The three first sections contain an updated, not-so-short account of a partly original approach to spinor geometry and field theories introduced by Jadczyk and myself; it is based on an intrisic treatment of 2-spinor geometry in which the needed background structures do not need to be assumed, but rather arise naturally from a unique geometric datum: a vector bundle with complex 2-dimensional fibres over a real 4-dimensional manifold. The two following sections deal with Dirac algebra and 4-spinor groups in terms of two spinors, showing various aspects of spinor geometry from a different perspective. The last section examines particle momenta in 2-spinor terms and the bundle structure of 4-spinor space over momentum space.
This paper is the second one of a series of three and it is the continuation of math-ph/0412074. We review some properties of the algebraic spinors in Cl(3,0) and Cl(0,3) and how Weyl, Pauli and Dirac spinors are constructed in Cl(3,0) (and Cl(0,3) in the case of Weyl spinors. A plane wave solution for the Dirac equation is obtained, and the Dirac equation is written in terms of Weyl spinors, and alternatively, in terms of Pauli spinors. Finally the covariant and contravariant undotted spinors in Cl(0,3), isomorphic to two copies of the quaternion algebra (H), are constructed. We prove that th…
We study the interplay between basic Dirac operator and transverse Killing and twistor spinors. In order to obtain results for general Riemannian foliations with bundle-like metric we consider transverse Killing spinors that appear as natural extension of the harmonic spinors associated with the basic Dirac operator. In the case of foliations with basic-harmonic mean curvature it turns out that these Killing spinors and twistor spinors coincide with the standard definition. We obtain the corresponding version of classical results on closed Riemannian manifold with spin structure, extending som…
We reexamine the minimal coupling procedure in the Hestenes' geometric algebra formulation of the Dirac equation, where spinors are identified with the even elements of the real Clifford algebra of spacetime. This point of view, as we argue, leads naturally to a non-Abelian generalisation of the electromagnetic gauge potential.
We clarify the structure obtained in Hélein and Vey's proposition for a variational principle for the Einstein-Cartan gravitation formulated on a frame bundle starting from a structure-less differentiable 10-manifold (arXiv:1508.07765v2 [math-ph]). The obtained structure is locally equivalent to a frame bundle. We show that under global hypotheses the formalism allows to produce variational principles which induce a frame bundle structure with a fibration over a base manifold and a variational principle on the base manifold. Our perspective on the model allows us to extend the Einstein-Cartan …
It is known that the bundle of Dirac spinors is produced as a direct sum of two bundles - the bundle of chiral spinors and its Hermitian conjugate bundle. In this paper some aspects of metric connections for chiral and Dirac spinors are resumed and their relation is studied.
We present here a novel method of computing spectral distances in doubled Moyal plane in a noncommutative geometrical framework using Dirac eigen-spinors, while solving the Lipschitz ball condition explicitly through matrices. The standard results of longitudinal, transverse and hypotenuse distances between different pairs of pure states have been computed and Pythagorean equality between them have been re-produced. The issue of non-unital nature of Moyal plane algebra is taken care of through a sequence of projection operators constructed from Dirac eigen-spinors, which plays a crucial role t…
Operator fields in the bundle of Dirac spinors and their conversion to spatial fields are considered. Some commutator equations are studied with the use of the conversion technique.
