Spinor Solutions Research Articles

Weyl time evolution operator in planar Bianchi-type-I universes

In this paper, Dirac’s equation in anisotropic Bianchi-type-I background spacetimes is treated with respect to orthonormal frames. By specializing to the massless spinor case and metrics with power-law scale factors and planar symmetry, an analytical expression of the approximate time evolution operator (TEO) is derived. By use of this operator, all approximate spinor solutions can be generated. By construction, these solutions agree asymptotically and at early times with the exact solutions. The operator approach is also well adapted for studying the case of small deviations from conformally flat backgrounds. In particular, it can be shown that the limiting case of vanishing anisotropy renders the correct result.

Spinor solutions of a Chern–Simons model for the superconformal algebra

We present analytical solutions for homogenous and isotropic spaces of the supersymmetric Chern–Simons model with matter in the adjoint representation. The configurations that we found correspond to a gravitating spinor content and torsion is also present. The spinor behaves like dark energy in the sense that drives an exponential expansion. The solution found can be seen as an anisotropic fluid.

Generalized quaternionic free rotational Dirac equation and spinor solutions in the electromagnetic field

Starting with the quaternionic Minkowski space–time and its four-vector representation, a rotational analogue of the quaternionic Dirac equation in the electromagnetic field is developed, which includes not only the energy solutions but also the angular momentum solutions for rotating Dirac fermions. The striking feature of the quaternionic approach is that it depicts a unified representation of energy–momentum in a single framework as the space–time. Furthermore, we establish the generalized Schrodinger–Pauli-like equation associated with generalized electric and magnetic dipole energies that corresponds to their dipole moments. As such, the present analysis deals with the invariant behavior of the extended quaternionic rotational Dirac equation under Lorentz–Poincaré, Gauge, duality, and CPT invariance.

Scalar Pre-potentials for Spinor and Tensor Fields on Spacetime

We review a technique for solving a class of classical linear partial diferential systems of relevance to physics in Minkowski spacetime. All the equations are amenable to analysis in terms of complex solutions in the kernel of the scalar Laplacian and a complexifed Hertz potential. The complexifcation prescription ensures the existence of regular physical solutions with chirality and propagating, non-singular, pulse-like characteristics that are bounded in all three spatial dimensions. The technique is applied to the source-free Maxwell, Bopp-Landé-Podolsky and linearised Einstein feld systems, and particular solutions are used for constructing classical models describing single-cycle laser pulses and a mechanism is discussed for initiating astrophysical jets. Our article concludes with a brief introduction to spacetime Cliford algebra ideals that we use to represent spinor felds. We employ these to demonstrate how the same technique used for tensor felds enables one to construct new propagating, chiral, non-singular, pulse-like spinor solutions to the massless Dirac equation in Minkowski spacetime.

Twistorial description of Bondi-Metzner-Sachs symmetries at null infinity

We describe a novel twistorial construction of the asymptotic BMS symmetries at null infinity for asymptotically flat spacetimes. We define BMS twistors as spinor solutions to some set of components of the usual spacetime twistor equation restricted to null infinity. The space of BMS twistors is infinite-dimensional. We show that given two BMS twistors their symmetric tensor product can be used to generate (complex) vector fields which are the infinitesimal BMS symmetries of null infinity. In this sense BMS twistors are "square roots" of BMS symmetries. We also show that these BMS twistor equations can be written a pair of covariant spinor-valued equations which are completely determined by the intrinsic universal structure of null infinity.

A square-integrable spinor solution to non-interacting Dirac equations

We consider the Dirac equation written in polar form, without any external potential but equipped with a non-zero tensorial connection, and we find a new type of solution that is localized around the origin with a decreasing exponential behavior in the radial coordinate.

Spinor field solutions in F(B2) modified Weyl gravity

We consider modified Weyl gravity where a Dirac spinor field is nonminimally coupled to gravity. It is assumed that such modified gravity is some approximation for the description of quantum gravitational effects related to the gravitating spinor field. It is shown that such a theory contains solutions for a class of metrics which are conformally equivalent to the Hopf metric on the Hopf fibration. For this case, we obtain a full discrete spectrum of the solutions and show that they can be related to the Hopf invariant on the Hopf fibration. The expression for the spin operator in the Hopf coordinates is obtained. It is demonstrated that this class of conformally equivalent metrics contains the following: (a) a metric describing a toroidal wormhole without exotic matter; (b) a cosmological solution with a bounce and inflation and (c) a transition with a change in metric signature. A physical discussion of the results is given.

