Covariant Equations Research Articles

Lévy flights and nonlocal quantum dynamics

We develop a fully fledged theory of quantum dynamical patterns of behavior that are nonlocally induced. To this end we generalize the standard Laplacian-based framework of the Schrödinger picture quantum evolution to that employing nonlocal (pseudodifferential) operators. Special attention is paid to the Salpeter (here, m ⩾ 0) quasirelativistic equation and the evolution of various wave packets, in particular to their radial expansion in 3D. Foldy's synthesis of “covariant particle equations” is extended to encompass free Maxwell theory, which however is devoid of any “particle” content. Links with the photon wave mechanics are explored.

Covariant action and equations of motion for the eleven dimensional multiple M0-brane system

We study the properties of the covariant supersymmetric and $\ensuremath{\kappa}$-symmetric action for a system of N nearly coincident M0-branes (mM0 system) in flat 11-dimensional (11D) superspace and obtain supersymmetric equations for this dynamical system. Although a single M0-brane is the massless 11D superparticle, center-of-energy motion of the mM0 system is characterized by a non-negative constant mass $M$ constructed from the matrix fields describing the relative motion of mM0 constituents. We show that a bosonic solution of the mM0 equations can be supersymmetric iff this effective mass vanishes, ${M}^{2}=0$, and that all the supersymmetric bosonic solutions preserve just one half of the 11D supersymmetry.

On the relation between the Einstein field equations and the Jacobi–Ricci–Bianchi system

The 1 + 3 covariant equations, embedded in an extended tetrad formalism and describing a spacetime with an arbitrary energy–momentum distribution, are reconsidered. It is shown that, provided the 1 + 3 splitting is performed with respect to a generic time-like congruence with a tangent vector u, the Einstein field equations can be regarded as the integrability conditions for the Jacobi and Bianchi equations together with the Ricci equations for u. The same conclusion holds for a generic null congruence in the Newman–Penrose framework.

A SIMPLER SOLUTION OF THE NON-UNIQUENESS PROBLEM OF THE COVARIANT DIRAC THEORY

Although the standard generally covariant Dirac equation is unique in a topologically simple spacetime, it has been shown that it leads to non-uniqueness problems for the Hamiltonian and energy operators, including the non-uniqueness of the energy spectrum. These problems should be solved by restricting the choice of the Dirac gamma field in a consistent way. Recently, we proposed to impose the value of the rotation rate of the tetrad field. This is not necessarily easy to implement and works only in a given reference frame. Here, we propose that the gamma field should change only by constant gauge transformations. To get that situation, we are naturally led to assume that the metric can be put in a space-isotropic diagonal form. When this is the case, it distinguishes a preferred reference frame. We show that by defining the gamma field from the "diagonal tetrad" in a chart in which the metric has that form, the uniqueness problems are solved at once for all reference frames. We discuss the physical relevance of the metric considered and our restriction to first-quantized theory.

Octonionic Non-Abelian Gauge Theory

We have made an attempt to describe the octonion formulation of Abelian and non-Abelian gauge theory of dyons in terms of 2×2 Zorn vector matrix realization. As such, we have discussed the U(1) e ×U(1) m Abelian gauge theory and U(1)×SU(2) electroweak gauge theory and also the SU(2) e ×SU(2) m non-Abelian gauge theory in term of 2×2 Zorn vector matrix realization of split octonions. It is shown that SU(2) e characterizes the usual theory of the Yang Mill’s field (isospin or weak interactions) due to presence of electric charge while the gauge group SU(2) m may be related to the existence of ’t Hooft-Polyakov monopole in non-Abelian Gauge theory. Accordingly, we have obtained the manifestly covariant field equations and equations of motion.

Class of probe brane universes

I examine the idea that our world is a branelike object of negligible influence on the bulk spacetime. No particular action functional is specified. Instead, the probe brane dynamics is derived from the universally valid stress-energy conservation equations. As an example, I study the case of a three-sphere in the five-dimensional Minkowski spacetime. The resulting four-universe turns out to depend on the equation of state of the brane constituent matter but differs from that of general relativity. One observes that, for every general relativistic universe, there exists a brane universe of equal geometry but a slightly different equation of state. As for the stability of brane universes, I find that no equation of state leads to a stable solution. However, the situation changes if the bulk spacetime has a Euclidean signature. In this case, a class of stable cosmological solutions is obtained. The brane vibrations are guided by the covariant Klein-Gordon equation with an effective metric of the Minkowski signature. It is this effective metric that local observers actually detect. As a consequence, the detected Universe is of bouncing type, with an accelerating expanding phase.