Almost all presentations of Dirac theory in first or second quantization in Physics (and Mathematics) textbooks make use of covariant Dirac spinor fields. An exception is the presentation of that theory (first quantization) offered originally by Hestenes and now used by many authors. There, a new concept of spinor field (as a sum of non homogeneous even multivectors fields) is used. However, a carefully analysis (detailed below) shows that the original Hestenes definition cannot be correct since it conflicts with the meaning of the Fierz identities. In this paper we start a program dedicated t…
Dual-helicity eigenspinors of the charge conjugation operator (ELKO spinor fields) belong, together with Majorana spinor fields, to a wider class of spinor fields, the so-called flagpole spinor fields, corresponding to the class (5), according to Lounesto spinor field classification based on the relations and values taken by their associated bilinear covariants. There exists only six such disjoint classes: the first three corresponding to Dirac spinor fields, and the other three respectively corresponding to flagpole, flag-dipole and Weyl spinor fields. This paper is devoted to investigate and…
The issue of general covariance of spinors and related objects is reconsidered. Given an oriented manifold $M$, to each spin structure $σ$ and Riemannian metric $g$ there is associated a space $S_{σ, g}$ of spinor fields on $M$ and a Hilbert space $\HH_{σ, g}= L^2(S_{σ, g},\vol{M}{g})$ of $L^2$-spinors of $S_{σ, g}$. The group $\diff{M}$ of orientation-preserving diffeomorphisms of $M$ acts both on $g$ (by pullback) and on $[σ]$ (by a suitably defined pullback $f^*σ$). Any $f\in \diff{M}$ lifts in exactly two ways to a unitary operator $U$ from $\HH_{σ, g} $ to $\HH_{f^*σ,f^*g}$. The canonical…
A novel method is developed to derive the original Dirac equation and demonstrate that it is the only Poincare invariant dynamical equation for 4-component spinor wavefunctions. New Poincare invariant generalized Dirac and Klein-Gordon equations are also derived. In the non-relativistic limit the generalized Dirac equation gives the generalized Levy-Leblond equation and the generalized Pauli-Schrodinger equation. The main difference between the original and generalized Dirac equations is that the former and latter are obtained with zero and nonzero phase functions, respectively. Otherwise, bot…
In "Part I: Vector Analysis of Spinors", the author studied the geometry of two component spinors as points on the Riemann sphere in the geometric algebra of three dimensional Euclidean space. Here, these ideas are generalized to apply to four component Dirac spinors on the complex Riemann sphere in the complexified geometric algebra of spacetime, which includes Lorentz transformations. The development of generalized Pauli matrices eliminate the need for the traditional Dirac gamma matrices. We give the discrete probability distribution of measuring a spin 1/2 particle in an arbitrary spin sta…
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…
We study Kohn-Dirac operators $D_θ$ on strictly pseudoconvex CR manifolds with ${\rm spin}^{\mathbb C}$ structure of weight $\ell\in{\mathbb Z}$. Certain components of $D_θ$ are CR invariants. We also derive CR invariant twistor operators of weight $\ell$. Harmonic spinors correspond to cohomology classes of some twisted Kohn-Rossi complex. Applying a Schrödinger-Lichnerowicz-type formula, we prove vanishing theorems for harmonic spinors and (twisted) Kohn-Rossi groups. We also derive obstructions to positive Webster curvature.
The main objective of this paper is to clarify the ontology of Dirac-Hestenes spinor fields (DHSF) and its relationship with sum of even multivector fields, on a general Riemann-Cartan spacetime admitting a spin structure and to give a mathematically rigorous derivation of the so called Dirac-Hestenes equation (DHE) when spacetime is a Lorentzian. To this aim we introduce the Clifford bundle of multivector fields (Cl(M,g)) and the left (Cl_{Spin_{1,3}^{e}}^{l}(M)) and right (Cl_{Spin_{1,3}^{e}}^{r}(M)) spin-Clifford bundles on the spin manifold (M,g) The relation between left ideal algebraic s…
Properties of tensors equivalent to the direct product of two different 4-spinors are investigated. It is shown that the tensors obey additional 8 nonlinear restrictions, those are presented in Lorentz covariant form. In the context of the Dirac-Kähler theory, such a property can be interpreted as follows: if one wishes to consider the Dirac-Kähler field as consisting of two 4-spinor fields, one must impose additional restrictions on tensors of the Dirac-Kähler field, which leads to a non-linear wave equation for a complex boson field (composed on the base of two 4-spinor fields). Instead, the…
We show that higher degree Dirac currents of twistor and Killing spinors correspond to the hidden symmetries of the background spacetime which are generalizations of conformal Killing and Killing vector fields respectively. They are the generalizations of 1-form Dirac currents to higher degrees which are used in constructing the bosonic supercharges in supergravity theories. In the case of Killing spinors, we find that the equations satisfied by the higher degree Dirac currents are related to Maxwell-like and Duffin-Kemmer-Petiau equations. Correspondence between the Dirac currents and harmoni…
Two explicit formulas for metric connections in the bundle of Dirac spinors are studied. Their equivalence is proved. The explicit formula relating the spinor curvature tensor with the Riemann curvature tensor is rederived.
