The Einstein-Dirac Equation in Robertson-Walker Space-Time Does Not Admit Standard Solutions

The Einstein-Dirac equation is considered in the Robertson-Walker space-time. Solutions of the equation are looked for in the class of standard solutions of the Dirac equation. It is shown that the Einstein-Dirac equation does not have standard solutions for both massive and massless Dirac field. Also superpositions of massive standard solutions are not solutions of the Einstein-Dirac equation. The result, that is briefly commented, is coherent and complementary to other existing results.     

Exact Solutions of Dirac Equation and Particle Creation in (1+3)-Dimensional Robertson-Walker Spacetime

The exact solutions of the Dirac equation are discussed for a Robertson-Walker spacetime with asymptotically Minkowskian in and out regions. We obtain the mode solutions which reduce to positive and negative Minkowskian spinors in asymptotically regions. Using the obtained solutions we compute the density of created particles.     

Scalar field equation in Robertson-Walker space-time

The quantization of the scalar field is reconsidered in some of its basic elements in the context of the Robertson-Walker space-time. The integration of the generalized Klein-Gordon equation is performed by preliminary separation of the equation with the usual separation method. The orthonormal mode solutions are determined by the explicit integration of the resulting angular and radial equations and by standard properties of the time equation. The time evolution given by the standard cosmological model is briefly discussed.     

The Dirac Equation in the Bertotti-Robinson Space-Time

The Dirac equation is considered in the uniform electromagnetic field space of Bertotti-Robinson with charge coupling. The methods of separation of variables and decoupling are easily achieved. The separated axial equation is reduced to a rare Riccati type of differential equation. The behaviour of potentials, their asymptotic solutions and the conserved currents of the Dirac equation are found.     

The time dependent Ginzburg-Landau equation in fractal space-time

We use the hydrodynamic formulation of Scale Relativity Theory to analyze the TDGL equation. As a result, London equations come naturally from the system, when equating to zero the real velocity, the imaginary one turns real, the superconducting fluid act as a subquantum medium energy accumulator, the vector potential, the real and the imaginary velocity are all written in terms of the elliptic function. When solving the resulted system by means of WKBJ method, we get tunneling and quantization. In other words, scale transformation laws produce, on the motion equation of particles governed by the TDGL equation, under some peculiar assumptions, effects which are analogous to those of a “macroscopic quantum mechanics”.     

Invariance properties of the Dirac equation with external electro-magnetic field

In this paper, we attempt to obtain the nature of the external field such that the Dirac equation with external electro-magnetic field is invariant. The Poincaré group, which is the maximal symmetry group for field free case, is constrained by the presence of the external field. Introducing infinitesimal transformation ofx and ψ, we apply Lie’s extended group method to obtain the class of external field which admit of the invariance of the equation. It is important to note that the constraints for the existence of invariance are explicity on the electric and magnetic field, though only potentials explicity appears in the equation. Presented at the Sixth Chittagong Conference on Mathematical Physics, January 2001.     

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.     

Proca fields interpretation of spin 1 equation in Robertson-Walker space-time

The general scheme for massive spin 1 equation in curved space-time is specialized to describe the Proca fields. The expressions of the Proca tensor fields are detailed in the Robertson-Walker space-time by means of the solutions of the spin 1 equation in a given tetrad and by the components of the tetrad itself. Asymptotic behaviours of the fields are discussed in the flat, closed and open space-time cases.Also GNFM, Italy     

Higher dimensional supersymmetric quantum mechanics and Dirac equation

We exhibit the supersymmetric quantum mechanical structure of the full 3+1 dimensional Dirac equation considering ‘mass’ as a function of coordinates. Its usefulness in solving potential problems is discussed with specific examples. We also discuss the ‘physical’ significance of the supersymmetric states in this formalism.     

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.     

A remark on the Dirac equation

We give a simple deductive derivation of the Dirac equation for a free particle. Our construction provides a clear distinction between what are the physical contents and what are purely mathematical expedients in the formalism of quantum mechanics (QM).     

Yangian symmetries and integrability of the Dirac equation with spin symmetry

We show that a Yangian symmetry, namely, Y(su(2))$Y\left(su\left(2\right)\right)$, exists in the Dirac equation with spin symmetry when the potential term takes a Coulomb form. We construct the generators of Y(su(2))$Y\left(su\left(2\right)\right)$ explicitly and get the energy spectrum of this model from the representation theory for Y(su(2))$Y\left(su\left(2\right)\right)$. We also show that this model is integrable, from RTT relations.     

A simple efficient methodology for Dirac equation in minimal length quantum mechanics

We solve the modified Dirac equation by adding a harmonic oscillator potential and implementing the Nikiforov–Uvarov technique. The closed forms of solutions are reported in a quite simple and systematic manner.     

Reduced Dirac equation and Lamb shift as off-mass-shell effect in quantum electrodynamics

Based on the accurate experimental data of energy-level differences in hydrogen-like atoms, especially the 1S--2S transitions of hydrogen and deuterium, the necessity of introducing a reduced Dirac equation with reduced mass as the substitution of original electron mass is stressed. Based on new cognition about the essence of special relativity, we provide a reasonable argument for the reduced Dirac equation to have two symmetries, the invariance under the (newly defined) space--time inversion and that under the pure space inversion, in a noninertial frame. By using the reduced Dirac equation and within the framework of quantum electrodynamics in covariant form, the Lamb shift can be evaluated (at one-loop level) as the radiative correction on a bound electron staying in an off-mass-shell state---a new approach eliminating the infrared divergence. Hence the whole calculation, though with limited accuracy, is simplified, getting rid of all divergences and free of ambiguity.     

Spin symmetric solutions of Dirac equation with PSschl-Teller potential

The approximate analytical solutions of the Dirac equation with the Poeschl-Teller potential is presented for arbitrary spin-orbit quantum number κ within the framework of the spin symmetry concept. The energy eigenvalues and the corresponding two Dirac spinors are obtained approximately in closed forms. The limiting cases of the energy eigenvalues and the two Dirac spinors are briefly discussed.     

Quantum-classical correspondence of the Dirac equation with a scalar-like potential

Quantum matrix elements of the coordinate, momentum and the velocity operator for a spin-1/2 particle moving in a scalar-like potential are calculated. In the large quantum number limit, these matrix elements give classical quantities for a relativistic system with a position-dependent mass. Meanwhile, the Klein-Gordon equation for the spin-0 particle is discussed too. Though the Heisenberg equations for both the spin-0 and spin-1/2 particles are unlike the classical equations of motion, they go to the classical equations in the classical limit.      

Exact solution to the one-dimensional Dirac equation of linear potential

In this paper the one-dimensional Dirac equation with linear potential has been solved by the method of canonical transformation. The bound-state wavefunctions and the corresponding energy spectrum have been obtained for all bound states.     

Solution of Dirac equation around a charged rotating black hole

The aim of this paper is to solve the radial parts of a Dirac equation in Kerr-Newman （KN） geometry. The potential is replaced by a collection of step functions, then the reflection and transmission coefficients as well as the solution of the wave equation are obtained by using a quantum mechanical method. The result shows that the waves with different values of mass will be scatted off very differently.