Lanczos Potential for the van Stockung Space-Time

The Lanczos Potential is a theoretical useful tool to find the conformal Weyl curvature tensor C _abcd of a given relativistic metric. In this paper we find the Lanczos potential L _abc for the van Stockung vacuum gravitational field. Also, we show how the wave equation can be combined with spinor methods in order to find this important three covariant index tensor.     

A class of field theories in two-dimensional space-time with aU 1×U 1 symmetry

The derivative coupling of massless pseudoscalar neutral particles with a charged spinor field in two-dimensional space-time is reduced to a self-interacting spinor field and a free pseudoscalar field.More generally, it is shown that any given local field theory with a conserved vector current and without massless particles can be extended to a local theory with an additional pseudoscalar field and with aU ₁×U ₁ symmetry.     

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.     

Vector-spinor space and field equations

Generalizing the work of Einstein and Mayer, it is assumed that at each point of space-time there exists a vector-spinor space with N_v vector dimensions and N_s spinor dimensions, where N_v=2k and N_s=2 ^k, k3. This space is decomposed into a tangent space with4 vector and4 spinor dimensions and an internal space with N_v–4 vector and N_s–4 spinor dimension. A variational principle leads to field equations for geometric quantities which can be identified with physical fields such as the electromagnetic field, Yang-Mills gauge fields, and wave functions of bosons and fermions.     

On “Conformal spinor geometry”: An attempt to “Understand” internal symmetry

The natural homomorphism of pure spinors corresponding to a given Clifford algebraC _2n to polarized isotropicn-planes of complex Euclidean spaceE _2n ^c is taken as a starting point for the construction of a geometry called spinor geometry where pure spinors are the only elements out of which all tensors have to be constructed (analytically as bilinear polynomials of the components of a pure spinor).C ₄ andC ₆ spinor geometry are analyzed, but it seems that C₈ spinor geometry is necessary to construct Minkowski spaceM ^3,1.C ₆ spinor field equations give rise in Minkowski space to a pair of Dirac equations (for conformal semispinors) presenting ansu(2) internal symmetry algebra. Mass is generated by breaking spontaneously the originalO(4,2) symmetry of the spinor equation.Invited talk presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.     

Is hypercharge U(1) in Weinberg's model redundant?

The U₁ × U₁(loc) × SU₂(loc) invariant Weinberg lagrangian can be reduced to a U₁ × SU₂(loc) invariant lagrangian without hypercharge group involving only a single left-handed Weyl spinor field and an isotriplet gauge vector field. Spontaneous symmetry breakdown is induced by condensation of fermion pairs. Charge appears simply as “isospin charge”.     

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.     

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.     

The classical spin-1/2 tachyon field

We present the classical theory of a set of two spinor fields patterned closely after the Dirac theory in two-component form, but each field obeys a modified Klein-Gordon equation, in which the sign ofm ² has been changed. The solution contains parts with real and imaginary frequencies, that contribute differently to conserved quantities such as the charge and the energy-momentum vector. We also show how the minimal interaction with the electromagnetic field is obtained.     

Anti-de Sitter BPS black holes in N=2 gauged supergravity

Electrically charged solutions breaking half of the supersymmetry in Anti-de Sitter four dimensional N=2 supergravity coupled to vector supermultiplets are constructed. These static black holes live in an asymptotic AdS₄ space time. The Killing spinor, i.e., the spinor for supersymmetry variation is explicitly constructed for these solutions.     

Massive gauge field in source theory. I

The vacuum polarization is calculated for a massive gauge field according to the methods of source theory. The spectral integral turns out to be convergent and the numerical coefficient, which is (1/3)(e²/4π²) in spinor electrodynamics, is here (3f/16)(g²/4π²) where f depends on the group and f = 1 for SU(2).     

On the Charge Form Factor and Eigenvalue of Fermion Energy in a Three-Dimensional Electrodynamics

A static field and self-energy of a particle are considered for a particle charge distributed throughout a 2 + 1-measurement space. The potential of the static field for r has the same asymptotics as for the delta form factor, provided an account is taken of the contribution from vacuum polarization; at the origin of coordinates, the above potential is regular. The proposed form factor allows a relation for the particle charge distribution to be derived in a closed form. The diagonal tension-tensor components of the particle-generated field are found to vanish and the particle field mass calculated using the classical method appears to be finite in the case where the proposed form factor is used. This mass coincides with that obtained through quantum calculations by the order of magnitude.     