It is shown that Maxwell's equation cannot be put into a spinor form that is equivalent to Dirac's equation. First of all, the spinor ψin the representation \vec{F} = ψ\vec{u} \barψ of the electromagnetic field bivector depends on only three independent complex components whereas the Dirac spinor depends on four. Second, Dirac's equation implies a complex structure specific to spin 1/2 particles that has no counterpart in Maxwell's equation. This complex structure makes fermions essentially different from bosons and therefore insures that there is no physically meaningful way to transform Maxw…
We give a streamlined account of $2$-spinors, up to and including the Dirac equation, using little more than the resources of linear algebra. We prove that the Dirac bundle is isomorphic to the associated bundles $\mathrm{SL}_2(\mathbb{C}) \times_{\mathrm{SU}_2} S$ and $\mathrm{SL}_2(\mathbb{C}) \times_{\mathrm{SU}_2} \bar{S}$. A solution of the Dirac equation determines a pair of conjugate $2$-spinor fields over the mass shell $X_m$.
The SL(3,C)-covariant 9-dimensional equation for a free 3-spinor particle is transformed into the Dirac-like form (p_Aδ^A - M)Ψ=0. However, the corresponding δmatrices do not satisfy the Dirac algebra. It is shown that δ^A lead to a Finslerian generalization of the Duffin-Kemmer algebra. The Appendix contains an explicit realization of the δmatrices.
A common view is that generalization of a wave equation on Riemannian space-time is substantially determined by what a particle is - boson or fermion. As a rule, they say that tensor equations for bosons are extended in a simpler way then spinor equations for fermions. In that context, a very interesting problem is of extension a wave equation for Dirac--Kähler field (Ivanenko--Landau field was historically first term, also the term a vector field of general type was used). The article relates a generally covariant tensor formalism to a spinor one when these both are applied to description of …
When utilizing a cluster decomposible relativistic scattering formalism, it is most convenient that the covariant field equations take on a linear form with respect to the energy and momentum dispersion on the fields in the manner given by the Dirac form for spin \half systems. A general spinor formulation is given for arbitrary spins by minimally extending the Lorentz algebra to include operators whose matrix representation give general Dirac matrices. The forms of these matrices are explicitly demonstrated for spin \half and spin 1 fields.
Generalized Dirac equation with operator mass term is presented. Its solutions are non standard (ELKO) spinors which are eigenvectors of the charge conjugation and dual helicity operators. It is demonstrated that in spite of their non covariant nature ELKO can serve as a carrier space of a representation of Poincaré group. However, the corresponding boost generators are not manifestly covariant and generate non-local momentum dependent transformations which are presented explicitly. These results present a new look on group-theoretical grounds of ELKO theories.
We prove a classification result for ground state solutions of the critical Dirac equation on $\mathbb{R}^n$, $n\geq2$. By exploiting its conformal covariance, the equation can be posed on the round sphere $\mathbb{S}^n$ and the non-zero solutions at the ground level are given by Killing spinors, up to conformal diffeomorphisms. Moreover, such ground state solutions of the critical Dirac equation are also related to the Yamabe equation for the sphere, for which we crucially exploit some known classification results.
An integrated approach to Lie derivatives of spinors, spinor connections and the gravitational field is presented, in the context of a previously proposed, partly original formulation of a theory of Einstein-Carta-Maxwell-Dirac fields based on "minimal geometric data": all the needed underlying structure is geometrically constructed from the unique assumption of a complex vector field $S\to M$ with 2-dimensional fibers. The Lie derivatives of objects of all considered types, with respect to a vector field $X:M\to TM$, are well-defined without making any special assumption about $X$, and fulfil…
For the description of space-time fermions, Dirac-Kähler fields (inhomogeneous differential forms) provide an interesting alternative to the Dirac spinor fields. In this paper we develop a similar concept within the symplectic geometry of phase-spaces. Rather than on space-time, symplectic Dirac-Kähler fields can be defined on the classical phase-space of any Hamiltonian system. They are equivalent to an infinite family of metaplectic spinor fields, i.e. spinors of Sp(2N), in the same way an ordinary Dirac-Kähler field is equivalent to a (finite) mulitplet of Dirac spinors. The results are int…
Spinor representations of surfaces immersed into 4-dimensional pseudo-riemannian manifolds are defined in terms of minimal left ideals and tensor decompositions of Clifford algebras. The classification of spinor fields and Dirac operators on the immersed surfaces is given. The Dirac-Hestenes spinor field on surfaces immersed into Lorentzian manifolds and on surfaces conformally immersed into Minkowski spacetime is defined.
