A modified gravity model coupled to a Dirac field in 2D space–times with quadratic nonmetricity and curvature

After summarizing the basic concepts for the exterior algebra, we first discuss the gauge structure of the bundle over base manifold for deciding the form of the gravitational sector of the total Lagrangian in any dimensions. Then we couple minimally a Dirac spinor field to our gravitational Lagrangian 2-form which is quadratic in the nonmetricity and both linear and quadratic in the curvature in two dimensions. Subsequently, we obtain field equations by varying the total Lagrangian with respect to the independent variables. Finally, we find some classes of solutions of the vacuum theory and then a solution of the Dirac equation in a specific background and analyze them.

First steps in classical field theory

Classical field theory, as it is applied to the most simple scalar, vector and spinor fields in flat spacetime, is described. The Klein-Gordan, Weyl and Dirac equations are obtained, and some features of their solutions are discussed. The Yukawa potential, the plane wave solutions, and the conserved currents are obtained. Spinors are introduced, both through physical pictures (flagpole and flag) and algebraic defintions (complex vectors). The relationship between spinors and four-vectors is given, and related to the Lie groups SU(2) and SO(3). The Dirac spinor is introduced.

Dirac–Kratzer problem with spin and pseudo-spin symmetries in curved space–time

In this work, the Dirac–Kratzer problem with spin and pseudo-spin symmetries in a deformed nucleus is analyzed. Thus, the Dirac equation in curved space–time was considered, with a line element given by [Formula: see text], where [Formula: see text] is a scalar potential, coupled to vector [Formula: see text] and tensor [Formula: see text] potentials. Defining the vector and scalar potentials of the Kratzer type and the tensor potential given by a term centrifugal-type term plus a term cubic singular at the origin, we obtain the Dirac spinor in a quasi-exact way and the eigenenergies numerically for the spin and pseudo-spin symmetries, so that these symmetries are removed due to the coupling of an Coulomb-type effective tensor potential coming from the curvature of space, however, when such potential is null the symmetries return. The probability densities were analyzed using graphs to compare the behavior of the system with and without spin and pseudo-spin symmetries.

Modified general relativity and quantum theory in curved spacetime

With appropriate modifications, the multi-spin Klein–Gordon (KG) equation of quantum field theory can be adapted to curved space–time for spins 0, 1, 1/2. The associated particles in the microworld then move as a wave at all space–time coordinates. From the existence in a Lorentzian space–time of a line element field [Formula: see text], the spin-1 KG equation [Formula: see text] is derived from an action functional involving [Formula: see text] and its covariant derivative. The spin-0 KG equation and the KG equation of the outer product of a spin-1/2 Dirac spinor and its Hermitian conjugate are then constructed. Thus, [Formula: see text] acts as a fundamental quantum vector field. The symmetric part of the spin-1 KG equation, [Formula: see text], is the Lie derivative of the metric. That links the multi-spin KG equation to Modified General Relativity (MGR) through its energy–momentum tensor of the gravitational field. From the invariance of the action functionals under the diffeomorphism group Diff(M), which is not restricted to the Lorentz group, [Formula: see text] can instantaneously transmit information along [Formula: see text]. That establishes the concept of entanglement within a Lorentzian formalism. The respective local/nonlocal characteristics of MGR and quantum theory no longer present an insurmountable problem to unify the theories.

Fermion localization in braneworld teleparallel f(T, B) gravity

We study a spin 1/2 fermion in a thick braneworld in the context of teleparallel f(T, B) gravity. Here, f(T, B) is such that $$f_1(T,B)=T+k_1B^{n_1}$$ f 1 ( T , B ) = T + k 1 B n 1 and $$f_2(T,B)=B+k_2T^{n_2}$$ f 2 ( T , B ) = B + k 2 T n 2 , where $$n_{1,2}$$ n 1 , 2 and $$k_{1,2}$$ k 1 , 2 are parameters that control the influence of torsion and the boundary term. We assume Yukawa coupling, where one scalar field is coupled to a Dirac spinor field. We show how the $$n_{1,2}$$ n 1 , 2 and $$k_{1,2}$$ k 1 , 2 parameters control the width of the massless Kaluza–Klein mode, the breadth of non-normalized massive fermionic modes and the properties of the analogue quantum-potential near the origin.

Gravitational interactions in the Dirac spinor fields and possible structure of local space-time of fermions

In the framework of general relativity, possible effects of the gravitational interactions in the Dirac spinor field are considered. It is shown that these interactions manifest locally as contact spin-spin interaction of the gravitational and spinor fields. This interaction leads to the classical rotation of particles with spin ħ /2. As a result, it leads to appearance of local internal space-time with specific geometric properties for each particle. New effect of an increase of the mass of spinor particles due to this interaction is found. Also, an explanation of the existence of a magnetic moment in Dirac spinor particles as a result of a local electro-spin-spin interaction has been proposed.

Possible structure of a local space-time of charged fermions

In the framework of general relativity, the properties of the local space-time of fermions with an electric charge are considered. It is known that the Dirac spinor field acts locally, as a contact spin-spin interaction of the gravitational and spinor fields and that may lead to some specific properties of spacetime structure. It is shown that this interaction causes real rotation of particles with spin ħ /2. It was found that this rotation is a source of intrinsic magnetic moment for electrically charged particles. As it was found, in this case the local spacetime of charged particles can have the spacetime structure of a passable «wormhole».

Model of a spin-1/2 electric charge in F(B2) modified Weyl gravity

Within [Formula: see text] modified Weyl gravity, we consider a model of a spin-[Formula: see text] electric charge consisting of interior and exterior regions. The interior region is determined by quantum gravitational effects whose approximate description is carried out using Weyl gravity nonminimally coupled to a massless Dirac spinor field. The interior region is embedded in exterior Minkowski spacetime, and the joining surface is a two-dimensional torus. It is shown that mass, electric charge, and spin of the object suggested may be the same as those for a real electron.