On the mass spectrum in Aγ 5- and SU3-invariant nonlinear field theory

A method of determining the mass spectrum in a ₅- and SU_n-invariant unquantized, nonlinear, spinor field theory is developed. The mass spectrum for a ₅-, SU₂-, and SU₃-invariant theory is considered in detail.I wish to thank Prof. D. Ivanenko, and also A. I. Naumov for their discussion of this problem.     

Gravitation and nonlinear spinor theory

In the framework of a single model of nonlinear spinor theory of matter, an attempt is made to describe the gravitational interaction. It is shown that the scattering amplitude of two primary fermions contains a contribution corresponding to one-graviton exchange. A relation between the gravitational constant and the mass of a subparticle is obtained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 75–81, September, 1974.I thank Professor D. D. Ivanenko and A. I. Naumov for their interest and helpful discussions.     

The nonlinear group representation method in the nonlinear spinor theory

Nonlinear spinor equations are derived in the paper by the nonlinear symmetry-group representation method. The basic field transforms according to the linear spinor representation of the orthochronous Lorentz group Lt. The internal symmetry group G^r is realized as a group of nonlinear transformations of the field . The invariant nonlinear spinor equations constituting the group G^r are found in terms of the covariant derivative. The group SU(2) is considered as an example.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 49–55, December, 1972.The author thanks D. D. Ivanenko and D. F. Kurdgelaidze for stimulating discussions and support.     

Spinor fields in Bianchi type-I universe

A system of minimally coupled nonlinear spinor and scalar fields within the scope of a Bianchi type-I (BI) cosmological model in the presence of a perfect fluid and a cosmological constant (Λ term) is studied, and solutions to the corresponding field equations are obtained. The problem of initial singularity and the asymptotical isotropization process of the Universe are thoroughly studied. The effect of the Λ term on the character of evolution is analyzed. It is shown that some special choice of spinor field nonlinearity generates a regular solution, but the absence of singularity results in violating the dominant energy condition in the Hawking-Penrose theorem. It is also shown that a positive Λ, which denotes an additional gravitational force in our case, gives rise to an oscillatory or a non-periodic mode of expansion of the Universe depending on the choice of problem parameter. The regular oscillatory mode of expansion violets the dominant energy condition if the spinor field nonlinearity occurs as a result of self-action, whereas, in the case of a linear spinor field or nonlinear one that occurs due to interaction with a scalar field, the dominant condition remains unbroken. A system with time-varying gravitational (G) and cosmological (Λ) constants is also studied to some extent. The introduction of magneto-fluid in the system generates nonhomogeneity in the energy-momentum tensor and can be exactly solved only under some additional condition. Though in this case, we indeed deal with all four known fields, i.e., spinor, scalar, electromagnetic, and gravitational, the over-all picture of evolution remains unchanged.     

Plane-Symmetric Solitons of Spinor and Scalar Fields

We consider a system of nonlinear spinor and scalar fields with minimal coupling in general relativity. The nonlinearity in the spinor field Lagrangian is given by an arbitrary function of the invariants generated from the bilinear spinor forms S= and P=i⁵; the scalar Lagrangian is chosen as an arbitrary function of the scalar invariant = _,^,, that becomes linear at 0. The spinor and the scalar fields in question interact with each other by means of a gravitational field which is given by a plane-symmetric metric. Exact plane-symmetric solutions to the gravitational, spinor and scalar field equations have been obtained. Role of gravitational field in the formation of the field configurations with limited total energy, spin and charge has been investigated. Influence of the change of the sign of energy density of the spinor and scalar fields on the properties of the configurations obtained has been examined. It has been established that under the change of the sign of the scalar field energy density the system in question can be realized physically iff the scalar charge does not exceed some critical value. In case of spinor field no such restriction on its parameter occurs. In general it has been shown that the choice of spinor field nonlinearity can lead to the elimination of scalar field contribution to the metric functions, but leaving its contribution to the total energy unaltered.     

