On Fibonacci spinors

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.

The geometry of null-like disformal transformations

Motivated by the hindrance of defining metric tensors compatible with the underlying spinor structure, other than the ones obtained via a conformal transformation, we study how some geometric objects are affected by the action of a disformal transformation in the closest scenario possible: the disformal transformation in the direction of a null-like vector field. Subsequently, we analyze symmetry properties such as mutual geodesics and mutual Killing vectors, generalized Weyl transformations that leave the disformal relation invariant, and introduce the concept of disformal Killing vector fields. In most cases, we use the Schwarzschild metric, in the Kerr–Schild formulation, to verify our calculations and results. We also revisit the disformal operator using a Newman–Penrose basis to show that, in the null-like case, this operator is not diagonalizable.

Kinetic Theory of Systems with SU(2) Structure

Systems with spin-orbit coupling and magnetic fields exhibit a SU(2) structure. Large classes of materials and couplings can be written into an effective spin-orbit coupled Hamiltonian with Pauli structure. Appropriate kinetic equations are derived keeping the quantum spinor structure. It results in coupled kinetic equations of scalar and vector distributions. The spin-orbit coupling, the magnetic field and the vector part of the selfenergy can be written in terms of an effective Zeeman field which couples both distributions. The currents and linear response are derived and the anomalous parts due to the coupling of the occurring band splitting are discussed. The response in magnetic fields reveals subtle retardation effects from which the classical and quantum Hall effect result as well as anomalous Hall effects. As application the dynamical conductivity of grapheme is successfully calculated and compared to the experiments.

Spin–curvature interaction from curved Dirac equation: Application to single-wall carbon nanotubes

The spin–curvature interaction (SCI) and its effects are investigated based on curved Dirac equation. Through the low-energy approximation of curved Dirac equation, the Hamiltonian of SCI is obtained and depends on the geometry and spinor structure of manifold. We find that the curvature can be considered as field strength and couples with spin through Zeeman-like term. Then, we use dimension reduction to derive the local Hamiltonian of SCI for cylinder surface, which implies that the effective Hamiltonian of single-wall carbon nanotubes results from the geometry and spinor structure of lattice and includes two types of interactions: one does not break any symmetries of the lattice and only shifts the Dirac points for all nanotubes, while the other one does and opens the gaps except for armchair nanotubes. At last, analytical expressions of the band gaps and the shifts of their positions induced by curvature are given for metallic nanotubes. These results agree well with experiments and can be verified experimentally.

FERMION DYNAMICS BY INTERNAL AND SPACETIME SYMMETRIES

This paper is devoted to introduce a gauge theory of the Lorentz group based on the ambiguity emerging in dealing with isometric diffeomorphism-induced Lorentz transformations. The behaviors under local transformations of fermion fields and spin connections (assumed to be ordinary world vectors) are analyzed in flat spacetime and the role of the torsion field, within the generalization to curved spacetime, is briefly discussed. The fermion dynamics is then analyzed including the new gauge fields and assuming time-gauge. Stationary solutions of the problem are also analyzed in the non-relativistic limit, to study the spinor structure of an hydrogen-like atom.

MAXIMALLY SYMMETRIC MINIMAL UNIFICATION MODEL SO(32) WITH THREE FAMILIES IN TEN-DIMENSIONAL SPACETIME

Based on a maximally symmetric minimal unification hypothesis and a quantum charge-dimension correspondence principle, it is demonstrated that each family of quarks and leptons belongs to the Majorana–Weyl spinor representation of 14 dimensions that relate to quantum spin-isospin-color charges. Families of quarks and leptons attribute to a spinor structure of extra six dimensions that relate to quantum family charges. Of particular, it is shown that ten dimensions relating to quantum spin-family charges form a motional ten-dimensional quantum spacetime with a generalized Lorentz symmetry SO (1, 9), and ten dimensions relating to quantum isospin-color charges become a motionless ten-dimensional quantum intrinsic space. Its corresponding 32-component fermions in the spinor representation possess a maximal gauge symmetry SO (32). As a consequence, a maximally symmetric minimal unification model SO (32) containing three families in ten-dimensional quantum spacetime is naturally obtained by choosing a suitable Majorana–Weyl spinor structure into which quarks and leptons are directly embedded. Both resulting symmetry and dimensions coincide with those of type I string and heterotic string SO (32) in string theory.

SEMICLASSICAL THEORY OF EXCITONIC POLARITONS IN A PLANAR SEMICONDUCTOR MICROCAVITY

We present a comprehensive theoretical description of quantum well exciton–polaritons imbedded in a planar semiconductor microcavity. The exact nonlacal dielectric response of the quantum well exciton is treated in detail. The 4-spinor structure of the hole subband in the quantum well is considered, including the pronounced band mixing effect. The scheme is self-contained and can be used to treat different semiclassical aspects of the microcavity properties. As an example, we analyze the "selection" rules for the exciton–cavity mode coupling for different excitons.