Harmonic symplectic spinors on Riemann surfaces

Symplectic spinor fields were considered already in the 70th in order to give the construction of half-densities in the context of geometric quantization. We introduced symplectic Dirac operators acting on symplectic spinor fields and started a systematical investigation. In this paper, we motivate the notion of harmonic symplectic spinor fields. We describe how many linearly independent harmonic symplectic spinors each Riemann surface admits. Furthermore, we calculate the spectrum of the symplectic spinor Laplacian on the complex projective space of complex dimension 1.     

On Conformal Invariants Built from Spinor Fields

By means of a conformal covariant differentiation process we construct generating systems for conformally invariant symmetric (r, s)–spinors in an arbitrary curved space–time. Extending this method to conformally invariant linear differential operators acting on symmetric spinor fields some classes of such operators are derived.     

Clifford Algebras and Euclid’s Parametrization of Pythagorean Triples

We show that the space of Euclid’s parameters for Pythagorean triples is endowed with a natural symplectic structure and that it emerges as a spinor space of the Clifford algebra R₂₁, whose minimal version may be conceptualized as a 4-dimensional real algebra of “kwaternions.” We observe that this makes Euclid’s parametrization the earliest appearance of the concept of spinors. We present an analogue of the “magic correspondence” for the spinor representation of Minkowski space and show how the Hall matrices fit into the scheme. The latter obtain an interesting and perhaps unexpected geometric meaning as certain symmetries of an Apollonian gasket. An extension to more variables is proposed and explicit formulae for generating all Pythagorean quadruples, hexads, and decuples are provided.     

Conservation laws for spinor fields on a Riemannian spacetime manifold

The analog of the polar decomposition theorem in Euclidean space is obtained in Minkowski space. The possibility of considering spinors in arbitrary frames is established by extending a Lorentz-group representation to a representation of the complete linear group in the space of spinors. The Lie derivative of spinors along arbitrary vector fields is constructed, and a Noether theorem for spinor fields is proved.Kazan State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 3, pp. 369–379, March, 1992.     

Skew Killing spinors

We study the existence of a skew Killing spinor on 2- and 3-dimensional Riemannian spin manifolds. We establish the integrability conditions and prove that these spinor fields correspond to twistor spinors in the two dimensional case while, up to a conformal change of the metric, they correspond to parallel spinors in the three dimensional case.     

On the Construction of a Geometric Invariant Measuring the Deviation from Kerr Data

This article contains a detailed and rigorous proof of the construction of a geometric invariant for initial data sets for the Einstein vacuum field equations. This geometric invariant vanishes if and only if the initial data set corresponds to data for the Kerr spacetime, and thus, it characterises this type of data. The construction presented is valid for boosted and non-boosted initial data sets which are, in a sense, asymptotically Schwarzschildean. As a preliminary step to the construction of the geometric invariant, an analysis of a characterisation of the Kerr spacetime in terms of Killing spinors is carried out. A space spinor split of the (spacetime) Killing spinor equation is performed to obtain a set of three conditions ensuring the existence of a Killing spinor of the development of the initial data set. In order to construct the geometric invariant, we introduce the notion of approximate Killing spinors. These spinors are symmetric valence 2 spinors intrinsic to the initial hypersurface and satisfy a certain second order elliptic equation—the approximate Killing spinor equation. This equation arises as the Euler-Lagrange equation of a non-negative integral functional. This functional constitutes part of our geometric invariant—however, the whole functional does not come from a variational principle. The asymptotic behaviour of solutions to the approximate Killing spinor equation is studied and an existence theorem is presented.     

Generalized Killing spinors on spheres

We study generalized Killing spinors on round spheres \(\mathbb {S}^n\) . We show that on the standard sphere \(\mathbb {S}^8\) any generalized Killing spinor has to be an ordinary Killing spinor. Moreover, we classify generalized Killing spinors on \(\mathbb {S}^n\) whose associated symmetric endomorphism has at most two eigenvalues and recover in particular Agricola–Friedrich’s canonical spinor on 3-Sasakian manifolds of dimension 7. Finally, we show that it is not possible to deform Killing spinors on standard spheres into genuine generalized Killing spinors.     

Spinors associated with theV 4 3 distribution and the generalized sen-witten equation

On the basis of the methods of nonholonomic differential geometry we introduce the concept of spinors associated with the V ₄ ³ distribution and their spatial covariant derivatives. We obtain the equations of the fundamental spinor fields under a local3+1-stratification of space-time and we study certain properties of their solutions. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 133–139.     

Spinoren in statischen Raum-Zeiten

In this paper, a spinor algebra and analysis adapted to static space-times is presented. Suitable SU(2)-bases are choosen in spinor space and it is shown, how these bases determine orthogonal systems in (three-dimensional) space. Some theorems on the curvature spinors of static space-times are proved by the help of the calculus of the connection spinors. The internal structure of the WEYL spinor as well as its connection with the RICCI tensor of the underlying (three-dimensional) space are examined. The presented calculus allows the computation of the NEWMAN-PENROSE spin coefficients and the canonical normal 1-spinors of the WEYL spinor with a relatively small expense, which is demonstrated on a sequence of examples.     

The Extended Fock Basis of Clifford Algebra

We investigate the properties of the Extended Fock Basis (EFB) of Clifford algebras [1] with which one can replace the traditional multivector expansion of ${\mathcal{C} \ell(g)}$ with an expansion in terms of simple (also: pure) spinors. We show that a Clifford algebra with 2m generators is the direct sum of 2 ^m spinor subspaces S characterized as being left eigenvectors of ??; furthermore we prove that the well known isomorphism between simple spinors and totally null planes holds only within one of these spinor subspaces. We also show a new symmetry between spinor and vector spaces: similarly to a vector space of dimension 2m that contains totally null planes of maximal dimension m, also a spinor space of dimension 2 ^m contains ??totally simple planes??, subspaces made entirely of simple spinors, of maximal dimension m.     

