q-Lorentz group and braided coaddition onq-Minkowski space

We present a new version ofq-Minkowski space, which has both a coaddition law and anSL _q (2, ) decomposition. The additive structure forms a braided group rather than a quantum one. In the process, we obtain aq-Lorentz group which coacts covariantly on thisq-Minkowski space.     

Wave equations with energy-dependent potentials

We study wave equations with energy-dependent potentials. Simple analytical models are found useful to illustrate difficulties encountered with the calculation and interpretation of observables. A formal analysis shows under which conditions such equations can be handled as evolution equation of quantum theory with an energy-dependent potential. Once these conditions are met, such theory can be transformed into ordinary quantum theory. This work was supported by the agreement between IN2P3 and ASCR (collaboration no. 97-13) and by the Grant Agency of ASCR (J.M., grant No.A1048305).     

Reflection equations andq-Minkowski space algebras

We express the defining relations of theq-deformed Minkowski space algebra as well as that of the corresponding derivatives and differentials in the form of reflection equations. This formulation encompasses the covariance properties with respect to the quantum Lorentz group action in a straightforward way. Different equivalences ofq-Minkowski algebras are pointed out.On leave of absence from the St. Petersburg's Branch of the Steklov Mathematical Institute of the Russian Academy of Sciences, Spain.     

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.     

Wave equations in linear viscoelastic materials

Conditions for writing wave equations in linear viscoelastic materials are investigated. The study is restricted to the infinitesimal theory and an application is suggested in modeling ultrasound propagation in soft biological tissues. First, a general wave equation is obtained for the displacement field in an inhomogeneous medium. Second, the propagation of "the mean principal stress" (i.e., minus the arithmetical mean of the principal stresses) is examined. That quantity is particularly relevant when the force per unit area is detected at the surface of a nondissipative coupling medium. If the material is homogeneous, a wave equation is always obtained for the mean principal stress. Otherwise, supplementary conditions have to be assumed on the material and possibly on the motion. Results are illustrated by examples which present linearly elastic perfect fluids and linearly elastic Newtonian viscous fluids as particular viscoelastic materials.     

Wave equations for matter force waves

Regarding phase space—time as a physical space for quantum matter particles, the wave equations are considered in the cases of scalar, vector, and spinor fields. Electrodynamics in pulsed space—time does not contain singularities in the force fields. Expressions are found for the particle propagators in momentum subspace. Order-of-magnitude estimates of force-wave emission and absorption for atomic and nuclear structures show that force waves cannot be neglected in nuclear processes. Force waves are not emitted by particles in a constant force field; the rate of emission is proportional to the square of the rate of force variation over time.NIIVTs Research Center, M. V. Lomosov Moscow State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 32–39, July, 1994.     

Discrete dynamical equations in Minkowski space

Discrete difference equations in Minkowski space are obtained and the discrete Minkowski force is shown to be a four-vector. A transformation from a discrete dynamical equation in Minkowski space to a Lorentz-invariant difference equation in one-dimensional space is given.     

Trilocal structures. V. Wave equations

In addition to the usual centroid-time wave equation, a trilocal structure will need to satisfy two relative-time wave equations. When the trilocal wave function is expanded in tree functions, each of the three wave equations becomes an infinite matrix equation, but when the four auxiliary conditions (defined in earlier articles in this series) are introduced, each wave equation reduces to a set of 16 linear homogeneous equations in 16 unknown expansion coefficients (the first 16 coefficients in the tree expansion). The 48 linear equations, in the 16 unknownC _j , are given explicitly. Every 16-by-16 determinant, formed from any 16 of these 48 linear homogeneous equations, must vanish if the trilocal structure is to be an acceptable solution; this requirement will be used in later calculations.     

Ion-pressure equations derived from measurements in space

Cluster measurements at the bow shock, the magnetosheath, and the magnetospheric boundary layer are used to derive ion-pressure equations for hot anisotropic plasmas. It is demonstrated that both perpendicular and parallel ion pressures are well approximated by polybaric expressions is proportional to N(gamma)B(kappa), where N is the plasma density, B is the magnetic field, gamma is in the range 0.5 to 2, and kappa is between -2 and 0. The parameters derived from observations are distinctively different from those predicted by double-adiabatic theory and are shown to hold for pressure variations over 4 orders of magnitude and for a range of plasma beta (ratio of kinetic/magnetic pressures) between 10(-4) and 10. The results are relevant for simulations and theories of astrophysical, solar, interplanetary, and magnetospheric processes based on MHD equations.     

The Navier-Stokes equations in space dimension four

Solutions to the Navier-Stokes equations in four space dimensions are continuous except for a closed set whose three dimensional Hausdorff measure is finite.This research was supported in part by the National Science Foundation Grant MCS 75-23332-A02     

Construction of spinor-field equations in cosmological space

The embedding method is employed to construct the spinor-field equation in curved space. The equivalence conditions are given for spinor-field equations in the original curved 4-space and in the flat 6-space of embedding.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, Vol. 15, No. 10, pp. 20–25, October, 1972.The authors would like to express their gratitude to Professor D. Ivanenko for discussing this problem with them.     

Second-order scalar-tensor field equations in a four-dimensional space

Lagrange scalar densities which are concomitants of a pseudo-Riemannian metric-tensor, a scalar field and their derivatives of arbitrary order are considered. The most general second-order Euler-Lagrange tensors derivable from such a Lagrangian in a four-dimensional space are constructed, and it is shown that these Euler-Lagrange tensors may be obtained from a Lagrangian which is at most of second order in the derivatives of the field functions.     

Field equations for a Riemann space with torsion

It is shown that a unique system of equations suitable for the purposes of combining gravitation and electromagnetism and differing by the fact that its compatibility is assured by two identities of different order in differentiation exists in the Einstein theory of absolute parallelism. The system turns out to be irregular in the case of the usual dimensionality of space. The possibility is noted of solutions with a zero electromagnetic field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 83–87, September, 1990.     

On the Maxwell equations in a Riemann-Cartan space

Considering the vacuum polarization effect in the presence of torsion, the modifications of the Maxwell equations are explicitly computated to first order in the spin-torsion coupling constant.     

Integrals of motion for some equations onq-minkowski space

The conservation laws for second order linear equation with constant coefficients on braided linear space are derived. As an example we study conserved currents connected with symmetry operators for scalar and spinor wave equations onq-Minkowski space. Then we formulate set of conditions for integral over spatial dimensions and use the postulated integral in construction of constant of motion for arbitrary conserved current onq-Minkowski space. Presented at the 6th International Colloquium on Quantum Groups “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997.     

Killing pairs and the empty space field equations

The empty space field equations are investigated for each of the canonical forms obtained previously for the metrics of space-times admitting a surface generating Killing pair, one member of which is hypersurface orthogonal. It is found that the rational first integral of the geodesic equation, corresponding to the Killing pair, is always necessarily the ration of two linear first integrals.