Dirac equation in Robertson-Walker metric

We recast the Dirac relativistic equation within the theoretical framework of the Robertson-Walker metric, using spatial hypersurfaces that are essentially curved, and hence more general, as compared to the flat ones employed by Barut and Duru.     

Solution of the Dirac equation in plane wave metric backgrounds

We present a new solution of the Dirac equation in the background of a plane wave metric. We examine the relation between sections of the exterior and Clifford bundles of a (pseudo-)Riemannian manifold. A spinor calculus is established and used to investigate a new solution of the Dirac equation lying in a minimal left ideal characterized by a certain idempotent projector.     

Dirac equation in curved spacetime with the metric in the Kerr-Schield form

The solution of the Dirac equation in the wave package form in a slightly curved spacetime (compared with the size of the wave package) was studied. For the metric in the Kerr-Schield form, a system of common differential equations describing spin conditions of massive neutral Dirac particles (neutrinos) was obtained. The effect of depolarization of the massive neutrinos in a gravitational field are discussed. This effect allows a considerable similarity between the theoretical and observed solar neutrino flows to be established if $m_{\nu _e } \geqslant 10^{ - 4} $ .     

On the Dirac equation

A connection is established between solutions of the Dirac equation (for an electron in a constant magnetic field) and those of the wave equation for a particle with zero mass.Translated from Izvestiya VUZ. Fizika, No. 12, pp. 96–100, December, 1973.     

Dirac equation and the Ivanenko-Landau-Kähler equation

We consider spinor theory within the framework of an inhomogeneous differential forms formalism. We also consider the possibility of describing fermions with the Ivanenko-Landau-Kähler equation. The relations between these two equations are studied.     

Six-dimensional Dirac equation

The Dirac equation in six-dimensional relativity (three space and three time) is considered and shown to correspond to particles which have spatial spin-1/2 and temporal spin-1/2. Explicit forms of the spinor transformation are found. Plane wave solutions are obtained and their properties are given in terms of spatial and temporal spins and helicities. An expression is found for the charge conjugation operator.     

Two-body Dirac equation versus KDP equation

A brief review of two-body Dirac and Kemmer-Duffin-Petiau approaches for the bound state problem of two fermions is presented from an algebraic point of view in a comparative manner. Reduction of the direct product of two Dirac spaces is discussed.Work supported by the T.B.T.A.K. under TBAG/CG-1.     

Linear and nonlinear Dirac equation

Using the usual matrix representation of Clifford algebra of spacetime, quantities independent of the choice of a representation in the Dirac theory are examined, relativistic invariance of the theory is discussed, and a nonlinear equation is proposed. The equation presents no negative energy waves and gives the same results as the linear theory for hydrogen atom.     

Relation between the Kähler equation and the Dirac equation

The formal analogy and the substantial differences between the Kähler equation and the Dirac equation are explained in terms of the relativistic compatibility of a common differential operator on the Clifford algebraC with two distinct representations of the Lorentz Lie algebra onC.     

Bosonic symmetries of the Dirac equation

We have found on the basis of the symmetry analysis of the standard Dirac equation with nonzero mass the new physically meaningful features of this equation. The new bosonic symmetries of the Dirac equation in both the Foldy-Wouthuysen and the Pauli-Dirac representations are found, among which (together with the 32-dimensional pure matrix algebra of invariance) the new spin s=(1,0) multiplet Poincaré symmetry is proved. In order to carry out the corresponding proofs a 64-dimensional extended real Clifford-Dirac algebra is put into consideration.     

Integration method for the Dirac equation

An integration method for the Dirac equation is proposed. The method, based on diagonalization, reduces the problem to one of integration of independent second-order differential equations.     

Tensorial representation of the Dirac equation

We discuss tensor representations of the Dirac equation using a geometric approach. We find that the mass zero Dirac equations can be represented by Maxwell equations having a source which obeys the empty space wave equation. We also obtain a relation for the source in terms ofE andH. In the case of mass not equal to zero a difficulty is encountered in removing the constant spinors¯ _Aand¯ _A.We find that the arbitrary constant spinors can be eliminated in a spinor theory based on the Klein-Gordon 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.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 84–89, February, 1972.The author is grateful to V. N. Shapovalov for discussions and valuable suggestions.     

Fluid-dynamical representations of the Dirac equation

A relative kinetic mass operator is defined bym =c ⁻²·(E −eΦ), and it is shown that bt using it in a symmetric form one can correlate the (charge) velocity operatorα in the Dirac theory exactly with the general quantum mechanical momentum —ih∇. Then the net force, defined as the rate of change of the relative momentum with time, is exactly equal to the Lorentz force. The contribution due to the time variation of mass equals the negative of space variation of the scalar potential, the Newtonian force, whereas the time variation of the charge current absorbs the entire vector potential dependence. The analogous Euler equations can be written either in terms of the charge current or in terms of the mass current. For a many particle system one needs the usual net single particle parameters and the consideration of both the direct and exchange contributions of the two particle interaction. These Euler equations yield two different conditions of the stationary state. It is shown that the charge-current condition is necessary but not sufficient, whereas the mass-current condition retains the appropriate scalar potential dependence. These two conditions are compared for the spherically symmetric case. The charge density, charge current and relative mass current are tabulated for atomic spinors. Differences between the quantum and classical forces for the H ₂ ⁺ molecular ion exhibit the inadequacy of ordinary atomic spinor basis in forming molecular spinors.