Poincaré gauge invariant spinor theory and the gravitational field. II

The nonlinear equation for an abstract noncanonical 2-component Weyl spinor field — as used with the inclusion of internal symmetries in Heisenberg's nonlinear spinor theory of elementary particles — which is invariant under scale, phase, and Poincaré transformations is modified in such a way as to become invariant under spacetime dependent phase gauge and Poincaré gauge transformations. In such an equation a phase gauge field B _m , six Lorentz gauge fields A[]m and four translation gauge fields g_m have to be introduced. It is demonstrated that all these fields can be identified as certain combinations of the Weyl spinor field, and hence should be considered in a rough sense as bound states of this spinor field. In particular the electromagnetic field B_m and the gravitational field g m appear as S-states and P-states of a spinor-antispinor system. The noncanonical property and the operator character of the spinor field is essential for this result. The relation between the translation gauge field and the spinor field involves a fundamental length. In a classical geometrical interpretation this relation leads to Einstein's equation of gravitation without cosmological term in a Riemannian space without torsion if the fundamental length is identified with Planck's length. It is shown that this equation is covariant under the larger symmetry group of phase gauge and Poincaré gauge transformations. The modified nonlinear equation constructed solely from a single 2-component Weyl field hence seems to incorporate in an extremely compact way electromagnetic and gravitational interaction in addition to non-mass-zero interactions. In this equation no arbitrary dimensionless constants enter. The considerations can be generalized to Dirac spinor fields and to spinor fields involving additional interior degress of freedom.An abridged version of this paper was presented at the International Conference on Gravitation and Relativity, Copenhagen, July 1971.     

Spinor geometry

In this paper the construction of the geometry begins with the assignment of a spinor (spinor ether) and the coordinates x are constructed as a spinor product. It is shown that the corresponding space is a Friedmann space and the coordinates x are Friedmann coordinates. The system of gravitational and field equations is closed. The theory contains eight real functions which specify both the reference system and the coordinate grid. The theory admits quantization of space-time and is free of the difficulties associated with inertia and the absolute character of flat space-time.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 7–12, February, 1977.     

Derivation of Treder-type gravitational equations from a variational principle

A single variation principle for the gravitational field and nongravitating matter leading to Treder-type gravitational equations is formulated. It is shown that when Treder's theory is modified in this manner the resulting special-relativistic theory of gravitation with 16-component gravitational potentials h _µ ⁽⁾ predicts 7/12 of the perihelion advance and 1/2 of the light deflection and light delay predicted by general relativity.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 36–39, January, 1976.I thank Professor D. D. Ivanenko for a helpful discussion.     

Spinor Field Nonlinearity and Space-Time Geometry

Within the scope of Bianchi type VI,VI0,V, III, I, LRSBI and FRW cosmological models we have studied the role of nonlinear spinor field on the evolution of the Universe and the spinor field itself. It was found that due to the presence of non-trivial non-diagonal components of the energy-momentum tensor of the spinor field in the anisotropic space-time, there occur some severe restrictions both on the metric functions and on the components of the spinor field. In this report we have considered a polynomial nonlinearity which is a function of invariants constructed from the bilinear spinor forms. It is found that in case of a Bianchi type-VI space-time, depending of the sign of self-coupling constants, the model allows either late time acceleration or oscillatory mode of evolution. In case of a Bianchi VI₀ type space-time due to the specific behavior of the spinor field we have two different scenarios. In one case the invariants constructed from bilinear spinor forms become trivial, thus giving rise to a massless and linear spinor field Lagrangian. This case is equivalent to the vacuum solution of the Bianchi VI₀ type space-time. The second case allows non-vanishing massive and nonlinear terms and depending on the sign of coupling constants gives rise to accelerating mode of expansion or the one that after obtaining some maximum value contracts and ends in big crunch, consequently generating space-time singularity. In case of a Bianchi type-V model there occur two possibilities. In one case we found that the metric functions are similar to each other. In this case the Universe expands with acceleration if the self-coupling constant is taken to be a positive one, whereas a negative coupling constant gives rise to a cyclic or periodic solution. In the second case the spinor mass and the spinor field nonlinearity vanish and the Universe expands linearly in time. In case of a Bianchi type-III model the space-time remains locally rotationally symmetric all the time, though the isotropy of space-time can be attained for a large proportionality constant. As far as evolution is concerned, depending on the sign of coupling constant the model allows both accelerated and oscillatory mode of expansion. A negative coupling constant leads to an oscillatory mode of expansion, whereas a positive coupling constant generates expanding Universe with late time acceleration. Both deceleration parameter and EoS parameter in this case vary with time and are in agreement with modern concept of space-time evolution. In case of a Bianchi type-I space-time the non-diagonal components lead to three different possibilities. In case of a full BI space-time we find that the spinor field nonlinearity and the massive term vanish, hence the spinor field Lagrangian becomes massless and linear. In two other cases the space-time evolves into either LRSBI or FRW Universe. If we consider a locally rotationally symmetric BI(LRSBI) model, neither the mass term nor the spinor field nonlinearity vanishes. In this case depending on the sign of coupling constant we have either late time accelerated mode of expansion or oscillatory mode of evolution. In this case for an expanding Universe we have asymptotical isotropization. Finally, in case of a FRW model neither the mass term nor the spinor field nonlinearity vanishes. Like in LRSBI case we have either late time acceleration or cyclic mode of evolution. These findings allow us to conclude that the spinor field is very sensitive to the gravitational one.     

