Supersymmetrization of Quaternion Dirac Equation for Generalized Fields of Dyons

The quaternion Dirac equation in presence of generalized electromagnetic field has been discussed in terms of two gauge potentials of dyons. Accordingly, the supersymmetry has been established consistently and thereafter the one, two and component Dirac Spinors of generalized quaternion Dirac equation of dyons for various energy and spin values are obtained for different cases in order to understand the duality invariance between the electric and magnetic constituents of dyons.     

Quaternion-Octonion Analyticity for Abelian and Non-Abelian Gauge Theories of Dyons

Einstein-Schrödinger (ES) non-symmetric theory has been extended to accommodate the Abelian and non-Abelian gauge theories of dyons in terms of the quaternion-octonion metric realization. Corresponding covariant derivatives for complex, quaternion and octonion spaces in internal gauge groups are shown to describe the consistent field equations and generalized Dirac equation of dyons. It is also shown that quaternion and octonion representations extend the so-called unified theory of gravitation and electromagnetism to the Yang-Mill’s fields leading to two SU(2) gauge theories of internal spaces due to the presence of electric and magnetic charges on dyons.     

Hypermass generalization of Einstein's gravitation theory

The curvilinear invariant quaternion formalism is examined for curved space time. Einstein's gravitation equation is shown to have a sample and natural form in this notation. The hypermass generalization of particle mass, which was generated in our studies of the Dirac equation, is incorporated in gravitation by generalizing Einstein's equation. Covariance requires that the gravitational constant be generalized to an invariant quaternion when the mass is. The modification appears minor and of no importance cosmologically, unless one begins considering time and mass dependence ofG.NASA-ASEE Summer Faculty Fellow, 1972 and 1973. Address 1972–1973, Department of Physics, Oregon State University, Corvallis, Oregon 97331.     

A classical Klein—Gordon particle

An elementary particle is described as a spherically symmetric solution of the Klein-Gordon equation and the Einstein equations of general relativity. It is found that it has a mass of the order of the Planck mass. If one assumes that the motion of its center of mass is determined by the Dirac equations, then it has a spin of 1/2.     

Mass term variations in the Dirac hydrogen atom

The extended quaternion algebra allowstwo forms for the Dirac equation mass term. We show that the definition of angular momentum in the relativistic hydrogen atom suggests that either is physically allowed, but not a combination of both. A combination of both mass termscould still give the usual mass term in the nonrelativistic limit as is shown.     

Exact Spin and Pseudospin Symmetry Solutions of the Dirac Equation for Mie-Type Potential Including a Coulomb-like Tensor Potential

We solve the Dirac equation for Mie-type potential including a Coulomb-like tensor potential under spin and pseudospin symmetry limits with arbitrary spin–orbit coupling quantum number κ. The Nikiforov–Uvarov method is used to obtain analytical solutions of the Dirac equation. Since it is only the wave functions which are obtained in a closed exact form; as for the eigenvalues, only the eigenvalue equations have been given and they have been solved numerically. It is also shown that the degeneracy between spin doublets and pseudospin doublets is removed by tensor interaction.     

The necessity of torsion in gravity

It is shown that torsion is required for a complete theory of gravitation, and that without it, the equations of gravitation violate fundamental laws. In the first case, we are reminded that, in the absence of external forces, the correct conservation law of total angular momentum arises only if torsion, whose origin is intrinsic spin, is included into gravitation. The second case considers the “mass reversal” transformation. It has been known that under a global chiral transformation and “mass to negative mass” transformation, the Dirac equation is invariant. But global transformations violate special relativity, so this transformation must be made local. It is shown that the torsion is the gauge field for this local invariance.     

Relativistic symmetry of position-dependent mass particle in Coulomb field including tensor interaction

The spatially-dependent mass Dirac equation is solved exactly for attractive scalar and repulsive vector Coulomb potentials including a tensor interaction under the spin and pseudospin symmetric limit. Closed forms of the energy eigenvalue equation and wave functions are obtained for arbitrary spin-orbit quantum number κ. Some numerical results are given too. The effect of the tensor interaction on the bound states is presented. It is shown that the tensor interaction removes the degeneracy between two states in the spin doublets. We also investigate the effects of the spatially-dependent mass on the bound states under the conditions of the spin symmetric limit in the absence of tensor interaction.     

