In this paper using the Clifford bundle (Cl(M,g)) and spin-Clifford bundle (Cl_{Spin_{1,3}^{e}}(M,g)) formalism, which permit to give a meaningfull representative of a Dirac-Hestenes spinor field (even section of Cl_{Spin_{1,3}^{e}}(M,g)) in the Clifford bundle , we give a geometrical motivated definition for the Lie derivative of spinor fields in a Lorentzian structure (M,g) where M is a manifold such that dimM =4, g is Lorentzian of signature (1,3). Our Lie derivative, called the spinor Lie derivative (and denoted £_{ξ}) is given by nice formulas when applied to Clifford and spinor fields, and moreoverl £_{ξ}g=0 for any vector field ξ. We compare our definitions and results with the many others appearing in literature on the subject.
Kosmann-Lie derivatives in the bundle of Weyl spinors are considered. It is shown that the basic spin-tensorial fields of this bundle are constants with respect to these derivatives.
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.
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…
Starting from the general concept of a Lie derivative of an arbitrary differentiable map, we develop a systematic theory of Lie differentiation in the framework of reductive G-structures P on a principal bundle Q. It is shown that these structures admit a canonical decomposition of the pull-back vector bundle i_P^*(TQ) = P\times_Q TQ over P. For classical G-structures, i.e. reductive G-subbundles of the linear frame bundle, such a decomposition defines an infinitesimal canonical lift. This lift extends to a prolongation Gamma-structure on P. In this general geometric framework the concept of a…
We show that the covariant derivative of a spinor for a general affine connection, not restricted to be metric compatible, is given by the Fock-Ivanenko coefficients with the antisymmetric part of the Lorentz connection. The projective invariance of the spinor connection allows to introduce gauge fields interacting with spinors. We also derive the relation between the curvature spinor and the curvature tensor for a general connection.
We investigate using Clifford algebra methods the theory of algebraic dotted and undotted spinor fields over a Lorentzian spacetime and their realizations as matrix spinor fields, which are the usual dotted and undotted two component spinor fields. We found that some ad hoc rules postulated for the covariant derivatives of Pauli sigma matrices and also for the Dirac gamma matrices in General Relativity cover important physical meaning, which is not apparent in the usual matrix presentation of the theory of two components dotted and undotted spinor fields. We also discuss some issues related to…
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…
We establish, via geometric quantization of the supercotangent bundle sM of (M,g), a correspondence between its conformal geometry and those of the spinor bundle. In particular, the Kosmann Lie derivative of spinors is obtained by quantization of the comoment map, associated to the new Hamiltonian action of conf(M,g) on sM. We study then the conf(M,g)-module structures induced on the space of differential operators acting on spinor densities and on its spaces of symbols (functions on sM). In the conformally flat case, we classify their conformal invariants, including the conformally odd powers…
We determine the holonomy of generalized Killing spinor covariant derivatives of the form $D= \nabla + Ω$ on pseudo-Riemannian reductive homogeneous spaces in a purely algebraic and algorithmic way, where $Ω: TM \rightarrow Λ^*(TM)$ is a left-invariant homomorphism. This is essentially an application of the theory of invariant principal bundle connections defined over homogeneous principal bundles. Moreover, the algorithm allows a computation of the associated Killing superalgebra in certain cases. The procedure is demonstrated by determining the supersymmetries of certain homogeneous M2 duals…
Let (M^n,g) be a Riemannian spin manifold. The basic equations in supergravity models of type IIa string theory with 4-form flux involve a 3-form T, a 4-form F, a spinorial covariant derivative \nabla depending on \nabla^g, T, F, and a \nabla-parallel spinor field Ψ. We classify and construct many explicit families of solutions to this system of spinorial field equations by means of non-integrable special geometries. The latter include α-Sasakian structures in dimensions 5 and 7, almost Hermitian structures in dimension 6 and cocalibrated G_2-structures in dimension 7. We show that there are s…
Results on symplectic spinors and their higher spin versions, concerning representation theory and cohomology properties are presented. Exterior forms with values in the symplectic spinors are decomposed into irreducible modules including finding the hidden symmetry (Schur--Weyl--Howe type duality) given by a representation of the Lie superalgebra $\mathfrak{osp}(1|2)$ in this case. We also determine ranges of the induced exterior symplectic spinor derivatives when they are restricted to bundles induced by the irreducible submodules mentioned. This duality is used to decompose curvature tensor…
We generalize the symmetry superalgebras of isometries and geometric Killing spinors on a manifold to include all the hidden symmetries of the manifold generated by Killing spinors in all dimensions. We show that bilinears of geometric Killing spinors produce special Killing-Yano and special conformal Killing-Yano forms. After defining the Lie algebra structure of hidden symmetries generated by Killing spinors, we construct the symmetry operators as the generalizations of the Lie derivative on spinor fields. All these constructions together constitute the structure of generalized symmetry supe…
