The Geometry of the Dirac Equation

We discuss the problem of the derivation and the interpretation of metric tensors and generalized equations of motion for test particles from quasilinear spinor equations.     

Derivation of the Dirac Equation by Conformal Differential Geometry

A rigorous ab initio derivation of the (square of) Dirac’s equation for a particle with spin is presented. The Lagrangian of the classical relativistic spherical top is modified so to render it invariant with respect conformal changes of the metric of the top configuration space. The conformal invariance is achieved by replacing the particle mass in the Lagrangian with the conformal Weyl scalar curvature. The Hamilton-Jacobi equation for the particle is found to be linearized, exactly and in closed form, by an ansatz solution that can be straightforwardly interpreted as the “quantum wave function” of the 4-spinor solution of Dirac’s equation. All quantum features arise from the subtle interplay between the conformal curvature acting on the particle as a potential and the particle motion which affects the geometric “pre-potential” associated to the conformal curvature itself. The theory, carried out here by assuming a Minkowski metric, can be easily extended to arbitrary space-time Riemann metric, e.g. the one adopted in the context of General Relativity. This novel theoretical scenario appears to be of general application and is expected to open a promising perspective in the modern endeavor aimed at the unification of the natural forces with gravitation.     

Quantum-like Interpretation of Dirac Wave Equation in?Robertson-Walker Geometry

The Dirac wave equation is separated in the Robertson-Walker metric. The resulting radial equation is interpreted as a one dimensional quantum-like equation that is explicitly solved. There results that the energy spectrum, that is determined in the flat, open and closed universe, is independent of the mass of the particle. Moreover it is the same of the massless neutrino case previously studied. In the closed metric case the discrete positive spectrum is asymptotically determined. The separation of the energy levels is however very far from being experimentally tested.     

Quaternion Dirac Equation and Supersymmetry

Quaternion Dirac equation has been analyzed and its supersymmetrization has been discussed consistently. It has been shown that the quaternion Dirac equation automatically describes the spin structure with its spin up and spin down components of two component quaternion Dirac spinors associated with positive and negative energies. It has also been shown that the supersymmetrization of quaternion Dirac equation works well for different cases associated with zero mass, nonzero mass, scalar potential and generalized electromagnetic potentials. Accordingly we have discussed the splitting of supersymmetrized Dirac equation in terms of electric and magnetic fields.     

Quaternionic Lorentz Group and Dirac Equation

We formulate Lorentz group representations in which ordinary complex numbers are replaced by linear functions of real quaternions and introduce dotted and undotted quaternionic one-dimensional spinors. To extend to parity the space-time transformations, we combine these one-dimensional spinors into bi-dimensional column vectors. From the transformation properties of the two-component spinors, we derive a quaternionic chiral representation for the space-time algebra. Finally, we obtain a quaternionic bi-dimensional version of the Dirac equation.     

Dirac Equation at Finite Temperature

In this paper, we propose finite temperature Dirac equation, which can describe the quantum systems in an arbitrary temperature for a relativistic particle of spin-1/2. When the temperature T=0, it become Dirac equation. With the equation, we can study the relativistic quantum systems in an arbitrary temperature.     

A Search on Dirac Equation

The solutions, in terms of orthogonal polynomials, of Dirac equation with analytically solvable potentials are investigated within a novel formalism by transforming the relativistic equation into a Schrodinger-like one. Earlier results are discussed in a unified framework, and some solutions of a large class of potentials are given.     

Time Delay for the Dirac Equation

We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator \({\int\limits_{0} ^{\infty}{\rm e}^{iH_{0}t}\zeta(\frac{\vert x\vert }{R}) {\rm e}^{-iH_{0}t}{\rm d}t}\), as \({R \rightarrow \infty}\), is presented. Here, H₀ is the free Dirac operator and \({\zeta\left(t\right)}\) is such that \({\zeta\left(t\right) = 1}\) for \({0 \leq t \leq 1}\) and \({\zeta\left(t\right) = 0}\) for \({t > 1}\). This approach allows us to obtain the time delay operator \({\delta \mathcal{T}\left(f\right)}\) for initial states f in \({\mathcal{H} _{2}^{3/2+\varepsilon}(\mathbb{R}^{3};\mathbb{C}^{4})}\), \({\varepsilon > 0}\), the Sobolev space of order \({3/2+\varepsilon}\) and weight 2. The relation between the time delay operator \({\delta\mathcal{T}\left(f\right)}\) and the Eisenbud–Wigner time delay operator is given. In addition, the relation between the averaged time delay and the spectral shift function is presented.     

The Dirac Equation in Binary Geometrophysics

Within the framework of a program of the construction (derivation) of the classical space-time from a certain system of notions and relations (binary systems of complex relations) that show up in microcosm physics, the prototype of the Dirac equation for a free particle is considered, and the change from this prototype to the Dirac equation is followed first in the momentum and then in the coordinate representation. The procedure of changing from elementary bases to a macroinstrument in reference to which the generally accepted physical notions are formulated.     