Propagators of the Dirac fermions on spatially flat FLRW space–times

The general formalism of the free Dirac fermions on spatially flat (1[Formula: see text]+[Formula: see text]3)-dimensional Friedmann-Lemaître–Robertson–Walker (FLRW) space–times is developed in momentum representation. The mode expansions in terms of the fundamental spinors satisfying the charge conjugation and normalization conditions are used for deriving the structure of the anticommutator matrix-functions and, implicitly, of the retarded, advanced, and Feynman fermion propagators. The principal result is that the new type of integral representation we proposed recently in the de Sitter case can be applied to the Dirac fermions in any spatially flat FLRW geometry. Moreover, the Dirac equation of the left-handed massless fermions can be analytically solved finding a general spinor solution and deriving the integral representations of the neutrino propagators. It is shown that in the Minkowski flat space–time our new integral representation is up to a change of variable just like the usual Fourier representation of the fermion propagators. The form of the Feynman propagator of the massive fermions on a spatially flat FLRW space–time with a scale factor of Milne-type is also outlined.

Charge-Displacement Analysis via Natural Orbitals for Chemical Valence in the Four-Component Relativistic Framework.

We have recently introduced a simple yet powerful tool for analyzing quantitatively the coordination bond in terms of the donation and back-donation constituents of the Dewar-Chatt-Duncanson model. Our approach is based on the decomposition, via natural orbitals for chemical valence (NOCV), of the so-called charge-displacement (CD) function into additive chemically meaningful components (Bistoni et al. J. Chem. Phys. 2015, 142, 084112 ). The method, referred to as NOCV/CD, provides clear-cut measures of donation and back-donation charge flows following bond formation, and its robustness has been demonstrated by a tight correlation of the related charge-transfer estimates with experimental observables. In this paper we extend the NOCV/CD analysis scheme to the four-component relativistic framework, which includes spin-orbit coupling variationally. This formalism is incorporated into a recently developed, highly efficient parallel version of the relativistic Dirac-Kohn-Sham (DKS) program BERTHA (Rampino et al. J. Chem. Theory Comput. 2014, 10, 3766-3776 ). We test the accuracy and numerical stability of this new implementation through the analysis of the convergence properties of the basis sets employed to expand the DKS spinor solution and those used to linearize the electronic density in the density-fitting algorithm speeding up the evaluation of the DKS matrix. An illustration of NOCV/CD analysis in the relativistic framework is also given through a study of the metal-carbonyl coordination bond in a series of [M-CO]+ (M = Cu, Ag, Au) complexes of group 11 metals, where relativistic effects, including spin-orbit coupling, are found to play an important role.

Stability of Solitary Waves and Vortices in a 2D Nonlinear Dirac Model.

We explore a prototypical two-dimensional massive model of the nonlinear Dirac type and examine its solitary wave and vortex solutions. In addition to identifying the stationary states, we provide a systematic spectral stability analysis, illustrating the potential of spinor solutions to be neutrally stable in a wide parametric interval of frequencies. Solutions of higher vorticity are generically unstable and split into lower charge vortices in a way that preserves the total vorticity. These conclusions are found not to be restricted to the case of cubic two-dimensional nonlinearities but are found to be extended to the case of quintic nonlinearity, as well as to that of three spatial dimensions. Our results also reveal nontrivial differences with respect to the better understood nonrelativistic analogue of the model, namely the nonlinear Schrödinger equation.

