Basic Spinors Research Articles

Spurless Method of Calculating the Amplitudes of Reactions Involving Massless Fermions

A new method of calculating the amplitudes of processes involving massless fermions with a spin of 1/2 is suggested in the present paper. The method is based on the use of the isotropic tetrad in Minkowski space and the basic spinors related to it. As an illustration, the amplitudes of the electron-positron annihilation into a fermion pair and the collisions of two γ-quanta and their transformation into an electron-positron pair are obtained.

Spinor techniques for massive fermions with arbitrary polarization

We present a new variant of the spinor techniques for calculating the amplitudes of processes involving massive fermions with arbitrary polarization. It is relatively simple and leads to basic spinor products. Our procedure is not more complex than CALCUL spinor techniques for massless fermions. We obtain spinor Chisholm identities for massive fermions. As an illustration, expressions are given for the amplitudes of electron-positron annihilation into fermions pairs for several polarizations.

EXPLANATION OF THE DATA ON $\Gamma (Z^0\to b\bar b) /\Gamma (Z^0\to {\rm hadrons})$ AND THE INCLUSIVE b→sγ BRANCHING RATIO

Calculations show that the data on the Z0 hadronic branching ratio fraction to [Formula: see text] at the Z0 pole as well as the inclusive b→sγ branching ratio can be explained by an anomalous (γ-Z0)WW coupling having ∆κ=−1 and λ=0. Such a vertex was derived from a representation in spinor space some time ago. The agreement is analyzed and possible implication of that basic spinor symmetry are described. Finally, the recent data on diboson production and their interpretation are examined critically. The used and shortly presented electroweak unification scheme of “structureless compositeness” provides two natural explanations of the increase of the inclusive jet cross-section above 200 GeV jet transverse energy observed at FNAL recently.

Hadron magnetic moment in the spinor strong interaction model

Hadron magnetic moments are considered in the framework of the spinor strong interaction theory for hadron spectra proposed by the author. Expressions of magnetic moments of ground-state hadrons are derived. These differ from the conventional ones in that they are no longer phenomenological and are basically relativistic. Pseudoscalar mesons have no magnetic moment. Charged vector meson magnetic moment values are given. The magnetic moment operators operate in the internal space, so that the ground-state octet baryons have the same spin-space symmetry, including the A. A formula for the ground-state octet baryon magnetic moment is derived from the basic spinor strong interaction baryon equations previously given, essentially without approximation and in a way analogous to the way in which the electron magnetic moment is derived.

Majorana-Weyl spinors and self-dual gauge fields in 2+2 dimensions

The properties of Dirac gamma matrices in a four-dimensional space—time with the (2, 2) signature are studied. The basic spinors are classified, and the existence of Majorana-Weyl spinors is noted. Supersymmetry in 2+2 dimensions is discussed, and the existence of the real chiral scalar supermultiplet is discovered. Supersymmetric self-dual Yang-Mills theories and self-dual supergravity model in 2+2 dimensions, that are apparently relevant to integrable systems, are formulated for the first time.

Supergauge independence of Witten's gravitational energy expression

Witten's proof of positive gravitational energy involved an apparent choice of supergauge, through a constraint on the basic spinor parameter. We show that the value of the energy is, as it must be on supergravity grounds, independent of this choice.

Conditions for the relativistic virial theorem for one electron

Conditions for the usual form of one-electron relativistic virial theorem to be satisfied are derived and illustrated. These can be applied to generate quality basic spinors for solving a one-electron Dirac equation.

Branching rules and even-dimensional rotation groups SO2k

Unambiguous methods are developed for calculating branching rules for the classical subgroups of the even-dimensional rotation group SO2k. Complete results are given for the subgroups SUk*U1, SO2k-2*U1, SO2p*SO2q and SO2p+1*SO2q+1. A number of examples relevant to problems in supergravity and unification theories are given. A complete resolution of the antisymmetric powers of the basic spinor irrep of SO10 is given and the results extended to SO11.

Non-abelian conformal gravity as the origin of strong interactions

A metric tensor of gravity originating as a composite structure from basic spinor fields naturally carries also the internal-symmetry properties of the underlying spinor fields. The closeness of the grand-unification breakdown mass to Planck's mass suggests an interpretation of the strong lepton-quark symmetry,e.g. SU4, as an internal group also relevant for the dilatation. Spontaneous breakdown of dilatation and lepton-quark symmetry are closely related. Indefinite metric, dangerous for energies of the order of Planck's mass, seems to be an unavoidable consequence of a spontaneous dilatation breakdown.

Symmetrised powers of rotation group representations

The results given permit the unambiguous evaluation of all possible Kronecker products of the irreducible representations (tensor and spinor) of On and SOn for n=2 nu and n=2 nu +1. A complete resolution of the second and third powers of the basic spinor representations of SO2 nu and SO2 nu +1 is given, together with a prescription for analysing the fourth power of these representations. Detailed application is made to the enumeration of properties of SO10 relevant to grand unified theories, and sufficient information given to resolve the fourth power of any representation of SO10.