Coulomb-Like Interactions in the Dirac Equation

The Dirac equation for the Coulomb-like problem is modified by incorporating minimal interactions into the Dirac Hamiltonian, that keep the 1/r potential dependence. We determine the general energy eigenvalues and the corresponding eigenfunctions.     

Dirac–Kähler Equation

Tensor, matrix, and quaternion formulations of Dirac–Kähler equation for massive and massless fields are considered. The equation matrices obtained are simple linear combinations of matrix elements in the 16-dimensional space. The projection matrix-dyads defining all the 16 independent equation solutions are found. A method of computing the traces of 16-dimensional Petiau–Duffin–Kemmer matrix product is considered. We show that the symmetry group of the Dirac–Kähler tensor fields for charged particles is SO(4, 2). The conservation currents corresponding this symmetry are constructed. We analyze transformations of the Lorentz group and quaternion fields. Supersymmetry of the Dirac–Kähler fields with tensor and spinor parameters is investigated. We show the possibility of constructing a gauge model of interacting Dirac–Kähler fields where the gauge group is the noncompact group under consideration.     

Vector Representation of Interacting Dirac Equation

Using the Clifford algebra, a vectorial equation for the Dirac spinorial equation is constructed and the relation with the Klein—Gordon equation becomes transparent. The equation interacting with the electromagnetic field leads to a nontrivial generalization for the interacting Klein—Gordon equation. The Lagrangian density for this interaction is given.     

Hermitian Geometry and Complex Space-Time

We consider a complex Hermitian manifold of complex dimensions four with a Hermitian metric and a Chern connection. It is shown that the action that determines the dynamics of the metric is unique, provided that the linearized Einstein action coupled to an antisymmetric tensor is obtained, in the limit when the imaginary coordinates vanish. The unique action is of the Chern-Simons type when expressed in terms of the Kähler form. The antisymmetric tensor field has gauge transformations coming from diffeomorphism invariance in the complex directions. The equations of motion must be supplemented by boundary conditions imposed on the Hermitian metric to give, in the limit of vanishing imaginary coordinates, the low-energy effective action for a curved metric coupled to an antisymmetric tensor.     

Non-Lie and Discrete Symmetries of the Dirac Equation

Abstract

New algebras of symmetries of the Dirac equation are presented, which are formed by linear and antilinear first–order differential operators. These symmetries are applied to decouple the Dirac equation for a charged particle interacting with an external field.     

Quintessence and Dark Matter Created by Weyl–Dirac Geometry

A spatially closed universe undergoing at present accelerated expansion, having a non-vanishing cosmological constant, and filled with luminous- and dark matter is described in terms of the Integrable Weyl–Dirac theory. It is shown that, during the dust-dominated period, dark matter and the quintessence pressure, the latter giving rise to acceleration: both are created by the Dirac gauge function. The behavior of two models: a nearly flat one and a well closed are considered in appropriate gauges, and plausible scenarios are obtained. The outcome of the present paper, together with results of a previous work,⁽³¹⁾ provide a geometrically based, classical, singularity-free model of the universe, that has originated from a pure geometric Weyl–Dirac entity, passed a prematter period, the radiation-dominated era, and continues its development in the present dust period.     

Particle Spectrum Implied by the Dirac Equation

There is a process that starts from the Lagrangian of a set of field equations and leads to a spectrum of particle states. The process is applied in this article to a Lagrangian for the Dirac equation. It leads to a differential equation with solutions that describe particles with definite mass, angular momentum J, charge, and isotopic spin I, having I = J. There is no suggestion of strangeness. The theory is in rough agreement with the masses of many of the 0⁺ (0^–+) and 0⁺ (0⁺⁺) mesons and with the masses of the nonstrange 1/2 (1/2⁺) and 3/2 (3/2⁺) baryons.     

5D Dirac Equation in Induced-Matter Theory

According to an induced-matter approach, Liu and Wesson obtained the rest mass of a typical particle from the reduction of a 5D Klein–Gordon equation to a 4D one. Introducing an extra-dimension momentum operator identified with the rest mass eigenvalue operator, we consider a way to generalize the 4D Dirac equation to 5D. An analogous normal Dirac equation is gained when the generalization reduces to 4D. We find the rest mass of a particle in curved space varies with spacetime coordinates and check this for the case of exact solitonic and cosmological solution of the 5D vacuum gravitational field equations.     

Dirac Equation in an Axial Coulomb Potential

We consider solutions to the Dirac equation in the presence of an external axial vector potential coupled to the spinor field psi through the interaction term . There turn out to be no bound-state energies in this system consistent with a normalizable wave function.     

Some Studies of the Neutron's Dirac Equation

In this paper, we continue the discussion for the neutron's Dirac equation and relevant problems after Ref.[1]. We consider the neutron's Dirac equation with the electric moment besides the magnetic moment, solve rigorously the neutron's Dirac equation in a uniform electromagnetic field. We also set up a relativistic neutron's spin-echo theory with a magnetic moment.