Nonlinear spinor fields in Bianchi-I space: Exact self-consistent solutions

Calculations are performed to obtain exact self-consistent solutions of nonlinear spinor-field equations with self-action terms in Bianchi-I space. The latter terms are arbitrary functions of the invariant . A detailed examination is made of equations with exponential nonlinearity, when the nonlinear term in the Lagrangian of the spinor field L_n=sⁿ. Here, is the nonlinearity parameter, n>1. It is shown that these equations have finite solutions and solutions that are singular at the initial moment of time. The singularity is absent in the case of solutions that describe systems of fields for which the energy dominance condition is violated. It is further shown that if the mass parameter m0 in the spinor-field equation, expansion of Bianchi-I space becomes isotropic as t . This does not occur when m=0. Specific examples of solutions of linear and nonlinear spinor-field equations are presented.Russian University of International Fellowship. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 40–45, July, 1994.     

Dirac Equation in an Axial Coulomb Potential

We consider solutions to the Dirac equation in the presence of an external axial vector potential coupled to the spinor field psi through the interaction term . There turn out to be no bound-state energies in this system consistent with a normalizable wave function.     

On the “Equivalence” of the Maxwell and Dirac Equations

It is shown that Maxwell's equation cannot be put into a spinor form that is equivalent to Dirac's equation. First of all, the spinor in the representation of the electromagnetic field bivector depends on only three independent complex components whereas the Dirac spinor depends on four. Second, Dirac's equation implies a complex structure specific to spin 1/2 particles that has no counterpart in Maxwell's equation. This complex structure makes fermions essentially different from bosons and therefore insures that there is no physically meaningful way to transform Maxwell's and Dirac's equations into each other.     

Exact self-consistent plane-symmetric solutions of the spinor-field equation with a nonlinear term dependent on the invariant P2

Exact self-consistent plane-symmetric solutions of the spinor-field equation with zero mass parameter and a nonlinear term that is an arbitrary function of the invariant , are obtained in gravitation theory. An equation with power-law nonlinearity in which the nonlinear term in the spinor-field Lagrangian has the form L_N=λP²ⁿ, where λ is the nonlinearity parameter and n=const, is investigated in detail. It is shown that λ=−Λ²<0, n>1, the original system of Einstein and nonlinear spinor-field equations has regular solutions with a localized spinor-field energy density. Here the soliton-like configuration of the fields possesses a negative energy. Exact solutions are also obtained for the above spinor-field equation in flat spacetime, and it is demonstrated that there are no soliton-like solutions in that case. Thus it is established that the proper gravitational field plays a decisive, controlling role in the formation of soliton-type solutions of the above nonlinear spinor-field equation. Russian International Friendship University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 48–53, July, 1997.     

Self-Gravitating String-Like Configurations of Nonlinear Spinor Fields

We consider the problem of the existence of soliton-like self-gravitating cylindrically symmetric configurations of a classical spinor field with the nonlinearity F(S) ( , F is an arbitrary function). Soliton-like configurations should have, by definition, a regular axis of symmetry and a flat or string-like geometry far from the axis (i.e., an asymptotically Minkowskian metric with a possible angular defect). It is shown that these conditions can be fulfilled if F(S) is finite as S and decreases faster than S ² as S 0. The set of field equations is entirely integrated, and some explicit examples are considered. A regularizing role of gravity is discussed.     

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.     

Nonperturbative Green's function approach to quantum field theory and arbitrariness of solutions

Green's function equations are considered for interacting spinor and (pseudo) scalar fields with interactions . These equations do not determine higher many-point functions if two-point functions are given as “input.” If vertex parts are given as input, two-point functions are determined but higher many-point functions are not determined.     

Regularity of weak solutions to a two-dimensional modified Dirac-Klein-Gordon system of equations

We show that solutions to the modified Dirac-Klein-Gordon system in standard notation

On the twistor-spinors

Two interesting conformal invariants which are constant on the manifold are given for twistor-spinors on a spin manifold following the notion of a twistor-spinor associated to a twisted spin bundle. For a twisted spin bundle corresponding to a flat Hermitian vector bundle, the associated twistor-spinors admit the same conformal invariants.An analysis is made of the twistor-spinors given by , where f is a complex-valued function. There is only one case where is not a Killing spinor. An example is given of a compact spin manifold for which the situation is realized.     

