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Ever since the first observations that we are living in an accelerating universe, it has been asked what dark energy is. There are various explanations, all of which have various drawbacks or inconsistencies. Here we show that using a dark spinor field it is possible to have an equation of state that crosses the phantom divide, becoming a dark phantom spinor which evolves into dark energy. This type of equation of state has been mildly favored by experimental data, however, in the past there were hardly any theories that satisfied this crossing without creating ghosts or causing a singularity which results in the universe essentially ripping. The dark spinor model converges to dark energy in a reasonable time frame avoiding the big rip and without attaining negative kinetic energy as it crosses the phantom divide.

Torsion propagation and torsion–spin coupling are studied in the perspective of the Velo–Zwanziger method of analysis; specifically, we write the most extensive dynamics of the torsion tensor and the most exhaustive coupling that is permitted between torsion and spinors, and check the compatibility with constraints and hyperbolicity and causality of field equations: we find that some components of torsion and many terms of the torsion–spin interaction will be restricted away and as a consequence, we will present the most general theory that is compatible with all restrictions.

One of the most important mathematical tools necessary for Quantum Field Theory calculations is the field propagator. Applications are always done in terms of plane waves and although this has furnished many magnificent results, one may still be allowed to wonder what is the form of the most general propagator that can be written. In this paper, by exploiting what is called polar form, we find the most general propagator in the case of spinors, whether regular or singular, and we give a general discussion in the case of vectors.

A gauge theory of the Lorentz group, based on the different behavior of spinors and vectors under local transformations, is formulated in a flat space-time and the role of the torsion field within the generalization to curved space-time is briefly discussed.

The spinor interaction with the new gauge field is then analyzed assuming the time gauge and stationary solutions, in the non-relativistic limit, are treated to generalize the Pauli equation.

We construct the Rarita–Schwinger basis vectors, $U^{\mu }$, spanning the direct product space, $U^{\mu }:=A^{\mu }\otimes u_M$, of a massless four-vector, $A^{\mu }$, with massless Majorana spinors, $u_M$, together with the associated field-strength tensor, ${{𝒯}}^{\mu \nu }:=p^{\mu }{U}^{\nu }-p^{\nu }{U}^{\mu }$. The ${{𝒯}}^{\mu \nu }$ space is reducible and contains one massless subspace of a pure spin-$3/2\in (3/2,0)\oplus (0,3/2)$. We show how to single out the latter in a unique way by acting on ${{𝒯}}^{\mu \nu }$ with an earlier derived momentum independent projector, ${{𝒫}}^{(3/2,0)}$, properly constructed from one of the Casimir operators of the algebra $so(1,3)$ of the homogeneous Lorentz group. In this way, it becomes possible to describe the irreducible massless $(3/2,0)\oplus (0,3/2)$ carrier space by means of the antisymmetric tensor of second rank with Majorana spinor components, defined as ${[w^{(3/2,0)}]}^{\mu \nu }:={[{{𝒫}}^{(3/2,0)}]}^{\mu \nu }{}_{\gamma \delta }{{𝒯}}^{\gamma \delta }$. The conclusion is that the $(3/2,0)\oplus (0,3/2)$ bi-vector spinor field can play the same role with respect to a $U^{\mu }$ gauge field as the bi-vector, $(1,0)\oplus (0,1)$, associated with the electromagnetic field-strength tensor, $F_{\mu \nu }$, plays for the Maxwell gauge field, $A_{\mu }$. Correspondingly, we find the free electromagnetic field equation, $p^{\mu }{F}_{\mu \nu }=0$, is paralleled by the free massless Rarita–Schwinger field equation, $p^{\mu }{[w^{(3/2,0)}]}_{\mu \nu }=0$, supplemented by the additional condition, ${\gamma }^{\mu }{\gamma }^{\nu }{[w^{(3/2,0)}]}_{\mu \nu }=0$, a constraint that invokes the Majorana sector.

In Quantum Field Theory (QFT), scattering amplitudes are computed from propagators which, for internal lines, are built upon spin/polarization-sum relationships. In turn, these are normally constructed upon plane-wave solutions of the free field equations. A question that may now arise is whether such spin/polarization-sums can be generalized. In the past, there has been a first attempt at generalizing spin sums for fermionic fields in terms of the Michel–Wightman identities. In this paper, we aim to find the most general spin sums for fermionic fields within the range of QFT.

This essay presents a new asymmetry that arises from the interplay of charge conjugation and Lense–Thirring effect. When applied to Majorana neutrinos, the effects predicts oscillations in gravitational environments with rotating sources. Parameters associated with astrophysical environments indicate that the presented effect is presently unobservable for solar neutrinos. But, it will play an important role in supernova explosions, and carries relevance for the observed matter–antimatter asymmetry in the universe.

By using the Nikiforov–Uvarov method, we give the approximate analytical solutions of the Dirac equation with the shifted Deng–Fan potential including the Yukawa-like tensor interaction under the spin and pseudospin symmetry conditions. After using an improved approximation scheme, we solved the resulting schrödinger-like equation analytically. Numerical results of the energy eigenvalues are also obtained, as expected, the tensor interaction removes degeneracies between spin and pseudospin doublets.

We give a necessary and sufficient condition for an n-dimensional Riemannian manifold to be isometrically immersed into one of the Lorentzian products 𝕊ⁿ × ℝ₁ or ℍⁿ × ℝ₁. This condition is expressed in terms of its first and second fundamental forms, the tangent and normal projections of the vertical vector field. As applications, we give an equivalent condition in a spinorial way and we deduce the existence of a one-parameter family of isometric maximal deformation of a given maximal surface obtained by rotating the shape operator.