Extended Einstein–Cartan theory à la Diakonov: The field equations

Diakonov formulated a model of a primordial Dirac spinor field interacting gravitationally within the geometric framework of the Poincaré gauge theory (PGT). Thus, the gravitational field variables are the orthonormal coframe (tetrad) and the Lorentz connection. A simple gravitational gauge Lagrangian is the Einstein–Cartan choice proportional to the curvature scalar plus a cosmological term. In Diakonov?s model the coframe is eliminated by expressing it in terms of the primordial spinor. We derive the corresponding field equations for the first time. We extend the Diakonov model by additionally eliminating the Lorentz connection, but keeping local Lorentz covariance intact. Then, if we drop the Einstein–Cartan term in the Lagrangian, a nonlinear Heisenberg type spinor equation is recovered in the lowest approximation.     

Mass of the fundamental fermion and degeneracy of the vacuum

Within the framework of the Heisenberg variant of the unified nonlinear spinor theory of elementary particles an attempt is made to calculate the mass of the fundamental fermion (nucleon). The guiding principle is the notion of the existence of a relation between the helicity properties of particles and their masses. In this case it proves necessary to take into account the possibility of degeneracy of the vacuum of the system of interacting fields in certain quantum numbers, A program is outlined for constructing a realistic scheme of elementary particles to include their Isotropic and strange properties. The possibility of eliminating divergences from the nonlinear theory, while preserving its applicability, by means of a somewhat modified perturbation theory is briefly discussed.The author is grateful to Professor D. Ivanenko for his constant interest and helpful advice.     

A new view of the ECSK theory with a Dirac spinor

A formulation of the ECSK (Einstein-Cartan-Sciama-Kibble) theory with a Dirac spinor is given in terms of differential forms with values in exterior vector bundles associated with a fixed principalSL(2, )-bundle over a 4-manifold. In particular, tetrad fields are represented as soldering forms. In this setting, both the scalar curvature (Einstein-Hilbert) action density and the Dirac action density are well-defined polynomial functions of the soldering form and an independentSL(2,)-connection form. Thus, these densities are defined even where the tetrad field is degenerate (e.g. when fluctuations in the gravitational field are large). A careful analysis of the initial-value problem (in terms of an evolving triad field, SU(2)-connection, second-fundamental form and spinor field) reveals a first-order hyperbolic system of 27 evolution equations (not including the 8 evolution equations for the Dirac spinor) and 16 constraints. There are 10 conservation equations (due to local Poincaré invariance) which team up with some of the evolution equations to guarantee that the 16 constraints are preserved under the evolution.     

Exact plane-symmetric solutions of the nonlinear equations of a spinor field in the theory of gravitation

We obtain exact plane-symmetric solutions of the spinor field equations with a nonlinear term that is an arbitrary functions of the invariant and with the self-gravitational field taken into account. Conditions are formulated for which the initial system of Einstein's equation and the spinor field equations with a power-law nonlinearity have regular solutions with localized (negative) spinor field energy density: so-called soliton-like solutions. Exact solutions of the spinor field equations are also obtained in flat space—time in this case and it is shown that the initial system of equations does not have soliton-like solutions. Hence the self-gravitational field plays a crucial (regularizing) in soliton-like solutions of the nonlinear spinor field equations.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 63–68, August, 1995.     

