distributions and differentiable, or analytic, functions onto appropriate spaces of distributions and functions of Lorentz invariant variables are also discussed.On leave of absence from the Institute of Atomic Physics, Bucharest, Romania.     

Strong gravitational waves in semiclosed electromagnetic universes. Statement and analysis of the problem in terms of the method of spin coefficients

The problem of integration is discussed for a complete system of Newman-Penrose equations for electrovacuum spaces of the general theory of relativity with nonzero cosmological constant. In terms of the method of spin coefficients, we formulate conditions on the electromagnetic and gravitational field variables, which distinguish a special class of Riemann spaces corresponding to strong gravitational waves in semiclosed Universes of Bertotti-Robinson type.Translated from Izvestiya Vysshikh Uchebnykh Zavederiii, Fizika, No. 11, pp. 74–78, November, 1987.     

The spaces of trial and generalized functions of infinite number of variables

In this article the spaces of trial and generalized functions of infinite number of variables closely connected with the equipment of Fock space are introduced. The spaces introduced are described in new terminology. The properties of continuity and differentiability of trial functions are studied.     

The five-dimensional Dirac equation in the theory of algebraic spinors

The Dirac equation is considered in five-dimensional spaces with signatures (2,3), (4,1) and (0,5). The algebraic spinor formalism with the application of fermionic variables is used as the basis of real Clifford algebras and the module over this algebra. It is shown that solutions to the five-dimensional Dirac equation in spaces with signatures (2,3) and (4,1) can be expanded over solutions with zero value of the fifth component of the generalized momentum, and the equation is equivalent to an equation in four-dimensional spacetime.     

Method of complete separation of complex variables in the Hamilton-Jacobi equation

A generalization of the Stekkel spaces is presented which allows a complete separation of complex variables in the Hamilton-Jacobi equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 23–27, September, 1988.     

Special Stâckel spaces of the electrovacuum

The investigation of the problem of classifying the spaces of the electrovacuum that admit a complete separation of variables in the Hamilton-Jacobi equation for a charged test particle is completed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 20–22, August, 1984.     

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.     

The topology of homophase misorientation spaces

Homophase misorientation spaces are investigated with a focus on the effect of symmetry operations on their topology and their minimum embedding dimensions in Euclidean space. Whereas the topology of rotation space is well established and requires a minimum of five variables for a one-to-one and continuous mapping, the spaces of orientations and misorientations are quotient spaces of the rotation space and are obtained by applying various equivalence relations. The equivalence relations for orientation spaces only involve the rotational symmetries of the underlying crystals. These spaces are classified under the three-dimensional manifolds called the spherical 3-manifolds, which have a non-trivial fundamental group, are not simply connected spaces, and do not embed in three-dimensional Euclidean space. In the case of homophase misorientation spaces, however, in addition to rotational symmetry operations there is a further ‘grain exchange symmetry’, which is shown to simplify the topology considerably. In some important cases this symmetry also reduces the number of Euclidean dimensions required to embed these misorientation spaces. The homophase misorientation spaces for the dihedral point groups D ₂(222), D ₄(422) and D ₆(622), the tetrahedral point group T(23), and the octahedral group O(432) are all found to be embeddable in only three dimensions, two dimensions less than required for rotations. Hence, these misorientation systems can be represented using three variables in a one-to-one and continuous manner.     

Polygons in Minkowski space and the Gelfand–Tsetlin method for pseudo-unitary groups

We study the symplectic geometry of the moduli spaces of polygons in the Minkowski 3-space. These spaces naturally carry completely integrable systems with periodic flows. We extend the Gelfand–Tsetlin method to pseudo-unitary groups and show that the action variables are given by the Minkowski lengths of non-intersecting diagonals.     

Caratheodory and the Foundations of Thermodynamics and Statistical Physics

Constantin Caratheodory offered the first systematic and contradiction free formulation of thermodynamics on the basis of his mathematical work on Pfaff forms. Moreover, his work on measure theory provided the basis for later improved formulations of thermodynamics and physics of continua where extensive variables are measures and intensive variables are densities. Caratheodory was the first to see that measure theory and not topology is the natural tool to understand the difficulties (ergodicity, approach to equilibrium, irreversibility) in the Foundations of Statistical Physics. He gave a measure-theoretic proof of Poincaré's recurrence theorem in 1919. This work paved the way for Birkhoff to identify later ergodicity as metric transitivity and for Koopman and von Neumann to introduce spectral analysis of dynamical systems in Hilbert spaces. Mixing provided an explanation of the approach to equilibrium but not of irreversibility. The recent extension of spectral theory of dynamical systems to locally convex spaces, achieved by the Brussels–Austin groups, gives new nontrivial time asymmetric spectral decompositions for unstable and/or non-integrable systems. In this way irreversibility is resolved in a natural way.     

Variational principles on principal fiber bundles: A geometry theory of Clebsch potentials and Lin constraints

The geometric theory of Lin constraints and variational principles in terms of Clebsch variables proposed recently by Cendra and Marsden [1987] will be generalized to include those systems defined not only on configuration spaces which are products of Lie groups and vector spaces but on configuration spaces which are principal bundles with structural group G. This generalization includes, for example, fluids with free boundaries, Yang-Mills fields, and it will be very useful, as it will be shown later, to illustrate some aspects of the theory of particles moving in a Yang-Mills field in both its variational and Hamiltonian aspects.     

A superspace path integral proof of the Gauss-Bonnet-Chern theorem

A rigorous theory of integration in the space of paths in super space is developed, by extending Berezin's method of integration to spaces of anticommuting variables with an uncountably high dimension. A Feynmam-Kac-Ito formula for the heat kernel of a wide class of superspace differential operators is established. This formula is then used to make rigorous the supersummetric proofs of the Gauss-Bonnet-Chern theorem [1, 2].     

The K-Orbits, Identities, and Classification of Four-Dimensional Homogeneous Spaces with the Group of Poincaré and de Sitter Transforms

The method of orbits traditionally applied to geometric quantization problems is used to study homogeneous spaces. Based on the proposed classification of the orbits of co-adjoint representation (K-orbits), a classification of homogeneous spaces is constructed. This classification allows one, in particular, to point out the explicit form of identities – functional relations between the transform-group generators – which are of great importance in applied problems (e.g., in the theory of separation of variables). All four-dimensional homogeneous spaces with the group of Poincaré and de Sitter transforms are classified and all independent identities on these spaces are given in explicit form.     

Noncommutative integration of the Dirac equation in Riemann spaces possessing a group of automorphisms

Applying the method of noncommutative integration for linear differential equations, we build exact solutions for the Dirac equation in 4-dimensional Riemann spaces, which have a 5-parameter group of automorphisms and where the Klein-Gordon and the Dirac equations are nonintegrable using the technique of complete separation of variables.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 43–46, September, 1991.     

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.     

Representations of non-commutative quantum mechanics and symmetries

We present a unified approach to representations of quantum mechanics on non-commutative spaces with general constant commutators of the phase-space variables. We find two phases and duality relations among them in arbitrary dimensions. Conditions for the physical equivalence of different representations of a given system are analyzed. Symmetries and classification of phase spaces are discussed. Especially, the dynamical symmetry of a physical system is investigated. Finally, we apply our analyses to the two-dimensional harmonic oscillator and the Landau problem. Received: 17 December 2002, Published online: 11 June 2003