Generalization of Dirac Conjugation in the Superalgebraic Theory of Spinors

Theoretical and Mathematical Physics - In the superalgebraic representation of spinors using Grassmann densities and the corresponding derivatives, we introduce a generalization of Dirac...     

Relativistically invariant representation of arbitrary bilinear combinations of Dirac spinors

It is shown that in the general case a bilinear combination of two arbitrary four-spinors with one of the 16 basic Dirac matrices cannot be expressed in terms of the momentum and spin four-vectors (Pauli-Lubaski vector). A universal general formula obtained in this paper also includes four-vectors corresponding to two three-dimensional unit vectors in the particle rest frame perpendicular to the direction of the particle spin. The transition to vanishing masses is considered. Special expressions that describe binary combinations of spinors in the case of elastic scattering for helicity and transversal amplitudes are given.Institute of High Energy Physics, Serpukhov. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 1, pp. 3–22, July, 1993.     

Triality,biquaternion and vector representation of the Dirac equation

The triality properties of Dirac spinors are studied, including a construction of the algebra of (complexified) biquaternion. A bilinear law of composition for biquaternion is defined by means of Levi-civita symbol and Lorentzian metric only. It is proved that there exists a vector-representation of Dirac spinors. The massive Dirac equation in the vector-representation is actually self-dual. A chiral transformations for spinors is equivalent to U(1) transformations for corresponding complex vectors. The Dirac’s idea of non-integrable phases is used to study the behavior of massive term.     

Rank of inclusion matrices and modular representation theory

We use results from the modular representation theory of the groupsS _n and GL _n (F _q ) to determine the rank of inclusion matrices.     

An asymptotic representation for products of random matrices

Let {X_k} be a stationary ergodic sequence of nonnegative matrices. It is shown in this paper that, under mild additional conditions, the logarithm of the i, jth element of X_t···X₁ is well approximated by a sum of t random variables from a stationary ergodic sequence. This representation is very useful for the study of limit behaviour of products of random matrices. An iterated logarithm result and an estimation result of use in the theory of demographic population projections are derived as corollaries.     

Commutation properties of anticommuting self-adjoint operators,spin representation and Dirac operators

LetA andB be anticommuting self-adjoint operators in a Hilbert space . It is proven thatiAB is essentially self-adjoint on a suitable domain and its closureC(A, B) anticommutes withA andB. LetU _s be the partial isometry associated with the self-adjoint operatorsS, i.e., the partial isometry defined by the polar decompositionS=U _S |S|. LetP _S be the orthogonal projection onto (KerS). Then the following are proven: (i) The operatorsU _A ,U _B ,U _C(A,B) ,P _A ,P _B , andP _A P _B multiplied by some constants satisfy a set of commutation relations, which may be regarded as an extension of that satisfied by the standard basis of the Lie algebra of the special unitary groupSU(2); (ii) There exists a Lie algebra associated with those operators; (iii) If is separable andA andB are injective, then gives a completely reducible representation of with each irreducible component being the spin representation of the Clifford algebra associated with ³; This result can be extended to the case whereA andB are not necessarily injective. Moreover, some properties ofA+B are discussed. The abstract results are applied to Dirac operators.     

An alternative representation of the generalized inverse of partitioned matrices

In this paper R.E. Cline's formula for the generalized inverse of the partitioned matrix (U, V) is simplified. The result is then applied to a particular case investigated by A. Ben-Israel.     

On the closed representation for the inverses of Hessenberg matrices

The general representation for the elements of the inverse of any Hessenberg matrix of finite order is here extended to the reduced case with a new proof. Those entries are given with proper Hessenbergians from the original matrix. It justifies both the use of linear recurrences of unbounded order for such computations on matrices of intermediate order, and some elementary properties of the inverse. These results are applied on the resolvent matrix associated to a finite Hessenberg matrix in standard form. Two examples on the unit disk are given.     

Probability representation of spin states and inequalities for unitary matrices

We review the probability representation of spin states described by probability distributions (tomograms). We use the relation of a tomogram to unitary group elements to obtain some inequalities for unitary matrices. We present the Cirel’son bound and the entropic inequalities for entangled spin states in the form of relations for functions on the unitary group.     

A note on the representation for the Drazin inverse of block matrices

In this note we consider representations of the Drazin inverse of 2×2 block matrices under conditions weaker than those used in recent papers on the subject, in particular in [D.S. Djordjević, P.S. Stanimirović, On the generalized Drazin inverse and generalized resolvent, Czechoslovak Math. J. 51 (126) (2001) 617–634; R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of 2×2 block matrix, SIAM J. Matrix Anal. Appl. 27 (2006) 757–771].     

A structure of the Bang-Bang representation for 3×3 embeddable matrices

Summary Necessary and sufficient conditions are given for a 3×3 stochastic matrix to be embeddable by 6 elementary stochastic matrices (Poisson matrices). For a 3×3 embeddable matrix, a structure of the minimal Bang-Bang representation, i.e. the one that contains the smallest number of elementary matrices, is obtained. Based on the minimal Bang-Bang representation an algorithm for determining the embeddability of a 3×3 stochastic matrix is given.I would like to thank Søren Johansen for helpful comments and stimulating discussions on the subject of this paper     

Dirac结构与Dirac流形

引入了Dirac结构的对偶特征对的概念,并给出了相应的可积性条件．利用这些结果,得到在Dirac流形的子流形上自然诱导出Dirac结构的条件,结果改进了Courant T．J．给出的相应条件;还得到Poisson流形在子流形上诱导出Poisson结构的条件,并改进了Weinstein A．和Courant T．J．所给出的相应条件;最后证明了预辛形式的可约Dirac结构与相应商流形上的辛结构之间存在一一对应的关系．