Covariant Equations Research Articles

A Liouville type theorem to an extension problem relating to the Heisenberg group

We establish a Liouville type theorem for nonnegative cylindrical solutions to the extension problem corresponding to a fractional CR covariant equation on the Heisenberg group by using the generalized CR inversion and the moving plane method.

Equivalence of cosmological observables in conformally related scalar tensor theories

Scalar tensor theories can be expressed in different frames, such as the commonly-used Einstein and Jordan frames, and it is generally accepted that cosmological observables are the same in these frames. We revisit this by making a detailed side-by-side comparison of the quantities and equations in two conformally related frames, from the actions and fully covariant field equations to the linearised equations in both real and Fourier spaces. This confirms that the field and conservation equations are equivalent in the two frames, in the sense that we can always re-express equations in one frame using relevant transformations of variables to derive the corresponding equations in the other. We show, with both analytical derivation and a numerical example, that the line-of-sight integration to calculate CMB temperature anisotropies can be done using either Einstein frame or Jordan frame quantities, and the results are identical, provided the correct redshift is used in the Einstein frame ($1+z\neq1/a$).

Magnetic connections in curved spacetime

The ideal MHD theorem on the conservation of the magnetic connections between plasma elements is generalized to relativistic plasmas in curved spacetime. The connections between plasma elements, which are established by a covariant connection equation, display a particularly complex structure in curved spacetime. Nevertheless, it is shown that these connections can be interpreted in terms of magnetic field lines alone by adopting a $3+1$ foliation of spacetime.

On the Relativistic Canonical Quantum Mechanics and Field Theory of Arbitrary Spin

The new relativistic equations of motion for the particles with arbitrary spin and nonzero mass, suggested by author in years 2014–2016, are under consideration. The complete version of brief paper in J. Phys: Conf. Ser., 670 (2016) 012047(1-16) is given. The axiomatic level description of the relativistic canonical quantum mechanics of an arbitrary mass and spin has been given. The 64-dimensional ClR(0,6) algebra in terms of Dirac gamma matrices has been suggested. The interpretation of the 28-dimensional gamma matrix representation of SO(8) algebra over the field of real numbers is given. The link between the relativistic canonical quantum mechanics of the arbitrary spin and the covariant local field theory in the form of extended Foldy–Wouthuysen transformation has been found. Different methods of the Dirac equation derivation have been reviewed. The manifestly covariant field equation for an arbitrary spin that follows from the corresponding equation of relativistic canonical quantum mechanics, has been considered. The found equations are without redundant components. The partial examples for spin s=3/2 and s=2 are presented. The covariant local field theory equations for spin s = (3/2,3/2) particle-antiparticle doublet and spin s = (2,2) particle-antiparticle doublet have been introduced. The Maxwell and slightly generalized Maxwell-like equations containing mass member have been considered as well.

General treatment of quantum and classical spinning particles in external fields

We develop the general theory of spinning particles with electric and magnetic dipole moments moving in arbitrary electromagnetic, inertial and gravitational fields. Both the quantum-mechanical and classical dynamics is investigated. We start from the covariant Dirac equation extended to a spin-${\frac 12}$ fermion with anomalous magnetic and electric dipole moments and then perform the relativistic Foldy-Wouthuysen transformation. This transformation allows us to obtain the quantum-mechanical equations of motion for the physical operators in the Schr\"odinger form and to establish the classical limit of relativistic quantum mechanics. The results obtained are then compared to the general classical description of the spinning particle interacting with electromagnetic, inertial and gravitational fields. The complete agreement between the quantum mechanics and the classical theory is proven in the general case. As an application of the results obtained, we consider the dynamics of a spinning particle in a gravitational wave and analyze the prospects of using the magnetic resonance setup to find possible manifestations of the gravitational wave on spin.

Covariant Field Equations in Supergravity

AbstractCovariance is a useful property for handling supergravity theories. In this paper, we prove a covariance property of supergravity field equations: under reasonable conditions, field equations of supergravity are covariant modulo other field equations. We prove that for any supergravity there exist such covariant equations of motion, other than the regular equations of motion, that are equivalent to the latter. The relations that we find between field equations and their covariant form can be used to obtain multiplets of field equations. In practice, the covariant field equations are easily found by simply covariantizing the ordinary field equations.