Higher Spin Quaternion Waves in the Klein-Gordon Theory

Electromagnetic interactions are discussed in the context of the Klein-Gordon fermion equation. The Mott scattering amplitude is derived in leading order perturbation theory and the result of the Dirac theory is reproduced except for an overall factor of sixteen. The discrepancy is not resolved as the study points into another direction. The vertex structures involved in the scattering calculations indicate the relevance of a modified Klein-Gordon equation, which takes into account the number of polarization states of the considered quantum field. In this equation the d’Alembertian is acting on quaternion-like plane waves, which can be generalized to representations of arbitrary spin. The method provides the same relation between mass and spin that has been found previously by Majorana, Gelfand, and Yaglom in infinite spin theories.     

The complex scaling method for Dirac resonances

The method of complex scaling, usually used in atomic and molecular resonance calculations, is generalized to the Dirac equation. It is shown that Dirac resonances are associated with nonreal eigenvalues of the scaled Dirac Hamiltonian. The perturbation theory for the resonance parameters is also discussed.     

Spin eigen-states of Dirac equation for quasi-two-dimensional electrons

Dirac equation for electrons in a potential created by quantum well is solved and the three sets of the eigen-functions are obtained. In each set the wavefunction is at the same time the eigen-function of one of the three spin operators, which do not commute with each other, but do commute with the Dirac Hamiltonian. This means that the eigen-functions of Dirac equation describe three independent spin eigen-states. The energy spectrum of electrons confined by the rectangular quantum well is calculated for each of these spin states at the values of energies relevant for solid state physics. It is shown that the standard Rashba spin splitting takes place in one of such states only. In another one, 2D electron subbands remain spin degenerate, and for the third one the spin splitting is anisotropic for different directions of 2D wave vector.     

Maximal symmetry and mass generation of Dirac fermions and gravitational gauge field theory in six-dimensional spacetime

The relativistic Dirac equation in four-dimensional spacetime reveals a coherent relation between the dimensions of spacetime and the degrees of freedom of fermionic spinors. A massless Dirac fermion generates new symmetries corresponding to chirality spin and charge spin as well as conformal scaling transformations. With the introduction of intrinsic W-parity, a massless Dirac fermion can be treated as a Majorana-type or Weyl-type spinor in a six-dimensional spacetime that reflects the intrinsic quantum numbers of chirality spin. A generalized Dirac equation is obtained in the six-dimensional spacetime with a maximal symmetry. Based on the framework of gravitational quantum field theory proposed in Ref. [1] with the postulate of gauge invariance and coordinate independence, we arrive at a maximally symmetric gravitational gauge field theory for the massless Dirac fermion in six-dimensional spacetime. Such a theory is governed by the local spin gauge symmetry SP(1,5) and the global Poincar′e symmetry P(1,5)= SO(1,5) P~(1,5) as well as the charge spin gauge symmetry SU(2). The theory leads to the prediction of doubly electrically charged bosons. A scalar field and conformal scaling gauge field are introduced to maintain both global and local conformal scaling symmetries. A generalized gravitational Dirac equation for the massless Dirac fermion is derived in the six-dimensional spacetime. The equations of motion for gauge fields are obtained with conserved currents in the presence of gravitational effects. The dynamics of the gauge-type gravifield as a Goldstone-like boson is shown to be governed by a conserved energy-momentum tensor, and its symmetric part provides a generalized Einstein equation of gravity. An alternative geometrical symmetry breaking mechanism for the mass generation of Dirac fermions is demonstrated.     