We investigate a model in which spinors are considered as being embedded within the Clifford algebra that operates on them. In Minkowski space $M_{1,3}$, we have four independent 4-component spinors, each living in a different minimal left ideal of $Cl(1,3)$. We show that under space inversion, a spinor of one left ideal transforms into a spinor of another left ideal. This brings novel insight to the role of chirality in weak interactions. We demonstrate the latter role by considering an action for a generalized spinor field $ψ^{αi}$ that has not only a spinor index $α$ but also an extra index…
In this paper we show how to describe the general theory of a linear metric compatible connection with the theory of Clifford valued differential forms. This is done by realizing that for each spacetime point the algebra of Clifford bivectors is isomorphic to the Lie algebra of Sl(2,C). In that way the pullback of the linear connection under a trivialization of the bundle is represented by a Clifford valued 1-form. That observation makes it possible to realize Einstein's gravitational theory can be formulated in a way which is similar to a Sl(2,C) gauge theory. Some aspects of such approach is…
The essentially unique torsionful version of the classical two-component spinor formalisms of Infeld and van der Waerden is presented. All the metric spinors and connecting objects that arise here are formally the same as the ones borne by the traditional formalisms. Any spin-affine connexion appears to possess a torsional part which is conveniently chosen as a suitable asymmetric contribution. Such a torsional affine contribution thus supplies a gauge-invariant potential that can eventually be taken to carry an observable character, and thereby effectively takes over the role of any trivially…
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.
The Bargmann-Wigner (BW) framework describes particles of spin-j in terms of Dirac spinors of rank 2j, obtained as the local direct product of n Dirac spinor copies, with n=2j. Such spinors are reducible, and contain also (j,0)+(0,j)-pure spin representation spaces. The 2(2j+1) degrees of freedom of the latter are identified by a projector given by the n-fold direct product of the covariant parity projector within the Dirac spinor space. Considering totally symmetric tensor spinors one is left with the expected number of 2(2j+1) independent degrees of freedom. The BW projector is of the order …
We present a new framework for defining fuzzy approximations to geometry in terms of a cutoff on the spectrum of the Dirac operator, and a generalization of it that we call the Dirac-Flux operator. This framework does not require a symplectic form on the manifold, and is completely rotation invariant on an arbitrary n-sphere. The framework is motivated by the formalism of Holographic Space-Time (HST), whose fundamental variables are sections of the spinor bundle over a compact Euclidean manifold. The strong holographic principle (SHP) requires the space of these sections to be finite dimension…
A view is taken up whereby the non-singular Trautman-Kopczyński big-bang creation of the Universe produced a highly torsional hot state at early stages of the cosmic evolution which particularly brought about the formation of a dark matter cloud. It is thus assumed that the combination of Einstein-Cartan's theory with the torsionful version of the two-component ε-formalism of Infeld and van der Waerden supplies a natural local description of dark matter in terms of uncharged spin-one massive fields. In the case of either handedness, the pertinent spinor field equation arises directly from a su…
A complete and explicit classification of all locally constructed conserved currents and underlying conserved tensors is obtained for massless linear symmetric spinor fields of any spin s>0 in four dimensional flat spacetime. These results generalize the recent classification in the spin s=1 case of all conserved currents locally constructed from the electromagnetic spinor field. The present classification yields spin s>0 analogs of the well-known electromagnetic stress-energy tensor and Lipkin's zilch tensor, as well as a spin s>0 analog of a novel chiral tensor found in the spin s=1 case. Th…
Reductive G-structures on a principal bundle Q are considered. It is shown that these structures, i.e. reductive G-subbundles P of Q, admit a canonical decomposition of the pull-back vector bundle $i_P^*(TQ) = P \times_Q TQ$ over P. For classical G-structures, i.e. reductive G-subbundles of the linear frame bundle, such a decomposition defines an infinitesimal canonical lift. This lift extends to a prolongation $Γ$-structure on P. In this general geometric framework the theory of Lie derivatives is considered. Particular emphasis is given to the morphisms which must be taken in order to state …
Motivated by BRST theory, we study generalized symmetries and supersymmetries depending on derivatives of dynamic variables in a most general setting. We state the first variational formula and conservation laws for higher order Lagrangian systems on fiber bundles and graded manifolds under generalized symmetries and supersymmetries of any order. Cohomology of nilpotent generalized supersymmetries are considered.
