Covariant spinor derivative associated with arbitrary linear connectedness

A covariant spinor derivative is constructed in arbitrary reference frames and also in coordinates, with specified arbitrary linear connectedness. Kazan State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 78–83, November, 1998.     

Conformal compacifications from spinor geometry

Compactified Minkowski spacetime is suggested by conformal covariance of Maxwell equations, while E. Cartan's definition of simple spinors leads to the idea of compactified momentum space. Assuming both diffeomorphic to (S ₃ × S ₁ )/Z ₂ , one may obtain in the conformally flat stereographic projection field theories both infrared and ultraviolet regularized. On the compact manifold themselves instead, Fourier integrals of wave-field oscillations would have to be replaced by Fourier series summed over indices of spherical eigenfunctions: n, l, m, m. Tentatively identifying those wave structures with spacetime itself (in the frame of Big-Bang) and/or with matter and radiation distribution, some large-scale (hydrogenic) and small-scale (lattice) space structures are conjectured.     

Projective spinor geometry and prespace

A method originally conceived by Bohm for abstracting key features of the metric geometry from an underlying spinor ordering is generalized to the projective geometry. This allows the introduction of the spinor into a projective context and the definition of an associated geometric algebra. The projective spinor may then be regarded as defining a pregeometry for the projective space.This article is dedicated to Professor David Bohm on the occasion of his 70th birthday. I would like to take this opportunity to offer him my warmest congratulations and also to express to him my sincerest appreciation for this kindness, guidance, and inspiration over many years.     

Covariant derivatives on null submanifolds

The degenerate nature of the metric on null hypersurfaces makes it difficult to define a covariant derivative on null submanifolds. Recent approaches using decomposition to define a covariant derivative on null hypersurfaces are investigated, with examples demonstrating the limitations of the methods. Motivated by Geroch’s work on asymptotically flat spacetimes, conformal transformations are used to construct a covariant derivative on null hypersurfaces, and a condition on the Ricci tensor is given to determine when this construction can be used. Several examples are given, including the construction of a covariant derivative operator for the class of spherically symmetric hypersurfaces.     

Grading of spinor bundles and gravitating matter in noncommutative geometry

The gravitating matter is studied within the framework of noncommutative geometry. The noncommutative Einstein-Hilbert action on the product of a four-dimensional manifold with discrete space gives models of matter fields coupled to the standard Einstein gravity. The matter multiplet is encoded in the Dirac operator which yields a representation of the algebra of universal forms. The general form of the Dirac operator depends on a choice of the grading of the corresponding spinor bundle. A choice is given, which leads to the nonlinear vectorσ-model coupled to the Einstein gravity.     

Covariant quantization of supersymmetric string theories: The spinor field of the Ramond-Neveu-Schwarz model

A brief review of superconformal field theory and superstrings is presented. The spacetime spinor contribution to the fermion vertex operator is constructed and the four-fermion amplitude is calculated using the differential equation method in the SO(1, 9) current algebra. The spinor field is also described as a vertex operator in the bosonization of the Ramond-Neveu-Schwarz fermions, and the two-cocycle for the fermion vertex is given.     

Covariant construction of the symmetric energy-momentum tensor of vector and spinor fields in an orthogonal frame of reference

A covariant method is devised to construct the symmetric energy-momentum tensor for vector fields in an orthogonal frame of reference. The method is then used to construct the symmetric energy-momentum tensor for spinor fields. Kazan’ University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 27–30, March, 1997.     

Gravitational interaction in a model of nonlinear spinor theory with derivatives of the fundamental fields

To eliminate the difficulties that arise when particles of high spin (J 2) are treated in nonlinear spinor field theory, it is proposed to include terms with derivatives of the fundamental fields in the self-interaction Lagrangian of the subparticles. It is shown that with this generalization the four-fermion model of nonlinear spinor theory describes the gravitational interaction of subparticles in the approximation of single-graviton exchange. A relation between the gravitational constant and the parameters of the model is found.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 79–85, August, 1977.I thank D. D. Ivanenko for a helpful discussion of this work.     

Creation of a monopole in a spinor condensate

We propose a method to create a monopole structure in a multicomponent condensate by applying the basic methods used to create vortices and solitons experimentally in single-component condensates. We also show that by using a two-component structure for a monopole, we can avoid many problems related to the previously suggested three-component monopole. We discuss the observation and dynamics of such a monopole structure, and note that the dynamics of the two-component monopole differs from the dynamics of the three-component monopole.     