Nonlinear Dirac equation in two-spinor form: Separation in static RW space-time

The Dirac equation with nonlinear terms induced by torsion is studied in Robertson-Walker (RW) space-time. An extension of a separation method of the equation, based on the Newman-Penrose formalism and previously applied to the nonlinear case, is considered. Accordingly the angular dependence of the Dirac spinor solution is separated, under a special assumption, in the general time-dependent RW metric. In the case of static RW metric the time dependence of the Dirac spinor factors out and one is left with a pair of two coupled nonlinear radial equations. The radial equations are disentangled by a suitable substitution of the spinor solution. The problem amounts then to the solution of a single second-order highly nonlinear differential equation. Some elementary considerations are done on the asymptotic behavior of the solution of the equation.

The nonlinear Dirac equation in Bose–Einstein condensates: I. Relativistic solitons in armchair nanoribbon optical lattices

We present a thorough analysis of soliton solutions to the quasi-one-dimensional (quasi-1D) nonlinear Dirac equation (NLDE) for a Bose–Einstein condensate in a honeycomb lattice with armchair geometry. Our NLDE corresponds to a quasi-1D reduction of the honeycomb lattice along the zigzag direction, in direct analogy to graphene nanoribbons. Excitations in the remaining large direction of the lattice correspond to the linear subbands in the armchair nanoribbon spectrum. Analytical as well as numerical soliton Dirac spinor solutions are obtained. We analyze the solution space of the quasi-1D NLDE by finding fixed points, delineating the various regions in solution space, and through an invariance relation which we obtain as a first integral of the NLDE. We obtain spatially oscillating multi-soliton solutions as well as asymptotically flat single soliton solutions using five different methods: by direct integration; an invariance relation; parametric transformation; a series expansion; and by numerical shooting. By tuning the ratio of the chemical potential to the nonlinearity for a fixed value of the energy–momentum tensor, we can obtain both bright and dark solitons over a nonzero density background.

Bohr-Sommerfeld quantization condition for Dirac states derived from an Ermakov-type invariant

It is shown that solutions of the second-order decoupled radial Dirac equations satisfy Ermakov-type invariants. These invariants lead to amplitude-phase-type representations of the radial spinor solutions, with exact relations between their amplitudes and phases. Implications leading to a Bohr-Sommerfeld quantization condition for bound states, and a few particular atomic/ionic and nuclear/hadronic bound-state situations are discussed.

From Generalized Dirac Equations to a Candidate for Dark Energy

We consider extensions of the Dirac equation with mass terms m1+iγ5m2 and im1+γ5m2. The corresponding Hamiltonians are Hermitian and pseudo-Hermitian (γ5 Hermitian), respectively. The fundamental spinor solutions for all generalized Dirac equations are found in the helicity basis and brought into concise analytic form. We postulate that the time-ordered product of field operators should yield the Feynman propagator (iϵ prescription), and we also postulate that the tardyonic as well as tachyonic Dirac equations should have a smooth massless limit. These postulates lead to sum rules that connect the form of the fundamental field anticommutators with the tensor sums of the fundamental plane-wave eigenspinors and the projectors over positive-energy and negative-energy states. In the massless case, the sum rules are fulfilled by two egregiously simple, distinguished functional forms. The first sum rule remains valid in the case of a tardyonic theory and leads to the canonical massive Dirac field. The second sum rule is valid for a tachyonic mass term and leads to a natural suppression of the right-handed helicity states for tachyonic particles and left-handed helicity states for tachyonic spin-1/2 antiparticles. When applied to neutrinos, the theory contains a free tachyonic mass parameter. Tachyons are known to be repulsed by gravity. We discuss a possible role of a tachyonic neutrino as a contribution to the accelerated expansion of the Universe “dark energy.”

Pseudo-Hermitian quantum dynamics of tachyonic spin-1/2 particles

We investigate the spinor solutions, the spectrum and the symmetry properties of a matrix-valued wave equation whose plane-wave solutions satisfy the superluminal (tachyonic) dispersion relation , where E is the energy, is the spatial momentum and m is the mass of the particle. The equation reads (iγμ ∂μ − γ5 m)ψ = 0, where γ5 is the fifth current. The tachyonic equation is shown to be invariant and invariant. The tachyonic Hamiltonian breaks parity and is non-Hermitian but fulfils the pseudo-Hermitian property , where P is the parity matrix and is the full parity transformation. The energy eigenvalues and eigenvectors describe a continuous spectrum of plane-wave solutions (which correspond to real eigenvalues for ) and evanescent waves, which constitute resonances and anti-resonances with complex-conjugate pairs of resonance eigenvalues (for ). In view of additional algebraic properties of the Hamiltonian which supplement the pseudo-Hermiticity, the existence of a resonance energy eigenvalue E implies that E*, −E and −E* also constitute resonance energies of H5.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.