Bilinear Spinor Representation of Space-Time

In this work a method of realizing the bilinear spinor representation of space-time 1s suggested, in which the usual translation transformation can be established. The basic spinors introduced are the quantities having the spinor structure with respect to the Lorentz transformation group and the isotopic transformation group as well. In the language of these spinor coordinates the field equation of motion and the Noether theorem are also considered. The problem of realizing the bilinear spinor representation of space-time has been examined first in the paper of Smrz.v The starting point of the author is the idea that the space-time is the most basic physical system and therefore the space-time four-vector should be generated by the more fundamental mathematical objects, i.e., spinors. Moreover, he has suggested that in the region occupied by an elementary particle the usual concept of space-time breaks dovvn and then we must work with its elementary constituents, the basic spinors. By introducing three different basic spinors and by the association of special role to the third element cj;, through which the space-time vector is expressed,

Photons, neutrinos, electrons and baryons in a unified spinor field theory of relativity

A mathematically precise model may be constructed by considering a distribution of basic spinors in a Kaluza 5-space, such that (1) The basic spinors W satisfy curl5 W=0 (2) The four-dimensional curvature is in accordance with Einstein's General Theory, taking the energy-momentum-tension tensor as TIJ = k1, Qi Qj + k2PiPj + k3gij(PkkPk) where\(Q_i = \widetilde{\bar W}TX_i W,{\text{ }}P_2 = \widetilde{\bar W}T\phi X_i W\), and the ratiosk1:k2:k3 are adjustable to give the best fit. (3) The five-dimensional curvature associated with the electromagnetic field is determined by the concomitant $$F_{ij} = \widetilde{\bar W}TX_i X_j W$$ In this model there can exist wave packets which behave in exactly the same way as the physical properties of photons, neutrinos, electrons and baryons.

Analytic spinor fields

An analogue of complex variable theory in the plane is obtained by considering a three-dimensional distribution of basic spinors. These satisfy the invariant linear differential equation denoted by curl W = 0, which plays the rôle of the Cauchy-Riemann equations. Spinor harmonics and associated spinor harmonics are evaluated. The analogues of Taylor series and Laurent series are obtained. Analytic continuation is discussed, and this leads to the analogue of a Riemann surface in three dimensions. These spinor fields are equivalent to the space-time spinor fields which describe elementary particles, the case considered here corresponding to zero time derivative.

Relativity and quantum theory. A unified model

By considering a distribution of complex basic spinors W, satisfying the invariant linear differential equation $$curl_5 W = 0,$$ quation> in a 5-dimensional Kaluza space such that the space curvatures are determined by the concomitants of the spinors, a precisely defined mathematical model is obtained, which displays the basic properties of the physical world, including gravitational, electromagnetics, and many, if not all, of the basic quantum effects.

Spinor space and strong interactions

In order to explain the existence of internal quantum numbers of elementary particles, the ordinary space–time position vector is assumed to be a bilinear combination of two basic spinors. It is shown that this assumption leads to the replacement of the inhomogeneous Lorentz group by the (4 + 1) de Sitter group. Using the idea of external and internal space for the description of strongly interacting particles, the assumption provides a simple way of introducing the internal symmetries. The internal symmetry group SU(3) appears naturally connected with the external space, yielding the Gell-Mann -Okubo mass formula.

On certain identities involving basic spinors and curvature spinors

The spinor analysis of Infeld and van der Waerden [1] is particularly well suited to the transcription of given flat space wave equations into forms which constitute possible generalizations appropriate to Riemann spaces [2]. The basic elements of this calculus are (i) the skew-symmetric spinor Γμν, (ii) the hermitian tensor-spinor σκ(img)ν(generalized Pauli matrices), and (iii) the curvature spinor Ρμνκι. When one deals with wave equations in Riemann spaces V4 one is apt to be confronted with expressions of somewhat bewildering appearance in so far as they may involve products of a large number of σ-symbols many of the indices of which may be paired in all sorts of ways either with each other or with the indices of the components of the curvature spinors. Such expressions are generally capable of great simplification, but how the latter may be achieved is often far from obvious. It is the purpose of this paper to present a number of useful relations between basic tensors and spinors, commonly known relations being taken for granted [3], [4], [5]. That some of these new relations appear as more or less trivial consequences of elementary identities is largely the result of a diligent search for their simplest derivation, once they had been obtained in more roundabout ways.

Book Reviews

BASIC spinors are defined with reference to the equation T-1γiT = Σaijγj, where γi represents a set of anti-commuting matrices, [ast] is the matrix of an orthogonal transformation, and T is the matrix of transformation for basic spinors. Dirac's equation for the electron is of the form ∇2ψ = k2ψ where ψ is a basic spinor. The same equation, but with ψ representing quantities of other types, occurs in other contexts in quantum theory and nuclear physics. On a Class of Linear Transformations Connected with Group Representations By Lars Gårding. (Meddelanden från Universitets Matematiska Seminarium, Band 6.) Pp. 125. (Lund: C. W. K. Gleerup, 1944.) n.p.