Spinor Approach Research Articles

Local freedom in the gravitational field revisited

Maartens et al (1997 Class. Quantum Grav. 14 1927) gave a covariant characterization, in a (1 + 3) formalism based on a perfect fluid's velocity, of the parts of the first derivatives of the curvature tensor in general relativity which are ‘locally free’, i.e. not pointwise determined by the fluid energy–momentum and its derivative. The full decomposition of independent curvature derivative components given in earlier work on the spinor approach to the equivalence problem enables analogous general results to be stated for any order: the independent matter terms can also be characterized. Explicit relations between the two sets of results are obtained. The 24 Maartens et al locally free data are shown to correspond to the ∇Ψ quantities in the spinor approach, and the fluid terms are similarly related to the remaining 16 independent quantities in the first derivatives of the curvature.

Superstrings with boundary fermions

The Green-Schwarz action for an open superstring with additional boundary fermions, representing Chan-Paton factors, is studied at the classical level. The boundary geometry is described by a bundle, with fermionic fibres, over the super worldvolume of a D-brane together with a map from the total space into the type II target superspace. This geometry is constrained by the requirement of kappa-symmetry on the boundary together with the use of the equations of motion for the fermions. There are two constraints which are formally similar to those that arise in the abelian case but which differ because of the dependence on the additional coordinates. The model, when quantised, would be a candidate for a fully kappa-symmetric theory of a stack of coincident D-branes including a non-abelian Born-Infeld sector. The example of the D9-brane in a flat background is studied. The constraints on the non-abelian field strength are shown to be in agreement with those derived from the pure spinor approach to the superstring. A covariant formalism is developed and the problem of quantisation is discussed.

On the b-antighost in the pure spinor quantization of superstrings

Recently Berkovits has constructed a picture raised, compound field bB which is used to compute higher loop amplitudes in the pure spinor approach of superstrings. On the other hand, in the twisted and gauge-fixed, superembedding approach with n=2 world-sheet (w.s.) supersymmetry that reproduces the pure spinor formulation, a field b appears quite naturally as the current of one of the two twisted charges of the w.s. supersymmetry, the other being the BRST charge. In this Letter we study the relation between b and bB. We shall show that bZ, where Z is a picture raising operator, and bB belong to the same BRST cohomological class. This result is of importance since it implies that the cumbersome singularity which is present in b, is in fact harmless if b is combined with Z.

Covariant one-loop amplitudes in [formula omitted

We generalize to the eleven-dimensional superparticle Berkovits' prescription for loop computations in the pure spinor approach to covariant quantization of the superstring. Using these ten- and eleven-dimensional results, we compute covariantly the following one-loop amplitudes: C ∧ X 8 in M-theory; B ∧ X 8 in type II string theory and F 4 in type I. We also verify the consistency of the formalism in eleven dimensions by recovering the correct classical action from tree-level amplitudes. As the superparticle is only a first approximation to the supermembrane, we comment on the possibility of extending this construction to the latter. Finally, we elaborate on the relationship between the present BRST language and the spinorial cohomology approach to corrections of the effective action.

Spinor approach to the rotation and reflection groups

We try to render the concept of a spinor less abstract by explaining that it is thecomplete image of a rotation, rather than that of some vector of Euclidean space.We take several construction paths through the landscape of the rotation group,which should allow the reader to become globally acquainted with this subjectmatter.

Spin-orbit splitting for single-particle and single-hole energies: Interplay of relativity and core polarization.

The effects of core polarization on the single-particle energies and single-hole energies relative to a closed major shell core are considered. Not only the familiar two-particle--one-hole and three-particle-- two-hole diagrams but also Hartree-Fock insertions are included. We further consider the effects of the Dirac spinor approach for nucleons inside the nucleus with a Dirac effective mass less than the bare mass and also for nucleons at the surface with an effective mass essentially equal to the bare mass. This relativistic effect tends to reduce the spin-orbit splitting of particle states as compared to that for hole states, which corresponds to the empirical data. The calculations are performed employing modern versions of the Bonn one-boson-exchange potentials with various strengths for the tensor component.

