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: (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. \end{abstract}
A spinor current-source is found in the Weyl non-Abelian gauge theory which does not contain the abstract gauge space. It is shown that the searched spinor representation can be constructed in the space of external differential forms and it is a 16-component quantity for which a gauge-invariant Lagrangian is determined. The connexion between the Weyl non-Abelian gauge potential and the Cartan torsion field and the problem of a possible manifestation of the considered interactions are considered.
The generalization of Sommers--Sen spinor connection for spinor fields, associated with a distribution $V_4^3$ and on this basis the equations for Weyl and Dirac null vector fields on complexificated $V_4^3$ are obtained. We interpret the obtained results by examining the interaction of spinor fields with the inertial forces.
We consider the wave equation for spinors in ${\cal D}$-dimensional Weyl geometry. By appropriately coupling the Weyl vector $ϕ_μ$ as well as the spin connection $ω_{μa b } $ to the spinor field, conformal invariance can be maintained. The one loop effective action generated by the coupling of the spinor field to an external gravitational field is computed in two dimensions. It is found to be identical to the effective action for the case of a scalar field propagating in two dimensions.
We investigate the spinor classification of the Weyl tensor in five dimensions due to De Smet. We show that a previously overlooked reality condition reduces the number of possible types in the classification. We classify all vacuum solutions belonging to the most special algebraic type. The connection between this spinor and the tensor classification due to Coley, Milson, Pravda and Pravdová is investigated and the relation between most of the types in each of the classifications is given. We show that the black ring is algebraically general in the spinor classification.
In spinor formalism, since any massless free-field spinor with spin higher than $1/2$ can be constructed with spin-1/2 spinors (Dirac-Weyl spinors) and scalars, we introduce a map between Weyl fields and Dirac-Weyl fields. We determine the corresponding Dirac-Weyl spinors in a given empty spacetime. Regarding them as basic units, other higher spin massless free-field spinors are then identified. Along this way, we find some hidden fundamental features related to these fields. In particular, for non-twisting vacuum Petrov type N solutions, we show that all higher spin massless free-field spinor…
A spinorial approach to 6-dimensional differential geometry is constructed and used to analyze tensor fields of low rank, with special attention to the Weyl tensor. We perform a study similar to the 4-dimensional case, making full use of the SO(6) symmetry to uncover results not easily seen in the tensorial approach. Using spinors, we propose a classification of the Weyl tensor by reinterpreting it as a map from 3-vectors to 3-vectors. This classification is shown to be intimately related to the integrability of maximally isotropic subspaces, establishing a natural framework to generalize the …
We consider the single-handed spinor field in interaction with its own gravitational field described by the set of field equations given by Weyl field equations written in terms of derivatives that are covariant with respect to the gravitational connection plus Einstein field equations soured with the energy tensor of the spinor: for the Weyl spinor and the ensuing spacetime of Weyl-Lewis-Papapetrou structure, we will find all exact solutions. The obtained solution for the metric tensor is that of a PP-wave spacetime while the spinor field is a flag-dipole.
We consider weighted parallel spinors in Lorentzian Weyl geometry in arbitrary dimensions, choosing the weight such that the integrability condition for the existence of such a spinor, implies the geometry to be Einstein-Weyl. We then use techniques developed for the classification of supersymmetric solutions to supergravity theories to characterise those Lorentzian EW geometries that allow for a weighted parallel spinor, calling the resulting geometries supersymmetric. The overall result is that they are either conformally related to ordinary geometries admitting parallel spinors (w.r.t. the …
We consider the problem of finding all space-time metrics for which all plane-wave Penrose limits are diagonalisable plane waves. This requirement leads to a conformally invariant differential condition on the Weyl spinor which we analyse for different algebraic types in the Petrov-Pirani-Penrose classification. The only vacuum examples, apart from actual plane waves which are their own Penrose limit, are some of the nonrotating type D metrics, but some nonvacuum solutions are also identified. The condition requires the Weyl spinor, whenever it is nonzero, to be proportional to a valence-4 Kil…
The well known Geodesic Equation of General Relativity is newly formulated in Weyl two-spinor language in a convenient way susceptible of being combined with a set of two-spinor equations, equivalent to the Lorentz Force of Electrodynamics, obtained in previous studies. General spinor equations of motion for a test charged spin 1/2 particle in arbitrary electromagnetic and gravitational fields fields are obtained describing not only spacetime trajectories but also spin precession. For the case of the Schwarzschild metric and radial trajectories, spinor equations unifying gravitational and elec…