A note onD(k, 0) Killing spinors

The equations for theD(k, 0) Killing spinor fields are integrated assuming that the left conformal curvature does not vanish and that eitherk 2, 4, 6, ..., or the Einstein vacuum field equations are satisfied.     

Neutrino production in a mixed-world cosmological model

Quantum-mechanical effects of a massless spinor field in an anisotropic metric of Bianchi type IX are analyzed. It is concluded from an analysis of the operator representing the second-quantized Hamiltonian of the field and the helicity operator that a pronounced initial anisotropy causes a change in the structure of the vacuum state of the spinor field, with the result that this state acquires a nonzero lepton charge. An analogy with a known situation in supercritical electric fields is pointed out.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 50–54, January, 1985.I wish to thank A. A. Grib and A. V. Nesteruk for useful discussions.     

Covariant extensions and the nonsymmetric unified field

The problem of generally covariant extension of Lorentz invariant field equations, by means of covariant derivatives extracted from the nonsymmetric unified field, is considered. It is shown that the contracted curvature tensor can be expressed in terms of a covariant gauge derivative which contains the gauge derivative corresponding to minimal coupling, if the universal constantp, characterizing the nonsymmetric theory, is fixed in terms of Planck's constant and the elementary quantum of charge. By this choice the spinor representation of the linear connection becomes closely related to the spinor affinity used by Infeld and Van Der Waerden in their generally covariant formulation of Dirac's equation.     

Relativistic orbits of classical charged bodies in a spherically symmetric electrostatic field

The orbits of a relativistic charged body in a static, spherically symmetric electric field are calculated and classified in the classical theory. Contrary to the nonrelativistic problem, we find that there is a limiting minimal value for the angular momentumL _c . Should the actual angular momentum of a charged test body be lower than this limit, the test particle will spiral into the central point charge instead of having (precessing) Keplerian orbits.     

On conformal covariance of spinor field equations

It is shown that the equation γ^μ_?μψ (χ) = 0 is covariant with respect to proper conformal transformations if the four-component spinor field ψ(χ) is a Cartan conformal semispinor of the first or second kind. If, instead, ψ is a Dirac spinor, then no less than two such equations are needed in a covariant field theory where the conformal group may be represented linearly. Some consequences are outlined.     

Scattering with vacuum field taken into account

We present the calculation of the photon-electron scattering by taking into account both the static and the dynamic vacuum fields. The vacuum field is represented by the configuration of the space between two uncharged conducting parallel plates which can be at rest (static vacuum field) or moving from each other at a constant velocity (dynamic vacuum field). Correction terms in the computedS-matrix and scattering cross section manifest the influence of the static and dynamic vacuum field.     

Fabrication of piezo-and pyrosensitive films of polyvinylidene fluoride by vacuum evaporation in an electric field

Films of polyvinylidene fluoride with tetrafluoroethylene several micron thick were prepared by vacuum electron-beam evaporation in a static electric field. It is shown that a polymer deposited on a negative electrode possesses a structure similar to that of uniaxially stretched and polarized films and has a high (up to 85%) content of the piezoactive β phase. The electromechanical coupling coefficient k ₃₃ and the pyrosensitivity of these samples are substantially higher than those of layers deposited under the same conditions in the absence of a field. Zh. Tekh. Fiz. 68, 101–103 (December 1998)     

Nonlinear Spinor Fields in Bianchi type-III Spacetime

Within the scope of Bianchi type-III spacetime we study the role of spinor field on the evolution of the Universe as well as the influence of gravity on the spinor field. In doing so we have considered a polynomial type of nonlinearity. In this case the spacetime remains locally rotationally symmetric and anisotropic all the time. It is found that depending on the sign of nonlinearity the models allows both accelerated and oscillatory modes of expansion. The non-diagonal components of energy-momentum tensor though impose some restrictions on metric functions and components of spinor field, unlike Bianchi type I, V and V I ₀ cases, they do not lead to vanishing mass and nonlinear terms of the spinor field.