Diracian theory of fields — Spinogravitation

The paper involves two essential parts. First, using the modern spinor approach, we prove that Maxwell and Yang-Mills fields, can be described by means of tensor products of some Dirac fields. An analogous method, using some properties of the complete lifts of Yano and Ishihara, from the space time manifold to its tangent bundle allows us to consider the Einstein equations (exterior case) as a consequence of Dirac equations for some spin 1 and massless spinor field. There is a natural association with gravitational field and some electromagnetic field. Our methods use two classical isomorphisms, essentially: the Kähler-Atiyah isomorphism between a Clifford algebra and an exterior algebra, and the isomorphism of the tensor product of a spinor space and its dual with the Clifford algebra. The application of the K-A isomorphism to spinors needs a symmetry breaking by means of some subgroup of the pseudo-orthogonal group, called “spinoriality group” defined with many details in [6b]. Note also in the second part the role of the harmonic coordinates. The reader has to know essential results about differential geometry and spinor bundles. The complete references are in [6b].

A spinor approach to the propagation in self-focusing fibers

The Helmholtz equation for the propagation of an e.m. wave in a self-focusing fiber is reduced to a Pauli-type two-component equation in the paraxial approximation. Thefiber spinor components are then shown to be related to the ray height and reduced slope. Finally, comments on the meaning of the paraxial approximation are presented.

Transfer matrix spectrum for the finite-width Ising model with adjustable boundary conditions: Exact solution

Using the spinor approach, we calculate exactly the complete spectrum of the transfer matrix for the finite-width, planar Ising model with adjustable boundary conditions. Specifically, in order to control the boundary conditions, we consider an Ising model wrapped around the cylinder, and introduce along the axis a “seam” of defect bonds of variable strength. Depending on the boundary conditions used, the mass gap is found to vanish algebraically or exponentially with the size of the system. These results are compared with recent numerical simulations, and with random-walk and capillary-wave arguments.

A Spinor Approach to Poincare Gauge Theory: In a Case of Massless Lorenz Gauge Field

On formule un formalisme spinoriel pour la theorie de jauge de Poincare dans le vide en approximation lineaire avec la condition α+(2|3) a=β-(2|3) a=γ+(3|2) a=0. Le formalisme spinoriel introduit la notion d'une tetrade nulle et est utilise pour etudier les champs sans masse. L'existence des modes sans masse dans un champ de jauge de Lorentz est etudiee dans la condition ci-dessus par deux groupes. L'un d'eux trouve deux modes sans masse se propageant avec une energie positive. Au contraire, l'autre groupe reclame la non-existence des modes sans masse. Afin d'eliminer cette situation confuse, on applique le formalisme a l'etude d'un «champ nul» dans le champ de jauge de Lorentz

Ising model with adjustable boundary conditions: Exact results for finite lattice mass gaps.

Using the spinor approach, we calculate exactly the complete spectrum of the transfer matrix for the finite-size, planar Ising model with adjustable boundary conditions. The mass gap is found to vanish exponentially or algebraically, depending on the boundary conditions used. These results are compared with recent numerical simulations, and with random-walk and capillary-wave arguments.

Spinor approach to gravitational motion and precession

The translational and rotational equations of motion for a small rigid body in a gravitational field are combined in a single spinor equation. Besides its computational advantages, this unifies the description of gravitational interaction in classical and quantum theory. Explicit expressions for gravitational precession rates are derived.

Invariant form of an integrable system of equations for type D electrovac gravitational fields

A spinor approach of the general theory of relativity is used to obtain a complete integrable system of equations in covariant derivatives for a set of geometric observables assigned by Petrov type D electrovac gravitational fields.

Positivity of energy inn-dimensional Lorentzian spaces

The spinor approach of Witten and Nester is used to show that in ann-dimensional Lorentzian spaceVn, the conserved total energy-momentum vectorPα is nonspace-like. It is shown thatPα may be expressed in terms of an integral of an antisymmetric tensorEαβ over ann − 2-dimensional subspace at space-like infinity inVn.Eαβ is expressed in terms of a spinor field inVn and its covariant derivatives. This tensor is a generalization of that used in the discussion of five-dimensional Lorentzian spaces.