Classification of 1st order symplectic spinor operators over contact projective geometries

We give a classification of 1st order invariant differential operators acting between sections of certain bundles associated to Cartan geometries of the so-called metaplectic contact projective type. These bundles are associated via representations, which are derived from the so-called higher symplectic (sometimes also called harmonic or generalized Kostant) spinor modules. Higher symplectic spinor modules are arising from the Segal-Shale-Weil representation of the metaplectic group by tensoring it by finite dimensional modules. We show that for all pairs of the considered bundles, there is at most one 1st order invariant differential operator up to a complex multiple and give an equivalence condition for the existence of such an operator. Contact projective analogues of the well known Dirac, twistor and Rarita-Schwinger operators appearing in Riemannian geometry are special examples of these operators.     

The Complex Algebra of Physical Space: A Framework for Relativity

A complex and, equivalently, hyperbolic extension of the algebra of physical space (APS) is discussed that allows one to distinguish space-time vectors from paravectors of APS, while preserving the natural origin of the Minkowski space-time metric. The CAPS formalism is Lorentz covariant and gives expression to persistent vectors in physical space as time-like planes in space-time. Commuting projectors ${P_{\pm} = \frac{1}{2} (1 \pm h)}$ project CAPS onto two-sided ideals, one of which is APS. CAPS has the same dimension as the space-time algebra (STA) if both are considered real algebras, and it distinguishes covariant roles of elements, as does STA. Its structure, however, is closer to APS, with a volume element that belongs to the center of the algebra and a simple relation between space-times of opposite signature. Furthermore, CAPS, unlike STA, distinguishes point-like space-time inversion of a Dirac spinor from a physical rotation. To illustrate its use, CAPS is applied to the Dirac equation and to the fundamental symmetry transformations of the equation and Dirac spinors. The physical interpretations of both the equation and the spinor are clarified, and it is seen that the space-time frame ${\{\gamma_{\mu}\}}$ arises fully from relative vectors and does not imply the existence of an absolute space-time frame.     

The twistor spinors of generic 2- and 3-distributions

Generic distributions on 5- and 6-manifolds give rise to conformal structures that were discovered by P. Nurowski resp. R. Bryant. We describe both as Fefferman-type constructions and show that for orientable distributions one obtains conformal spin structures. The resulting conformal spin geometries are then characterized by their conformal holonomy and equivalently by the existence of a twistor spinor which satisfies a genericity condition. Moreover, we show that given such a twistor spinor we can decompose a conformal Killing field of the structure. We obtain explicit formulas relating conformal Killing fields, almost Einstein structures and twistor spinors.     

Skew Killing spinors in four dimensions

Annals of Global Analysis and Geometry - This paper is devoted to the classification of 4-dimensional Riemannian spin manifolds carrying skew Killing spinors. A skew Killing spinor $$\psi $$ is a...     

Structures of Boson and Fermion Fock Spaces in the Space of Symmetric Functions

We realize the Weil representation of infinite-dimensional symplectic group and spinor representation of infinite-dimensional group GL by linear operators in the space of symmetric functions in infinite number of variables.     

A memoir on the projective geometry of spinors

Summary The classical methods of projective geometry are applied to a number of questions in general relativity, by using the Van der Waerden spinor analysis. These include a new geometric theory of spinors, refinements in the spinor calculus, the classification of electromagnetic and gravitational fields, Weyl-Maxwell fields, a classification of the Einstein spinor, and the projective geometry of the Bel-Petrov types. Dedicated to the memory of my teacher and friend Professor Dr.Vaclav Hlaváty (1894–1969) Entrata in Redazione il 14 marzo 1975.     

Spinor spaces of Clifford algebras

In this paper, we study the structures of Clifford algebras. We represent the pinor and spinors spaces as subspaces of Clifford algebras. With suitable bases of the Clifford algebras, we construct isomorphisms between Clifford algebras and matrix algebras. In doing these we develop some spinor calculus.     

用旋量方法规划机器人运动轨迹

以姿态旋量描述机器人的位置姿态,在对偶空间中通过姿态旋量映射的点规划机器人的终端轨迹,具有直观、简便的独特优点。规划中直接根据跟踪误差进行收敛,提高了轨迹运行的动态精度,并适合于冗余自由度操作器。     

Spinorial Formulations of Graph Problems

Beginning with an arbitrary finite graph, various spinor spaces are constructed within Clifford algebras of appropriate dimension. Properties of spinors within these spaces then reveal information about the structure of the graph. Spinor polynomials are introduced and the notions of degrees of polynomials and Fock subspace dimensions are tied together with matchings, cliques, independent sets, and cycle covers of arbitrary finite graphs. In particular, matchings, independent sets, cliques, cycle covers, and cycles of arbitrary length are all enumerated by dimensions of spinor subspaces, while sizes of maximal cliques and independent sets are revealed by degrees of spinor polynomials. The spinor adjacency operator is introduced and used to enumerate cycles of arbitrary length and to compute graph circumference and girth.     

Structure spinorielle associée à un espace vectoriel quaternionien à droite e sur H, muni d’une forme sesquilinéaire b non dégénérée H-anti hermitienne

This paper, self-contained, deals with the study of Clifford Algebras associated with n-dimensional skew-hermitian spaces over the skew field H. The different structures associated with the spaces S of corresponding spinors are given and the natural imbeddings of associated spinor groups are revealed.