Compact description is given of algebras of poly-numbers: quaternions, bi-quaternions, double (split-complex) and dual numbers. All units of these (and exceptional) algebras are shown to be represented by direct products of 2D vectors of a local basis defined on a fundamental surface. In this math medium a series of equalities identical or similar to known formulas of physical laws is discovered. In particular, a condition of the algebras' stability with respect to transformations of the 2D-basis turns out equivalent to the spinor (Schrödinger–Pauli and Hamilton–Jacobi) equations of mechanics. It is also demonstrated that isomorphism of SO(3, 1) and SO(3, ℂ) groups leads to formulation of a quaternion relativity theory predicting all effects of special relativity but simplifying solutions of relativistic problems in non-inertial frames. Finely it is shown that the Cauchy–Riemann type equations written for functions of quaternion variable repeat vacuum Maxwell equations of electrodynamics, while a quaternion space with non-metricity comprises main relations of Yang–Mills field theory.

The presence of spinor fields is considered in the framework of some extensions of teleparallel gravity, where the Weitzenböck connection is assumed. Some well-known models as the Chaplygin gas and its generalizations are reconstructed in terms of a spinor field in the framework of teleparallel gravity. In addition, the ΛCDM model is also realized with the presence of a spinor field where a simple self-interacting term is considered and the corresponding action is reconstructed. Other cosmological solutions and the reconstruction of the gravitational action in terms of the scalar torsion are studied.

We give a generalized Weierstrass formula for a Lorentz surface conformally immersed in the four-dimensional space ${ℝ}^{2,2}$ using spinors and Lorentz numbers. We also study the immersions of a Lorentzian surface in the Anti-de Sitter space (a pseudo-sphere in ${ℝ}^{2,2}$): we give a new spinor representation formula and deduce the conformal description of a flat Lorentzian surface in that space.

We consider generally relativistic gauge transformations for the spinorial fields finding two mutually exclusive but together exhaustive classes in which fermions are placed adding supplementary information to the results obtained by Lounesto, and identifying quantities analogous to the momentum vector and the Pauli–Lubanski axial vector. We discuss how our results are similar to those obtained by Wigner by taking into account the system of Dirac field equations. We will investigate the consequences for the dynamics and in particular we shall address the problem of getting the nonrelativistic approximation in a consistent way. We are going to comment on extensions.

We describe both the Hodge–de Rham and the spin manifold Dirac operator on the spheres ${\mathrm{\text{S}}}^3$ and ${\mathrm{\text{S}}}^2$, following the formalism introduced by Kähler, and exhibit a complete spectral resolution for them in terms of suitably globally defined eigenspinors.

In this paper, we will take into account the most complete background with torsion and curvature, providing the most exhaustive coupling for the Dirac field, we will discuss the integrability of the interaction of the matter field and the reduction of the matter field equations.

In this paper, we consider torsion gravity in the case of Dirac field, and by going into the rest frame, we study what happens when a uniform precession as well as a phase is taken into account for the spinor field; we discuss how partially conserved axial-vector currents and torsion-spin attractive potentials justify negative Takabayashi angle and energy smaller than mass: because in this instance the module goes to zero exponentially fast, we obtain stable and localized matter distributions suitable to be regarded as a description of particles.

Spinors are used in physics quite extensively. Basically, the forms of use include Dirac four-spinors, Pauli three-spinors and quaternions. Quaternions in mathematics are essentially equivalent to Pauli spin matrices which can be generated by regarding a quaternion matrix as compound. The goal of this study is also the spinor structure lying in the basis of the quaternion algebra. In this paper, first, we have introduced spinors mathematically. Then, we have defined Fibonacci spinors using the Fibonacci quaternions. Later, we have established the structure of algebra for these spinors. Finally, we have proved some important formulas such as Binet and Cassini formulas which are given for some series of numbers in mathematics for Fibonacci spinors.

‘Internal spacetime’ is a modification of general relativity that was recently introduced as an approximate spacetime geometric model of quantum nonlocality. In an internal spacetime, time is stationary along the worldlines of fundamental (dust) particles. Consequently, the dimensions of tangent spaces at different points of spacetime vary, and spin wavefunction collapse is modeled by the projection from one tangent space to another. In this paper, we develop spinors on an internal spacetime, and construct a new Dirac-like Lagrangian ${ℒ}=\bar{\psi }(i\partial /-\widehat{\omega })\psi$ whose equations of motion describe their couplings and interactions. Furthermore, we show that hidden within ${ℒ}$ is the entire standard model: ${ℒ}$ contains precisely three generations of quarks and leptons, the electroweak gauge bosons, the Higgs boson, and one new massive spin-$2$ boson; gluons are considered in a companion paper. Specifically, we are able to derive the correct spin, electric charge, and color charge of each standard model particle, as well as predict the existence of a new boson.

A composite model of the standard model particles was recently derived using the Dirac Lagrangian on a spacetime where time does not advance along the worldlines of fundamental dust particles, called an ‘internal spacetime’. The aim of internal spacetime geometry is to model certain quantum phenomena using (classical) degenerate spacetime metrics. For example, on an internal spacetime, tangent spaces have variable dimensions, and spin wavefunction collapse is modeled by the projection from one tangent space to another. In this article, we show that the combinatorial structure of the internal Dirac Lagrangian yields precisely the standard model trivalent vertices, together with two new additional (longitudinal) $Z$ vertices that generate the four-valent boson vertices. In particular, we are able to derive electroweak parity violation for both leptons and quarks. We also obtain new restrictions on the possible spin states that can occur in certain interactions. Finally, we determine the trivalent vertices of the new massive spin-$2$ boson predicted by the model.

This article will provide the reader a short introduction to dark spinors, which are ELKO spinors, eingenspinors of the charge conjugation operator, applied to dark matter/energy.