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 spinors

For a 2^n-dimensional complex Hermitian vector space S, we prove that any unitary basis of S can be explained as an augmented spinor structure on S. By using this explanation, a SpinC（2n）- action on S is equivalent to an action on a subset of augmented spinor structures. The latter action is a little easy to be understood, and is shown in the last part of this paper. Such kind of understanding could be of use to the discussions of Hermitian manifolds and spin manifolds, especially could help to find connections and elliptical operators.     

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.     

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.     

偶数维复Clifford代数中的Dirac旋量空间

本文引入了偶数维欧氏空间的复结构及Witt基,在此基础上讨论了偶数维复Clifford代数中的Dirac旋量空间.由Fock空间的结果我们得到了Dirac旋量空间视为复Clifford代数中极小左理想,最后我们研究了Dirac旋量空间的对偶空间.     

Clifford algebraic spinor and the Dirac wave equations

The hyperbolic complex (HC) space is congruent with Minkowski space time.HC is a special kind of non-Euclidean space with continuous odd-points. The Clifford algebraic spinor and the Dirac wave equation can be introduced in the hyperbolic complex space. The Clifford algebraic spinor contains eight independent elements and the Dirac wave equations 64 coefficients. For Dirac particles 4×8 and for antiparticles 4×8 variables which are Hermitian conjugate to each other (on four dimensional space-time).     

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.     

Antisymmetric tensor fields and the Weyl gauge theory

The aim of the present paper is to find a spinor current—a source—in the Weyl non-Abelian gauge theory whose distinguishing feature is that it involves no abstract gauge space. It is shown that the desired spinor representation of the Weyl gauge group can be constructed in the space of antisymmetric tensor fields in the form of a 16-component quantity for which a gauge-invariant Lagrangian is established. The relationship between the Weyl non-Abelian gauge potential and the Cartan torsion field, and the question of where the interactions in question could manifest are discussed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 1, pp. 112–123, October, 1997.     

Spinorial Method of Terminal Control of Spatial Rotations

In this paper, we introduce the notion of three-dimensional generalized rotations. We obtain relations between the parameters of the spinor representation of the group of three-dimensional generalized rotations and the coordinates of the initial and terminal points of rotation. Simple relations between elements of a three-dimensional orthogonal matrix of the basic representation and the Euler angles, on the one hand, and the coordinates of the initial and terminal points of rotation, on the other hand, were derived. The spinor method of solution of the inverse kinematic problem for spatial mechanisms with spherical pairs is developed and the corresponding algorithm is proposed. The obtained results allow one to reduce the three-dimensional problem of spatial motion control to the one-dimensional problem. Simple adaptive algorithms are suggested, by means of which various partial problems on the terminal control are solved under various terminal conditions. New algorithms of control of spatial rotations of manipulating robots are studied.     

On some relations between spinor and orthogonal groups

In this article we consider Clifford algebras over the field of real numbers of finite dimension. We define the operation of Hermitian conjugation for the elements of Clifford algebra. This operation allows us to define the structure of Euclidian space on the Clifford algebra. We consider pseudo-orthogonal group and its subgroups — special pseudo-orthogonal, orthochronous, orthochorous and special orthochronous groups. As we know, spinor groups are double covers of these orthogonal groups.We proved theorem that relates the norm of element of spinor group to the minor of matrix of the corresponding orthogonal group.     

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.     

Conditions of Equivalence of Spinor and Tensor Methods in the Problem of Positive Definiteness of the Gravitational Problem

It is demonstrated that J. Nester's tensor method for proving the theorem on the positive definiteness of gravitational energy in an asymptotically Minkowsky space is equivalent to E. Witten's spinor method, in which the Saint–Witten equation must be completed with a linear term that includes the gradient of a certain scalar potential. A new proof of the theorem on the positive definiteness of energy is proposed.     

Some Results on Augmented Lagrangians in Constrained Global Optimization via Image Space Analysis

The aim of this paper is to present some results for the augmented Lagrangian function in the context of constrained global optimization by means of the image space analysis. It is first shown that a saddle point condition for the augmented Lagrangian function is equivalent to the existence of a regular nonlinear separation in the image space. Local and global sufficient optimality conditions for the exact augmented Lagrangian function are then investigated by means of second-order analysis in the image space. Local optimality result for this function is established under second-order sufficiency conditions in the image space. Global optimality result is further obtained under additional assumptions. Finally, it is proved that the exact augmented Lagrangian method converges to a global solution–Lagrange multiplier pair of the original problem under mild conditions.     

对等式约束非线性规划问题的Hestenes-Powell增广拉格朗日函数的进一步研究

本文对用无约束极小化方法求解等式约束非线性规划问题的Hestenes-Powell 增广拉格朗日函数作了进一步研究．在适当的条件下,我们建立了Hestenes-Powell增广拉格朗日函数在原问题变量空间上的无约束极小与原约束问题的解之间的关系,并且也给出了Hestenes-Powell增广拉格朗日函数在原问题变量和乘子变量的积空间上的无约束极小与原约束问题的解之间的一个关系．因此,从理论的观点来看,原约束问题的解和对应的拉格朗日乘子值不仅可以用众所周知的乘子法求得,而且可以通过对Hestenes-Powell 增广拉格朗日函数在原问题变量和乘子变量的积空间上执行一个单一的无约束极小化来获得．     

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.     

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.     

ψ-序压缩算子的不动点定理

2005年,张宪在Banach空间中通过其中的锥所定义的半序引进了序压缩算子,证明了几个相应的定理．但是在一般的度量空间中,能否定义序压缩算子,能否得到类似的结论呢？本文在度量空间X中,通过X上的泛函ψ-所定义的半序,引进了ψ--序压缩算子,并且得到了相应的不动点定理．     

The inverse Penrose transform of a solution to the Maxwell-Dirac-Weyl field equations

An explicit family of solutions to the nonlinear coupled Maxwell-Dirac-Weyl equations in Minkowski space is presented. The abstract results of Henkin and Manin (Phys. Lett. B, 95 (1980), 405–408) show that these solutions are equivalent by the Penrose transform to a coupled system of cohomology classes and a complex line bundle on ambitwistor space, the space of null lines in Minkowski space. The explicit inverse Penrose transform of this family of solutions is computed giving explicit expressions for the line bundle (transform of the vector potential), the obstruction to extension (transform of the charge), and the two cohomology classes (transform of the Dirac-Weyl coupled spinor fields).     

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.