We characterize the Lie derivative of spinor fields from a variational point of view by resorting to the theory of the Lie derivative of sections of gauge-natural bundles. Noether identities from the gauge-natural invariance of the first variational derivative of the Einstein(--Cartan)--Dirac Lagrangian provide restrictions on the Lie derivative of fields.
A fundamental action, representing a mass dimension-transmuting operator between Dirac and ELKO spinor fields, is performed on the Dirac Lagrangian, in order to lead it into the ELKO Lagrangian. Such a dynamical transformation can be seen as a natural extension of the Standard Model that incorporates dark matter fields. The action of the mass dimension-transmuting operator on a Dirac spinor field, that defines and introduces such a mapping, is shown to be a composition of the Dirac operator and the non-unitary transformation that maps Dirac spinor fields into ELKO spinor fields, defined in a p…
Three out of four complex components of the Dirac spinor can be algebraically eliminated from the Dirac equation (if some linear combination of electromagnetic fields does not vanish), yielding a partial differential equation of the fourth order for the remaining complex component. This equation is generally equivalent to the Dirac equation. Furthermore, following Schrödinger (Nature, \textbf{169}, 538 (1952)), the remaining component can be made real by a gauge transform, thus extending to the Dirac field the Schrödinger's conclusion that charged fields do not necessarily require complex repr…
Dirac linear spinor fields were obtained from non-linear Heisenberg spinors, in the literature. Here we extend that idea by considering not only Dirac spinor fields but spinor fields in any of the Lounesto's classes. When one starts considering all these classes of fields, the question of providing a classification for the Heisenberg spinor naturally arises. In this work the classification of Heisenberg spinor fields is derived and scrutinized, in its interplay with the Lounesto's spinor field classification.
We introduce spinors, at a level appropriate for an undergraduate or first year graduate course on relativity, astrophysics or particle physics. The treatment assumes very little mathematical knowledge (mainly just vector analysis and some idea of what a group is). The SU(2)--SO(3) homomorphism is presented in detail. Lorentz transformation, chirality, and the spinor Minkowski metric are introduced. Applications to electromagnetism, parity violation, and to Dirac spinors are presented. A classical form of the Dirac equation is obtained, and the (quantum) prediction that $g=2$ for Dirac particl…
The triality properties of Dirac spinors are studied, including a construction of the algebra of (complexified) biquaternion. It is proved that there exists a vector-representation of Dirac spinors. The massive Dirac equation in the vector-representation is actually self-dual. The Dirac's idea of non-integrable phases is used to study the behavior of massive term.
We study conformal $Spin$-subgeometry of submanifolds in a semi-Riemannian $Spin$-manifold, focusing on conformal $Spin$-manifolds $(M,[h])$ and their Poincaré-Einstein metrics $(X,g_+)$. Our approach is based on the spectral theory of Dirac operator in the ambient $Spin$-manifold, and associated spinor valued meromorphic family of distributions with residues given by the residue family operators $\slashed{D}_N^{res}(h;λ)$ on spinors. We develop basic aspects and properties of $\slashed{D}_N^{res}(h;λ)$ including conformal covariance, factorization properties by conformally covariant operators…
Transformation properties of Dirac equation correspond to Spin(3,1) representation of Lorentz group SO(3,1), but group GL(4,R) of general relativity does not accept a similar construction with Dirac spinors. On the other hand, it is possible to look for representation of GL(4,R) in some bigger space, there Dirac spinors are formally situated as some "subsystem." In the paper is described construction of such representation, using Clifford and Grassmann algebras of 4D space.
In the first part of the paper we give a tensor version of the Dirac equation. In the second part we formulate and analyse a simple model equation which for weak external fields appears to have properties similar to those of the 2--dimensional Dirac equation.
The connection between wave functions in the Dirac and Foldy-Wouthuysen representations is found. When the Foldy-Wouthuysen transformation is exact, upper spinors in two representations differ only by constant factors, and lower spinors in the Foldy-Wouthuysen representation are equal to zero.