High-frequency perturbations of coupled gravitational and electromagnetic fields in a weakly ionized dust

Wavelike perturbations of a system consisting of a gravitational and an electromagnetic field and a dust with a small ionized component are studied. By using the spin coefficient formalism we expand the perturbations of the various quantities characterizing the system into asymptotic series. The whole dynamics of high-frequency waves is shown to be governed by four propagation equations for the expansion coefficients of the tetrad components and of perturbations of the Weyl tensor and Maxwell tensor in each order of the expansion. The perturbations of all the other variables can be derived without integration. The propagation equations are explicitly derived and discussed in the zeroth-order (geometrical optics) and in the first-order approximation. The influence of dust and plasma on the propagation is considered in the first-order equations.     

Yet another formulation of the Dirac equation

The 2-by-2 Pauli matrix algebra is used to write the 1-by-4 Dirac field in anequivalent 2-by-2 matrix . The current 4-vectors and *_µ are then compared and the latter is shown to not be easily interpretable as a probability density, and also tocontain .     

Exact static solutions of nonlinear spinor-field equations in a Hedel universe

Exact static solutions of spinor-field equations with nonlinear terms that are arbitrary functions of the invariant S=ψψ are obtained in the external gravitational field of a Hedel universe. The specific type of nonlinear Lagrangian that produces regular and localized distributions of spinor-field energy density is discussed. Exact solutions of the original equations are also obtained in plane spacetime. Here it is shown that irrespective of the form of the nonlinear Lagrangian, the energy density of the spinor field is constant, i.e., there is no localization. This means that the external gravitational field of a Hedel universe has a definite role in forming soliton-like configurations of the nonlinear spinor field. Russian University of International Amity. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 111–116, July, 1996.     

Discontinuities of solutions of the Einstein equations. II

Right zero-vectors of the characteristic matrix of the Einstein equations are constructed on isotropic cones. Relationships for the discontinuities of two derived functions of the field in the and surfaces are indicated. Quantities describing the weak discontinuities of solutions of the gravitational field equations are constructed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 38–41, February, 1982.     

Spinor two-point functions in maximally symmetric spaces

The two-point function for spinors on maximally symmetric four-dimensional spaces is obtained in terms of intrinsic geometric objects. In the massless case, Weyl spinors in anti de Sitter space can not satisfy boundary conditions appropriate to the supersymmetric models. This is because these boundary conditions break chiral symmetry, which is proven by showing that the order parameter for a massless Dirac spinor is nonzero. We also give a coordinate-independent formula for the bispinor introduced by Breitenlohner and Freedman [1], and establish the precise connection between our results and those of Burges, Davis, Freedman and Gibbons [2].     

Green's functions of spinor field in conformally plane space—time

A complete set of solutions of the Dirac equation is found in space-time with the metric ds² = ²(d ²- dx²- dy²- dz²). For the Green's functions in$$ " align="middle" border="0"> and in$$ " align="middle" border="0"> , representations are found in the form of contour integrals from the Schwinger-De Witt nucleus over various contours.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 35–40, April, 1985.It remains to thank Yu. Yu. Vol'fengaut for discussions in the course of the present work.     

Existence of localized solutions for a classical nonlinear Dirac field

We prove the existence of stationary states for nonlinear Dirac equations of the form: $$i\gamma ^\mu \partial _\mu \psi - m\psi + F(\bar \psi \psi )\psi = 0.$$ We seek solutions which are separable in spherical coordinates and we use a shooting method for solving the associated problem of ordinary differential equations.     

Exact self-consistent plane-symmetric solutions of the equations of two interacting fields (spinor field and scalar field)

A spinor field interacting with a zero-mass neutral scalar field is considered for the case of the simplest type of direct interaction, where the interaction Lagrangian has the formL _int =1/2 ϕ_αϕ_,α F(S) whereF(S) is an arbitrary function of the spinor field invariantS=ψψ. Exact solutions of the corresponding systems of equations that take into account the natural gravitational field in a plane-symmetric metric are obtained. It is proved that the initial system of equations has regular localized soliton-type solutions only if the energy density of the zero-mass scalar field is negative as it “disengages” from interaction with the spinor field. In two-dimensional space-time the system of field equations we are studying describes the configuration of fields with constant energy densityT ₀₀ , i.e., no soliton-like solutions exist in this case. Russian People’s Friendship University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 69–75, July, 1998.     

Gauge choice in Witten's energy expression

Witten's equation can be interpreted as a gauge fixing condition for classical supergravity. We rigorously prove the existence of asymptotically constant solutions of the more general gauge condition for almost all endomorphismsA of the spin bundle. Each gives an expression for the gravitational energy similar to Witten's. These include the choice , which yields the particularly elegant energy expression first noticed by Deser.Partially supported by N. S. F. Grant MCS-82-02018 at Harvard University