A massless Weyl-invariant dynamics of a scalar, a Dirac spinor, and electromagnetic fields is formulated in a Weyl space, $W_4$, allowing for conformal rescalings of the metric and of all fields with nontrivial Weyl weight together with the associated transformations of the Weyl vector fields $\ka_μ$ representing the D(1) gauge fields with D(1) denoting the dilatation group. To study the appearance of nonzero masses in the theory the Weyl-symmetry is broken explicitly and the corresponding reduction of the Weyl space $W_4$ to a pseudo-Riemannian space $V_4$ is investigated assuming the breakin…
A simple modification to Einstein's theory of gravity in terms of a non-Riemannian connection is examined. A new tensor-variational approach yields field equations that possess a covariance similar to the gauge covariance of electromagnetism. These equations are shown to possess solutions analogous to those found in the Einstein-Maxwell system. In particular one finds gravi-electric and gravi-magnetic charges contributing to a spherically symmetric static Reissner-Nordström metric. Such Weyl ``charges'' provide a source for the non-Riemannian torsion and metric gradient fields instead of the e…
The Weyl double copy relates vacuum solutions in general relativity to Abelian gauge fields in Minkowski spacetime. In a previous work, we showed how the Weyl double copy can be extended to provide a treatment of external gravitational sources consistent with the classical Kerr-Schild double copy. Using this generalization, here we provide a complete double copy analysis of electrovacuum Petrov type D spacetimes. This includes the first analysis of the charged C-metric, whose single copy interpretation invokes the two-potential formalism of electrodynamics. We also present the first double cop…
In Ref. \cite{Horava2005} Hořava suggested, that the multi - fermion many-body system with topologically stable Fermi surfaces may effectively be described (in a vicinity of the Fermi surface) by the theory with coarse-grained fermions. The number of the components of these coarse-grained fermions is reduced compared to the original system. Here we consider the $3+1$ D system and concentrate on the particular case when the Fermi surface has co-dimension $p=3$, i.e. it represents the Fermi point in momentum space. First we demonstrate explicitly that in agreement with Hořava conjecture, in the …
The renormalized quantum stress energy tensor $\left<T_μ^ν\right>_{ren}$ for a massive spinor field around global monopoles is constructed within the framework of Schwinger-DeWitt approximation, valid whenever the Compton length of the quantum field is much less than the characteristic radius of the curvature of the background geometry. The results obtained shows that the quantum massive spinor field in the global monopole spacetime violates all the pointwise energy conditions.
We explore the Euclidean supersymmetric solutions admitting the self-dual gauge field in the framework of ${\cal N}=2$ minimal gauged supergravity in four dimensions. According to the classification scheme utilizing the spinorial geometry or the bilinears of Killing spinors, the general solution preserves one quarter of supersymmetry and is described by the Przanowski-Tod class with the self-dual Weyl tensor. We demonstrate that there exists an additional Killing spinor, provided the Przanowski-Tod metric admits a Killing vector that commutes with the principal one. The proof proceeds by recas…
The goal of this paper is to define fermionic fields on causal set. This is done by the use of holonomies to define vierbines, and then defining spinor fields by taking advantage of the leftover degrees of freedom of holonomies plus additional scalar fields. Grassmann nature is being enforced by allowing measure to take both positive and negative values, and also by introducing a vector space to have both commutting dot product and anticommutting wedge product.
This is a doctoral thesis on the application of techniques originally developed in the programme of characterisation of supersymmetric solutions to Supergravity theories, to finding alternative backgrounds. We start by discussing the concept of a Killing spinor, and how these are paramount to the process of classifying the aforementioned supersymmetric solutions. Moreover, these geometric objects also have applications when considered in different scenarios (the 'beyond' in the title). In particular, techniques based on a different parallelising rule for a spinorial field can be used for obtai…
The Lounesto classification splits spinors in six classes: I, II, III are those for which at least one among scalar and pseudo-scalar bi-linear spinor quantities is non-zero, its spinors are called regular, and among them we find the usual Dirac spinor. IV, V, VI are those for which the scalar and pseudo-scalar bi-linear spinor quantities are identically zero, its spinors are called singular, and they are split in further sub-classes: IV has no further restrictions, its spinors are called flag-dipole; V is the one for which the spin axial-vector vanishes, its spinors are called flagpole, and a…
We calculate the renormalized vacuum average of the energy-momentum tensor of massless left-handed spinor field in the pointlike global monopole spacetime using point-separation approach. The general structure of the vacuum average of the energy-momentum tensor is obtained and expressed in terms of $<T^0_0>^{ren}$ component, explicit form of which is analyzed in great details for arbitrary solid angle deficit.