Evolution of matter density perturbations in [formula omitted] theories of gravity with non-minimal coupling between matter and geometry

In the context of f(R) theories of gravity with non-minimal coupling between matter and geometry, we study the evolution of scalar cosmological perturbations in the metric formalism. We not only derive the perturbation equations in the longitudinal gauge, but also obtain the equation of matter density perturbations by using both the perturbed covariant conservation equations which are subject to taking the Lagrangian density of matter as opposite to the energy density of a perfect fluid, and the subhorizon approximation. Furthermore, in order to investigate the behavior of matter density perturbations, we apply the obtained perturbation equation to a class of models. In these models, it is shown that 1) the stronger the non-minimal coupling between matter and geometry is, the more the growth of matter density perturbations slows down; 2) the small scale perturbations grows faster than the large ones; and 3) the more the deviation of parameter n or λ from zero gets, the more evident the scale dependence of the growth of matter density perturbations becomes.

Geometrization of the electromagnetic field and dark matter

Fundamental concepts, symmetries and dynamic equations of the theory of dark matter are derived from the simple relation: everything in the concept of space and the concept of space in everything. It is shown that the electromagnetic field is the singlet state of the dark matter field and, hence, the last may be considered as a generalized electromagnetic field (shortly gef) and a simple solution is given to the old problem of connecting the electromagnetic field with geometric properties of the physical manifold itself. It is shown that gauge fixing renders the generalized electromagnetic field effectively massive while the Maxwell electromagnetic field remains massless. To learn more about interactions between matter and dark matter on the microscopic level (and to recognize the fundamental role of internal symmetry in this case), the general covariant Dirac equation is derived and its natural generalization is considered. The experiment is suggested to test the formulated theory.

Effects of mechanical rotation and vibration on spin currents

We discuss theoretically the generation of spin currents in both rotationally and linearly accelerated systems. The spin-orbit interaction modified by inertial effects is derived from the low energy limit of the generally covariant Dirac equation. It is shown that the spin-orbit interaction is responsible for the generation of spin currents by mechanical rotation and vibration. We also study effects of impurity scattering on the mechanically induced spin current, and calculate the spin accumulation by solving the spin diffusion equation with the spin-source term originating from the inertial effects.

Covariant Uniform Acceleration

We derive a 4D covariant Relativistic Dynamics Equation. This equation canonically extends the 3D relativistic dynamics equation , where F is the 3D force and p = m0γv is the 3D relativistic momentum. The standard 4D equation is only partially covariant. To achieve full Lorentz covariance, we replace the four-force F by a rank 2 antisymmetric tensor acting on the four-velocity. By taking this tensor to be constant, we obtain a covariant definition of uniformly accelerated motion. This solves a problem of Einstein and Planck.We compute explicit solutions for uniformly accelerated motion. The solutions are divided into four Lorentz-invariant types: null, linear, rotational, and general. For null acceleration, the worldline is cubic in the time. Linear acceleration covariantly extends 1D hyperbolic motion, while rotational acceleration covariantly extends pure rotational motion.We use Generalized Fermi-Walker transport to construct a uniformly accelerated family of inertial frames which are instantaneously comoving to a uniformly accelerated observer. We explain the connection between our approach and that of Mashhoon. We show that our solutions of uniformly accelerated motion have constant acceleration in the comoving frame. Assuming the Weak Hypothesis of Locality, we obtain local spacetime transformations from a uniformly accelerated frame K' to an inertial frame K. The spacetime transformations between two uniformly accelerated frames with the same acceleration are Lorentz. We compute the metric at an arbitrary point of a uniformly accelerated frame.We obtain velocity and acceleration transformations from a uniformly accelerated system K' to an inertial frame K. We introduce the 4D velocity, an adaptation of Horwitz and Piron s notion of "off-shell." We derive the general formula for the time dilation between accelerated clocks. We obtain a formula for the angular velocity of a uniformly accelerated object. Every rest point of K' is uniformly accelerated, and its acceleration is a function of the observer's acceleration and its position. We obtain an interpretation of the Lorentz-Abraham-Dirac equation as an acceleration transformation from K' to K.

Clebsch Variational Principles in Field Theories and Singular Solutions of Covariant Epdiff Equations

This paper introduces and studies a field theoretic analogue of the Clebsch variational principle of classical mechanics. This principle yields an alternative derivation of the covariant Euler-Poincar\'e equations that naturally includes covariant Clebsch variables via multisymplectic momentum maps. In the case of diffeomorphism groups, this approach gives a new interpretation of recently derived singular peakon solutions of Diff(R)-strand equations, and allows for the construction of singular solutions (such as filaments or sheets) for a more general class of equations, called covariant EPDiff equations. The relation between the covariant Clebsch principle and other variational principles arising in mechanics and field theories, such as Hamilton-Pontryagin principles, is explained through the introduction of a class of covariant Pontryagin variational principles in field theories.

Singlet-triplet conversion and the long-range proximity effect in superconductor-ferromagnet structures with generic spin dependent fields.