The structure of 29-dimensional extended real Clifford-Dirac algebra, which has been introduced in our paper Phys. Lett. A, 2011, 375, 2479, is considered in brief. Using this algebra, the property of Fermi-Bose duality of the Dirac equation with nonzero mass is proved. It means that Dirac equation can describe not only the fermionic but also the bosonic states. The proof of our assertion based on the examples of bosonic symmetries, solutions and conservation laws is given. Some statistical aspects of the spinor field Fermi-Bose duality are discussed.
Reporting about the Wigner formalism for describing Dirac spinor structures through a covariant phase-space formulation, the quantum information quantifiers for purity and mutual information involving spin-parity (discrete) and position-momentum (continuous) degrees of freedom are consistently obtained. For Dirac spinor Wigner operators decomposed into Poincaré classes of $SU(2) \otimes SU(2)$ spinor couplings, a definitive expression for quantum purity is identified in a twofold way: firstly, in terms of phase-space positively defined quantities, and secondly, in terms of the {\em spin-parity…
The so called Inomata-McKinley spinors are a particular solution of the non-linear Heisenberg equation. In fact, free linear massive (or mass-less) Dirac fields are well known to be represented as a combination of Inomata-McKinley spinors. More recently, a subclass of Inomata-McKinley spinors were used to describe neutrino physics. In this paper we show that Dirac spinors undergoing this restricted Inomata-McKinley decomposition are necessarily of the first type, according to the Lounesto classification. Moreover, we also show that this type one subclass spinors has not an exotic counterpart. …
There are Poincare group representations on complex Hilbert spaces, like the Dirac spinor field, or real Hilbert spaces, like the electromagnetic field tensor. The Majorana spinor is an element of a 4 dimensional real vector space. The Majorana spinor field is a space-time dependent Majorana spinor, solution of the free Dirac equation. The Majorana-Fourier and Majorana-Hankel transforms of Majorana spinor fields are defined and related to the linear and angular momenta of a spin one-half representation of the Poincare group. We show that the Majorana spinor field with finite mass is an unitary…
A method to construct noncommutative instantons as deformations from commutative instantons was provided in arXiv:0805.3373. Using this noncommutative deformed instanton, we investigate the spinor zero modes of the Dirac operator in a noncommutative instanton background on noncommutative R^4, and we modify the index of the Dirac operator on the noncommutative space slightly and show that the number of the zero mode of the Dirac operator is preserved under the noncommutative deformation. We prove the existence of the Green's function associated with instantons on noncommutative R^4, as a smooth…
We solve for spectrum, obtain explicitly and study group properties of eigenfunctions of Dirac operator on the Riemann sphere $S^2$. The eigenvalues $λ$ are nonzero integers. The eigenfunctions are two-component spinors that belong to representations of SU(2)-group with half-integer angular momenta $l = |λ| - \half$. They form on the sphere a complete orthonormal functional set alternative to conventional spherical spinors. The difference and relationship between the spherical spinors in question and the standard ones are explained.
We apply a new general method of anholonomic frames with associated nonlinear connection structure to construct new classes of exact solutions of Einstein-Dirac equations in five dimensional (5D)gravity. Such solutions are parametrized by off-diagonal metrics in coordinate (holonomic) bases, or, equivalently, by diagonal metrics given with respect to some anholonomic frames (pentads, or funfbiends, satisfing corresponding constraint relations). We consider two possibilities of generalization of the Taub NUT metric in order to obtain vacuum solutions of 5D Einsitein equations with effective ren…
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…
Relying on the mathematical analogy of the pure states of a two-qubit system with four-component Dirac spinors, we provide an alternative consideration of quantum entanglement using the mathematical formulation of Cartan's pure spinors. A result of our analysis is that the Cartan equation of two qubits state is entanglement sensitive in a way that the Dirac equation for fermions is mass sensitive. The Cartan equation for unentangled qubits is reduced to a pair of Cartan equations for single qubits as the Dirac equation for massless fermions separates into two Weyl equations. Finally, we establ…
Dual-helicity eigenspinors of the charge conjugation operator (ELKO spinor fields) belong -- together with Majorana spinor fields -- to a wider class of spinor fields, the so-called flagpole spinor fields, corresponding to the class-(5), according to Lounesto spinor field classification based on the relations and values taken by their associated bilinear covariants. There exists only six such disjoint classes: the first three corresponding to Dirac spinor fields, and the other three respectively corresponding to flagpole, flag-dipole and Weyl spinor fields. Using the mapping from ELKO spinor f…