Quantum quenches in a spinor condensate

We discuss the ordering of a spin-1 condensate when quenched from its paramagnetic phase to its ferromagnetic phase by reducing the magnetic field. We first elucidate the nature of the equilibrium quantum phase transition. Quenching rapidly through this transition reveals XY ordering either at a specific wave vector, or the "light-cone" correlations familiar from relativistic theories, depending on the end point of the quench. For a quench proceeding at a finite rate the ordering scale is governed by the Kibble-Zurek mechanism. The creation of vortices through growth of the magnetization fluctuations is also discussed. The long-time dynamics again depends on the end point, conserving the order parameter in a zero field, but not at a finite field, with differing exponents for the coarsening of magnetic order. The results are discussed in the light of a recent experiment by Sadler et al.     

Knots in a spinor Bose-Einstein condensate

We show that knots of spin textures can be created in the polar phase of a spin-1 Bose-Einstein condensate, and discuss experimental schemes for their generation and probe, together with their lifetime.     

Exact solutions of a one-dimensional mixture of spinor bosons and spinor fermions

The exact solutions of a one-dimensional mixture of spinor bosons and spinor fermions with δ-function interactions are studied. Some new sets of Bethe ansatz equations are obtained by using the graded nest quantum inverse scattering method. Many interesting features appear in the system. For example, the wave function has the SU(2|2) supersymmetry. It is also found that the ground state of the system is partial polarized, where the fermions form a spin singlet state and the bosons are totally polarized. From the solution of Bethe ansatz equations, it is shown that all the momentum, spin and isospin rapidities at the ground state are real if the interactions between the particles are repulsive; while the fermions form two-particle bounded states and the bosons form one large bound state, which means the bosons condensed at the zero momentum point, if the interactions are attractive. The charge, spin and isospin excitations are discussed in detail. The thermodynamic Bethe ansatz equations are also derived and their solutions at some special cases are obtained analytically.     

A classical solution of the nonlinear pure spinor models with higher derivatives

We construct pure spinor models with higher derivatives in every dimension, which is an extension of the well-known Gursey model in four dimensions, and construct instanton-like solutions.Partially supported by the Grant-in-Aid for Scientific Research No. 6140005.     

Noncommutative differential geometry with higher-order derivatives

We build a toy model of differential geometry on the real line, which includes derivatives of the second order. Such construction is possible only within the framework of noncommutative geometry. We introduce the metric and briefly discuss two simple physical models of scalar field theory and gauge theory in this geometry.Partially supported by KBN grant 2 P302 168 4.     

Covariant electrodynamics in a medium,II

The energy-momentum and angular momentum emission rates for an arbitrarily moving charge (whose speed is less than that of light in the medium) in a uniform transparent medium are calculated in manifestly covariant form. The calculations are executed for three types of stress tensor: Minkowski, Abraham, and Marx. Among other things it is found that the energy-momentum emission rates for the latter two tensors are equal and differ from that of the former. Further, the angular momentum emission rates for all three tensors are found to be equal. Only for the Marx tensor is this rate independent of the orientation of the associated asymptotic space-like surface.     

Covariant representations in a fibre bundle framework

Canonical and covariant representations of Lie groups of the semidirect product form G = N⊙K with N Abelian, are analyzed in a fibre bundle framework. We exhibit first the relationship between both kinds of representations in such framework. Two complementary methods of selecting irreducible representations from the covariant ones are developed. The first one proceeds by restriction to an invariant subspace and is exemplified in the case of massive integer spin representations of the Poincaré group. The second method takes quotients and is particularly useful when we deal with reducible but indecomposable representations. A family of stepped gauge transformations is generated when the method is used to obtain the covariant massless integer helicity representations of the Poincaré group; the electromagnetic and gravitational gauge transformations are just the first two cases of such a family.     

Normal-superfluid interaction dynamics in a spinor bose gas

Coherent behavior of spinor Bose-Einstein condensates is studied in the presence of a significant uncondensed (normal) component. Normal-superfluid exchange scattering leads to near-perfect local alignment between the spin fields of the two components. We observe that, through this spin locking, spin-domain formation in the condensate is vastly accelerated as the spin populations in the condensate are entrained by large-amplitude spin waves in the normal component.     

Raman fingerprints on a spinor BEC

We present examples of ring structures that arise in a ⁸⁷Rb spinor condensate through the application of Raman laser pulses that couple the internal spin states of the F = 2 ground state manifold. The appearance of fringes within the context of the population transfer dynamics is discussed, and several pulse protocol options are compared.