The long-range proximity effect in superconductor-ferromagnet (S/F) hybrid nanostructures is observed if singlet Cooper pairs from the superconductor are converted into triplet pairs which can diffuse into the ferromagnet over large distances. It is commonly believed that this happens only in the presence of magnetic inhomogeneities. We show that there are other sources of the long-range triplet component (LRTC) of the condensate and establish general conditions for their occurrence. As a prototypical example, we consider first a system where the exchange field and spin-orbit coupling can be treated as time and space components of an effective SU(2) potential. We derive a SU(2) covariant diffusive equation for the condensate and demonstrate that an effective SU(2) electric field is responsible for the long-range proximity effect. Finally, we extend our analysis to a generic ferromagnet and establish a universal condition for the LRTC. Our results open a new avenue in the search for such correlations in S/F structures and make a hitherto unknown connection between the LRTC and Yang-Mills electrostatics.

Fermionic covariant prolongation structure theory for multidimensional super nonlinear evolution equation

The fermionic covariant prolongation structure theory is investigated. We extend the fermionic covariant prolongation structure technique to the multidimensional super nonlinear evolution equation and present the fermionic covariant fundamental equations determining the prolongation structure. Furthermore, we investigate a (2+1)-dimensional super nonlinear Schrödinger equation and analyze its integrability by means of this prolongation structure technique. We derive its Lax representation and Bäcklund transformation. Moreover, we present a solution of this multidimensional super integrable equation.

Covariant equations of motion for test bodies in gravitational theories with general nonminimal coupling

We present a covariant derivation of the equations of motion for test bodies for a wide class of gravitational theories with nonminimal coupling, encompassing a general interaction via the complete set of 9 parity-even curvature invariants. The equations of motion for spinning test bodies in such theories are explicitly derived by means of Synge's expansion technique. Our findings generalize previous results in the literature and allow for a direct comparison to the general relativistic equations of motion of pole-dipole test bodies.

Spin currents in non-inertial frames

The covariant Dirac equation and its solutions show that rotation and acceleration can be used to generate and control spin currents.

Nucleon Compton scattering in the Dyson-Schwinger approach

We analyze the nucleon's Compton scattering amplitude in the Dyson-Schwinger/Faddeev approach. We calculate a subset of diagrams that implements the nonperturbative handbag contribution as well as all $t$-channel resonances. At the quark level, these ingredients are represented by the quark Compton vertex whose analytic properties we study in detail. We derive a general form for a fermion two-photon vertex that is consistent with its Ward-Takahashi identities and free of kinematic singularities, and we relate its transverse part to the on-shell nucleon Compton amplitude. We solve an inhomogeneous Bethe-Salpeter equation for the quark Compton vertex in rainbow-ladder truncation and implement it in the nucleon Compton scattering amplitude. The remaining ingredients are the dressed quark propagator and the nucleon's bound-state amplitude which are consistently solved from Dyson-Schwinger and covariant Faddeev equations. We verify numerically that the resulting quark Compton vertex and nucleon Compton amplitude both reproduce the $\ensuremath{\pi}\ensuremath{\gamma}\ensuremath{\gamma}$ transition form factor when the pion pole in the $t$ channel is approached.

TUNNELING FROM REISSNER-NORDSTRÖM-DE SITTER BLACK HOLE WITH A GLOBAL MONOPOLE

This paper is devoted to investigate the Hawking radiation as a tunneling phenomenon from the Reissner-Nordström-de Sitter black hole with a global monopole. We use the semiclassical WKB approximation to the general covariant charged Dirac equation and evaluate tunneling probability as well as Hawking temperature. We also study the back reaction effects of the emitted spin particles and Bekenstein-Hawking entropy corrections of fermions tunneling through horizon.

Полевые уравнения и d-тензоры термоупругого континуума с «тонкой» микроструктурой

Полевые уравнения и d-тензоры термоупругого континуума с «тонкой» микроструктурой

Making the relativistic dynamics equation covariant: explicit solutions for motion under a constant force

We derive a four-dimensional (4D) covariant relativistic dynamics equation. This equation canonically extends the 3D relativistic dynamics equation , where F is the 3D force and p = m0γv is the 3D relativistic momentum. The standard 4D equation is only partially covariant. To achieve full Lorentz covariance, we replace the four-force F by a rank 2 antisymmetric tensor acting on the four-velocity. By taking this tensor to be constant, we obtain a covariant definition of uniformly accelerated motion. This solves a problem of Einstein and Planck. We compute explicit solutions for uniformly accelerated motion. The solutions are divided into four Lorentz-invariant types: null, linear, rotational and general. For null acceleration, the worldline is cubic in time. Linear acceleration covariantly extends 1D hyperbolic motion, while rotational acceleration covariantly extends pure rotational motion.

Charged fermions tunneling from regular black holes

We study Hawking radiation of charged fermions as a tunneling process from charged regular black holes, i.e., the Bardeen and ABGB black holes. For this purpose, we apply the semiclassical WKB approximation to the general covariant Dirac equation for charged particles and evaluate the tunneling probabilities. We recover the Hawking temperature corresponding to these charged regular black holes. Further, we consider the back-reaction effects of the emitted spin particles from black holes and calculate their corresponding quantum corrections to the radiation spectrum. We find that this radiation spectrum is not purely thermal due to the energy and charge conservation but has some corrections. In the absence of charge, e = 0, our results are consistent with those already present in the literature.