On neutrino Oscillations and Predicting the 125 GEV Two Photon Emission State from p-p Collisions Based on the 5D Homogeneous Space-Time Projection Model

Previously the 5D homogeneous space-time metric was introduced with explicitly given projection operators in matrix form which map the 5D space-time manifold into a Lorentzian space-time. Based on this projection model, vector field and spinor solutions are found to be expressible in terms of SU(2)xL and SU(3)xL, where L is the 4D Lorentz space-time group. The spinor solutions give the SU(2) leptonic states arising from space-time projection, whereas the SU(3) representation arises from conformal projection and gives the quarks, and due to gauge requirement leads to mesons and baryons. This process of mapping the 5D space-time manifold into the 4D space-time is at the basis of an analysis of the recent CERN experimental results, the presence of neutrino oscillations and the observed 125 GeV resonance in the p-p collisions, respectively. In fact, it is found that the spinor solution contains an oscillating phase, and the 125 GeV resonance is shown to be predictable, thereby 1) eliminating the need to introduce a Higgs vacuum, and 2) can be shown possibly to be an indicator for a missing heavy baryon octet.

Deriving Electromagnetic Fields From the Spinor Solution of the Massless Dirac Equation

In this paper, we present the explicit derivation of the electromagnetic (EM) field solution of Maxwell equations starting from the Dirac equation, used in describing the so-called spinor wave function of quantum particles. In particular, we show that if the four-component vector (spinor) solution of the Dirac equation for zero mass is identified with the four-potential of the EM field, then, under the Lorentz gauge, fields derived from that potential satisfy Maxwell equations. <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Vice versa</i> , the four-potential could be used to express a spinor solution, provided that the latter satisfies the Lorenz gauge. Some examples in the frequency domain clarify this connection. A crucial choice is needed: the EM potential has to be assumed as a linear combination of positive- and negative solutions of the spinor. This work may help to clarify the controversial relation between Maxwell and Dirac equations, while presenting an original way to derive the EM fields, leading, perhaps, to novel concepts in EM simulations.

Model Dirac-Maxwell equations with pseudounitary symmetry

We introduce a new system of equations called a model system of Dirac-Maxwell equations, reproducing the main properties of the standard system. At the same time, the model system of equations differs from the standard system in several ways; in particular, it is a tensor system and has a new symmetry with respect to the pseudounitary group. We also propose a version of the model system of Dirac-Maxwell equations with local (gauge) pseudounitary symmetry. We show that any spinor solution of the standard system of Dirac-Maxwell equations can be obtained from the corresponding tensor solution of the model system.

Spinor Derivation of Quasilocal Mean Curvature Mass in General Relativity

A spinor derivation is presented for quasilocal mean-curvature mass of spacelike 2-surfaces in General Relativity. The derivation is based on the Sen-Witten spinor identity and involves the introduction of novel nonlinear boundary conditions related to the Dirac current of the spinor at the 2-surface and the tangential flux of a boundary Dirac operator, as well the use of a spin basis adapted to the mean curvature frame of the 2-surface normal space. This setting may provide an alternative approach to a positivity proof for mean-curvature mass based on showing that Witten’s equation admits a spinor solution satisfying the proposed nonlinear boundary conditions.

Resonant states in an attractive one-dimensional cusp potential

We solve the two-component Dirac equation in the presence of a spatially one-dimensional symmetric attractive cusp potential. The components of the spinor solution are expressed in terms of Whittaker functions. We compute the bound states solutions and show that, as the potential amplitude increases, the lowest energy state sinks into the Dirac sea becoming a resonance. We characterize and compute the lifetime of the resonant state with the help of the phase shift and the Breit–Wigner relation. We discuss the limit when the cusp potential reduces to a delta point interaction.