We propose an unified theory for spinor fields on extended Weyl manifolds taking into account self-interactions to obtain the Relativistic dynamics on a general curved Riemannian background as continuation of the Relativistic Quantum Geometry program, recently introduced. We focuss our attention separately on both, massless and matter fields. We study an example of confined gravitons on a de Sitter (dS) background at Planckian scales.
We show that the Einstein-Hilbert, the Einstein-Palatini, and the Holst actions can be derived from the Quadratic Spinor Lagrangian (QSL), when the three classes of Dirac spinor fields, under Lounesto spinor field classification, are considered. To each one of these classes, there corresponds a unique kind of action for a covariant gravity theory. In other words, it is shown to exist a one-to-one correspondence between the three classes of non-equivalent solutions of the Dirac equation, and Einstein-Hilbert, Einstein-Palatini, and Holst actions. Furthermore, it arises naturally, from Lounesto …
We will consider the most general least-order derivative action for the torsional completion of gravitational backgrounds filled with left-handed and right-handed semi-spinorial fields, accounting for all parity-even as well as parity-odd contributions; we will proceed by performing the customary analysis, decomposing torsion and substituting it in terms of the semi-spinorial density currents, in order to obtain the effective action with the torsionally-induced self-interacting potentials among the chiral fermionic fields: we shall see that the resulting effective non-linear potentials will tu…
We study the relations of the positive frequency mode functions of Dirac field in 4-dimensional Minkowski spacetime covered with Rindler and Kasner coordinates, and describe the explicit form of the Minkowski vacuum state with the quantum states in Kasner and Rindler regions, and analytically continue the solutions. As a result, we obtain the correspondence of the positive frequency mode functions in Kasner region and Rindler region in a unified manner which derives vacuum entanglement.
Within $F\left( B^2\right)$ modified Weyl gravity, we consider a model of a spin-$1/2$ 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.
A special subclass of shear-free null congruences (SFC) is studied, with tangent vector field being a repeated principal null direction of the Weyl tensor. We demonstrate that this field is parallel with respect to an effective affine connection which contains the Weyl nonmetricity and the skew symmetric torsion. On the other hand, a Maxwell-like field can be directly associated with any special SFC, and the electric charge for bounded singularities of this field turns to be ``self-quantized''. Two invariant differential operators are introduced which can be thought of as spinor analogues of t…
We show that the Teukolsky connection, which defines generalized wave operators governing the behavior of massless fields on Einstein spacetimes of Petrov type D, has its origin in a distinguished conformally and GHP covariant connection on the conformal structure of the spacetime. The conformal class has a (metric compatible) integrable almost-complex structure under which the Einstein space becomes a complex (Hermitian) manifold. There is a unique compatible Weyl connection for the conformal structure, and it leads to the construction of a conformally covariant GHP formalism and a generaliza…
Under a rotation by an angle $\vartheta$, both the right- and left- handed Weyl spinors pick up a phase factor ${\exp(\pm\, i \vartheta/2)}$. The upper sign holds for the positive helicity spinors, while the lower sign for the negative helicity spinors. For $\vartheta = 2π$ radians this produces the famous minus sign. However, the four-component spinors are built from a direct sum of the indicated two-component spinors. The effect of the rotation by $2π$ radians on the eigenspinors of the parity - that is, the Dirac spinors -- is the same as on Weyl spinors. It is because for these spinors the…
It is shown that since the geometric spinors are elements of Clifford algebras, they must have the same transformation properties as any other Clifford number. In general, a Clifford number $Φ$ transforms into a new Clifford number $Φ'$ according to $Φ\to Φ' = {\rm{R}}\,Φ\,{\rm{S}}$, i.e., by the multiplication from the left and from the right by two Clifford numbers ${\rm R}$ and ${\rm S}$. We study the case of $Cl(1,3)$, which is the Clifford algebra of the Minkowski spacetime. Depending on choice of ${\rm R}$ and ${\rm S}$, there are various possibilities, including the transformations of v…