Positivity of energy in five-dimensional classical unified field theories

Five-dimensional classical unified field theories as well as in Yang-Mills theory with gauge group U(1), are described in terms of a Lorentzian five-dimensional space V5 with metric tensor y;; which admits a space-like Killing vector ξα. It is assumed that: (1) V5 has the topology of V4×S1, S1 is a circle and V4 is a four-dimensional Lorentzian space that is asymptotically flat and (2) the Einstein tensor Γαβ of V5 satisfies \(\Gamma _{\alpha \beta } u^\alpha v^\beta \leqslant 0\), where uα and vβ are future oriented time-like vectors with \(\gamma _{\alpha \beta } \upsilon ^\alpha \xi ^\beta = 0\). The spinor approach of Witten, Nester, and Moreschi and Sparling is used to show that the conserved five-dimensional energy momentum vector P; is nonspace-like. If P;=Γαβ=0 then V5 must admit a time-like Killing vector. Lichnerowicz's results then imply that V5 must be flat. A lower bound for P4 (the mass) similar to that found by Gibbons and Hull is obtained.

Positivity of energy in five dimensional classical unified field theories, II

Five dimensional classical unified field theories as well as Yang-Mills theory with gauge group U (1), are described in terms of a Lorentzian five dimensional space V 5 with metric tensor γαβ which admits a space-like Killing vector ζ α. It is assumed that: (1) V 5 has the topology of V 4 x S 1, S 1 is a circle and V 4 is a four dimensional Lorentzian space that is asymptotically flat and (2) the Einstein tensor Γ αβ of V 5 satisfies Γ αβ U α υ β ⩽ 0 where U α and υ α are future oriented time-like vectors with γ αβ υ α ζ β = 0. The spinor approach of Witten [4], Nester [3] and Moreschi and Sparling [5] is used to show that the conserved five dimensional energymomentum vector P α = ifΓ αβ = 0 then V 5 must admit a time-like Killing vector. Lichnerowicz's results [1] then imply that V 5 must be flat. A lower bound for P 4 (the mass) similar to that found by Gibbons and Hull [6] is obtained.

The geometry of theSL 2,C gauge formulation of general relativity

The formulation of Einstein’s general theory of relativity as anSL2,C gauge theory is considered. Use is made of the language of fibre bundles and general arguments are put forward in favour of theSL2,C approach to problems connected with the study of the space-time structure. The possibility of deriving the dynamics of the theory from a Yang-Mills-type Lagrangian density is discussed. Finally, the spinor approach is compared with other approaches to the problem of formulating Einstein’s theory as a gauge theory.

A spinor approach to the generalized singular electromagnetic field theory of Bel, Lapiedra, and Montserrat

A spinor approach to the generalized singular electromagnetic field of Bel, Lapiedra, and Montserrat is presented. We include new proofs of the basis existence theorems of the field theory using the method of spin coefficients, and interpretations of these theorems in terms of the optical scalars associated with a null geodesic congruence. We also exhibit general classes of solutions of the generalized singular electromagnetic field equations and classify them according to the optical properties of their associated null geodesic congruences.

A spinor approach to the Lanczos spin tensor

The van der Waerden spinor corresponding to the Lanczos spin tensor is obtained and shown to be a spinor of type (3,1). This spinor, which directly appears in the spinor equivalents of the Wey1 and Bell-Robinson tensors, is intimately related to the Petrov classification. By using the dyad formalism of Newman and Penrose this relationship is explicitly exhibited in terms of the spin coefficients, the optical scalars, and the dyad component of the Lanczos spinor. A possible physical interpretation of the Lanczos spinor is proposed.

A spinor approach to general relativity

A calculus for general relativity is developed in which the basic role of tensors is taken over by spinors. The Riemann-Christoffel tensor is written in a spinor form according to a scheme of Witten. It is shown that the curvature of empty space can be uniquely characterized by a totally symmetric four-index spinor which satisfies a first order equation formally identical with one for a zero rest-mass particle of spin two. However, the derivatives used here are covariant, so that on iteration, instead of the usual wave equation, a nonlinear “source” term appears. The case when a source-free electromagnetic field is present is also considered. (No quantization is attempted here.) The “gravitational density” tensor of Robinson and Bel is obtained in a natural way as a striking analogy with the spinor expression for the Maxwell stress tensor in the electromagnetic case. It is shown that the curvature tensor determines four gravitational principal null directions associated with flow of “gravitational density”, which supplement the two electromagnetic null directions of Synge. The invariants and Petrov type of the curvature tensor are analyzed in terms of these, and a natural classification of curvature tensors is given. An essentially coordinate-free method is outlined, by which any analytic solution of Einstein's field equations may, in principle, be found. As an elementary example the gravitational and gravitational-electromagnetic plane wave solutions are obtained.