Geometric significance of the spinor covariant derivative

The spinor covariant derivative through which the equations of quantum fields are generalized to include gravitational coupling has a direct and simple geometric significance. The formula for the difference of two spinor covariant derivatives taken in different order is derived geometrically; and the geometric proof of the covariant constancy of the spin-1/2 -matrices in curved space is given.     

Quantum gravity model with fundamental spinor fields

We discuss the possibility that gravitational potentials (metric, coframe and connection) may emerge as composite fields from more fundamental spinor constituents. We use the formalism of Poincaré gauge gravity as an appropriate theoretical scheme for the rigorous development of such an approach. We postulate the constitutive relations of an elastic Cosserat type continuum that models spacetime. These generalized Hooke and MacCullagh type laws consistently take into account the translational and Lorentz rotational deformations, respectively. The resulting theory extends the recently proposed Diakonov model. An intriguing feature of our theory is that in the lowest approximation it reproduces Heisenberg’s nonlinear spinor model.     

New spinor representation of Maxwell's equations. I. Generalities

A new representation of the electromagnetic field tensor has been found. In this representation it is shown that an intimate relationship exists between electromagnetism and spin; the duality rotation of the already unified theory is shown to coincide with the Touschek-Nishijima transformation of the theory of leptons. A nonlinear spinor equation equivalent to Maxwell's equations is deduced.     

Dark Matter versus Mach's Principle

Empirical and theoretical evidence show that the astrophysical problem of dark matter might be solved by a theory of Einstein-Mayer type. In this theory, up to global Lorentz rotations, the reference system is determined by the motion of cosmic matter. Thus, one is led to a Riemannian space with teleparallelism realizing a geometric version of the Mach-Einstein doctrine. The field equations of this gravitational theory contain hidden matter terms, where the existence of hidden matter is inferred solely from its gravitational effects. It is argued that, in the nonrelativistic mechanical approximation, they provide an inertia-free mechanics, where the inertial mass of a body is induced by the gravitational action of the comic masses. Interpreted from the Newtonian point of view, this mechanics shows that the effective gravitational mass of astrophysical objects depends on r such that one expects the existence of dark matter.     

On the “nonclassical many-valuedness” of the universe

Two problems connected with Dirac quantization of relativistic cosmological models are considered. The difficulties with this approach have been the lack of physical interpretation of the spinor wave functions and the difficulty in extracting the spinor square root with a variable mass term in the classical Hamiltonian. It is suggested here that the spinor wave functions may be connected with the temporal irreversibility of physical processes in the universe. The analytical difficulty of extracting the spinor square root suggests a modification of the classical theory, in analogy with the theory of the Dirac electron in an electromagnetic field.     

Soliton concept in general relativity

Soliton physics has made considerable progress in solving nonlinear problems. This paper is meant to relate the soliten concept to the stationary axisymmetric vacuum fields in general relativity. We present a functional transformation which, working as a nonlinear creation operator, generates gravitational fields of isolated sources. When applied to flat space-time (gravitational vacuum) this operation leads to a nonlinear superposition of an arbitrary number of Kerr particles. This superposition also includes the Tomimatsu-Sato fields. The functional transformations form an infinite-parameter group which contains the Kinnersley-Geroch group as a subgroup.This essay received the second award from the Gravity Research Foundation for the year 1980.-Ed.     

Algebraic structure of nonassociative spinor field

An analysis of the algebra of octonions, the algebraic structure of nonassociative spinors, is presented, and a spinor field theory that is completely identical to Dirac theory is constructed in an associative basis. A spinor covariance transformation is introduced, and it is shown that it coincides with the Poincaré group of 4-dimensional space. The field equation is introduced through a spinor invariance transformation. Constraints imposed by the field equation on the eigenvalues of the transformation generators are considered. It is proved that the particles in a system at rest which are nonzero are , the unit; , the energysign of the particle; and s ₆, one of the spin components of the particle. Tbilisi University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 59–65, November, 1998.