The Isotopic Field-Charge Assumption Applied to the Electromagnetic Interaction

This paper applies the isotopic field-charge spin theory (Darvas in Int. J. Theor. Phys. 50(10):2961–2991, 2011) to the electromagnetic interaction. First, a modified Dirac equation in the presence of a velocity dependent gauge field and isotopic field charges (namely Coulomb and Lorentz type electric charges, as well as gravitational and inertial masses) is derived. This equation is compared with the classical Dirac equation. It is shown that, since the presence of isotopic field-charges would distort the Lorentz invariance of the equation, there is a transformation, which together with the Lorenz transformation restores the invariance of the equation, in accordance with the conservation of the isotopic field-charge spin (Darvas in Concepts Phys. VI 1:3–16, 2009). The paper discusses conclusions derived from the extensions of the Dirac equation. It is shown that in semi-classical approximation the model returns the original Dirac equation, and at significantly relativistic velocities it approaches the Schrödinger equation. Among other conclusions, the clue gives physical meaning to the electric moment. The closing section summarises a few further conclusions and shows a few developments to be discussed in detail in a subsequent paper (Darvas in Int. J. Theor. Phys., 2013).     

Triality and Quantization of Singularities in Massive Fermion

It is proved that fermions can acquire the mass through the additional non-integrable exponential factor. For this propose the special vector potential associated with the spinor field was introduced. Such a vector potential has closk relation with the. triality property in Dirac spinors and plays crucial role in the construction of massive term. It is shown that the change in phase of a wavefunction round any closed curve with the possibility of there being singularities in our vector potential will lead to the law of quantization of physical constants including the mass. The triality properties of Dirac's spinors are studied and it leads to a double covering vector representation of Dirac spinor field. It is proved that massive Dirac equation in the bosonic representation is self-dual.     

电子的相对论平均场理论与一阶、二阶Rashba效应

用多体平均场意义下电子的Dirac方程讨论了电子自旋动力学及其相关问题. 在大分量Dirac方程的非相对论展开中讨论了电子自旋动力学的高阶效应,并且在二维情形下得到了包括一阶和二阶Rashba效应的电子自旋动力学哈密顿量,求出了相应的包括二阶Rashba效应的哈密顿量的能量和波函数的本征值解,由此讨论了二阶Rashba效应修正的物理含义和大小. 关键词： 二阶Rashba效应 自旋电子学 Dirac方程 相对论平均场理论     

Axial Anomaly for Eguchi-Hanson Metrics with Nonzero Total Mass

We compute the Dirac indexes for the two spin structures k₀ and k₁ for Eguchi-Hanson metrics with nonzero total mass. It shows that the Dirac indexes do not vanish in general, and axial anomaly exists. When the metric has zero total mass, the Dirac index vanishes for the spin structure k₀, and no axial anomaly exists in this case.     

RELATIVISTIC STUDY OF PROTON-NUCLEUS ELASTIC SCATTERING AT LOWER ENERGIES

The Lorentz invariant relativistic optical potential have been discussed at energies below 300MeV.The Dirac equation with scalar and vector potential is solved by exact partial wave method.The calculated results of proton ⁴⁰Ca at energy region 300—65MeV are presented and compared with the experimental data of differential cross section dσ/dΩ,analyzing power A_y(θ) and spin rotation function Q(θ).It is shown that the impoved relativistic optical potential fits the data well.     

BPS Monopoles in Moduli Space under SU(2) Gauge Potential

Solving Dirac equation for a BPS monopole moving in the field of another BPS monopole in moduli space, it has been shown that spin momentum of the interacting monopole behaves as an extra energy source. The possibilities of splitting of the energy levels of the system have been explored.     

Lagrangian and Hamiltonian formalisms for relativistic dynamics of a charged particle with dipole moment

The Lagrangian and Hamiltonian formulations for the relativistic classical dynamics of a charged particle with dipole moment in the presence of an electromagnetic field are given. The differential conservation laws for the energy-momentum and angular momentum tensors of a field and particle are discussed. The Poisson brackets for basic dynamic variables, which form a closed algebra, are found. These Poisson brackets enable us to perform the canonical quantization of the Hamiltonian equations that leads to the Dirac wave equation in the case of spin 1/2. It is also shown that the classical limit of the squared Dirac equation results in equations of motion for a charged particle with dipole moment obtained from the Lagrangian formulation. The inclusion of gravitational field and non-Abelian gauge fields into the proposed formalism is discussed.Received: 4 June 2005, Published online: 27 July 2005