The paper deals with the Weyl equation which is the massless Dirac equation. We study the Weyl equation in the stationary setting, i.e. when the spinor field oscillates harmonically in time. We suggest a new geometric interpretation of the stationary Weyl equation, one which does not require the use of spinors, Pauli matrices or covariant differentiation. We think of our 3-dimensional space as an elastic continuum and assume that material points of this continuum can experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of…
Recent observations of Gravitational Waves (GW) generated by black-hole collisions have opened a new window to explore the universe in diverse scales. Detection of primordial gravitational waves is expected to happen in the next years. However, the standard theory to describe these effects was developed for weak gravitational waves, when their dynamics can be linearized. In this work we develop a non-perturbative formalism to describe GW using the Unified Spinor Fields (USF) theory. The tensor index is calculated and we obtain that it must be $0.0283 <n_T < 0.0407$, in order for the $+$ and $\…
We construct a Lagrangian of Weyl spinors and gauge fields, which is invariant under the action of equivalent local transformations on the spinor algebra representations. A model of vacuum with a nontrivial gauge strength-tensor setting a scale and spontaneously breaking the gauge symmetry is suggested, so that the leading approximation at low energies is described by the Einstein--Hilbert Lagrangian of gravity. We consider a mechanism for the cancellation of cosmological constant due to a symmetry between the vacuum strength tensor and the tensor dual to it. The appearance of nonzero masses f…
We explore the cosmological viability of a theory of gravity defined by the Lagrangian $f(\mathcal{R})=\mathcal{R}^{n\left(\mathcal{R}\right)}$ in the Palatini formalism, where $n\left(\mathcal{R}\right)$ is a dimensionless function of the Palatini scalar curvature $\mathcal{R}$ that interpolates between general relativity when $n\left(\mathcal{R}\right)=1$ and a locally scale-invariant and superficially renormalizable theory when $n\left(\mathcal{R}\right)=2$. We refer to this model as asymptotically Weyl-invariant gravity (AWIG). We analyse perhaps the simplest possible implementation of AWI…
We investigate the coupling of matter to geometry in conformal quadratic Weyl gravity, by assuming a coupling term of the form $L_m\tilde{R}^2$, where $L_m$ is the ordinary matter Lagrangian, and $\tilde{R}$ is the Weyl scalar. The coupling explicitly satisfies the conformal invariance of the theory. By expressing $\tilde{R}^2$ with the help of an auxiliary scalar field and of the Weyl scalar, the gravitational action can be linearized, leading in the Riemann space to a conformally invariant $f\left(R,L_m\right)$ type theory, with the matter Lagrangian nonminimally coupled to the Ricci scalar.…
We consider Poincare gravity coupled in a nonminimal way to spinors. The gravitational action is considered that contains both Palatini and Holst terms. Due to torsion the effective four - fermion interactions appear that may lead to the left - right asymmetry and the condensation of fermions. When the mass parameter entering the mentioned terms of the gravitational action is at a Tev scale the given construction may provide the dynamical Electroweak symmetry breaking. This is achieved via an arrangement of all Standard Model fermions in the left - handed Dirac spinors while the right - handed…
Anti-self-dual metrics in the $(++--)$ signature which admit a covariantly constant real spinor are studied. It is shown that finding such metrics reduces to solving a fourth order integrable PDE, and some examples are given. The corresponding twistor space is characterised by existence of a preferred non-zero real section of $κ^{-1/4}$, where $κ$ is the canonical line bundle of the twistor space It is demonstrated that if the parallel spinor is preserved by a Killing vector, then the fourth order PDE reduces to the dispersionless Kadomtsev--Petviashvili equation and its linearisation. Einstei…
We consider a five-dimensional Einstein--Cartan spacetime upon which Dirac spinor fields can be defined. Dirac spinor fields in five and four dimensions share many features, like the fact that both are described by four-component spinor fields, but they are also characterized by strong differences, like the fact that in five dimensions we do not have the possibility to project on left-handed and right-handed chiral parts unlike what happens in the four-dimensional instance: we conduct a polar decomposition of the spinorial fields, so to highlight all similarities and discrepancies. As an appli…