A Unique Mathematical Derivation of the Fundamental Laws of Nature Based on a New Algebraic-Axiomatic (Matrix) Approach ‡

In this article, as a new mathematical approach to origin of the laws of nature, using a new basic algebraic axiomatic (matrix) formalism based on the ring theory and Clifford algebras (presented in Section 2), “it is shown that certain mathematical forms of fundamental laws of nature, including laws governing the fundamental forces of nature (represented by a set of two definite classes of general covariant massive field equations, with new matrix formalisms), are derived uniquely from only a very few axioms.” In agreement with the rational Lorentz group, it is also basically assumed that the components of relativistic energy-momentum can only take rational values. In essence, the main scheme of this new mathematical axiomatic approach to the fundamental laws of nature is as follows: First, based on the assumption of the rationality of D-momentum and by linearization (along with a parameterization procedure) of the Lorentz invariant energy-momentum quadratic relation, a unique set of Lorentz invariant systems of homogeneous linear equations (with matrix formalisms compatible with certain Clifford and symmetric algebras) is derived. Then by an initial quantization (followed by a basic procedure of minimal coupling to space-time geometry) of these determined systems of linear equations, a set of two classes of general covariant massive (tensor) field equations (with matrix formalisms compatible with certain Clifford, and Weyl algebras) is derived uniquely as well.

Classical and quantum aspects of particle propagation in external gravitational fields

In the study of covariant wave equations, linear gravity manifests itself through the metric deviation [Formula: see text] and a two-point vector potential [Formula: see text] itself constructed from [Formula: see text] and its derivatives. The simultaneous presence of the two gravitational potentials is noncontradictory. Particles also assume the character of quasiparticles and [Formula: see text] carries information about the matter with which it interacts. We consider the influence of [Formula: see text] on the dispersion relations of the particles involved, the particles’ motion, quantum tunneling through a horizon, radiation, energy–momentum dissipation and flux quantization.

Covariant chiral kinetic equation in the Wigner function approach

The covariant chiral kinetic equation (CCKE) is derived from the 4-dimensional Wigner function by an improved perturbative method under the static equilibrium conditions. The chiral kinetic equation in 3-dimensions can be obtained by intergation over the time component of the 4-momentum. There is freedom to add more terms to the CCKE allowed by conservation laws. In the derivation of the 3-dimensional equation, there is also freedom to choose coefficients of some terms in $dx_{0}/d\tau$ and $d\mathbf{x}/d\tau$ ($\tau$ is a parameter along the worldline, and $(x_{0},\mathbf{x})$ denotes the time-space position of a particle) whose 3-mometum integrals are vanishing. So the 3-dimensional chiral kinetic equation derived from the CCKE is not uniquely determined in the current approach. To go beyond the current approach, one needs a new way of building up the 3-dimensional chiral kinetic equation from the CCKE or directly from covariant Wigner equations.

Remarks on a gauge theory for continuous spin particles

We discuss in a systematic way the gauge theory for a continuous spin particle proposed by Schuster and Toro. We show that it is naturally formulated in a cotangent bundle over Minkowski spacetime where the gauge field depends on the spacetime coordinate {x^mu } and on a covector eta _mu . We discuss how fields can be expanded in eta _mu in different ways and how these expansions are related to each other. The field equation has a derivative of a Dirac delta function with support on the eta -hyperboloid eta ^2+1=0 and we show how it restricts the dynamics of the gauge field to the eta -hyperboloid and its first neighbourhood. We then show that on-shell the field carries one single irreducible unitary representation of the Poincaré group for a continuous spin particle. We also show how the field can be used to build a set of covariant equations found by Wigner describing the wave function of one-particle states for a continuous spin particle. Finally we show that it is not possible to couple minimally a continuous spin particle to a background abelian gauge field, and we make some comments about the coupling to gravity.

Polysymplectic formulation for topologically massive Yang–Mills field theory

We analyze the De Donder–Weyl covariant field equations for the topologically massive Yang–Mills theory. These equations are obtained through the Poisson–Gerstenhaber bracket described within the polysymplectic framework. Even though the Lagrangian defining the system of our interest is singular, we show that by appropriately choosing the polymomenta one may obtain an equivalent regular Lagrangian, thus avoiding the standard analysis of constraints. Further, our simple treatment allows us to only consider the privileged [Formula: see text]-forms in order to obtain the correct field equations, in opposition to certain examples found in the literature.

Covariant version of Verlinde’s emergent gravity

A generally covariant version of Erik Verlinde's emergent gravity model is proposed. The Lagrangian constructed here allows an improved interpretation of the underlying mechanism. It suggests that de-Sitter space is filled with a vector-field that couples to baryonic matter and, by dragging on it, creates an effect similar to dark matter. We solve the covariant equation of motion in the background of a Schwarzschild space-time and obtain correction terms to the non-covariant expression. Furthermore, we demonstrate that the vector field can also mimic dark energy.

Weyl invariance for generalized supergravity backgrounds from the doubled formalism

It has recently been shown that a set of the generalized type IIB supergravity equations follows from the requirement of kappa symmetry of the type IIB Green-Schwarz superstring theory defined on an arbitrary background. In this paper, we show that the whole bosonic part of the generalized type II supergravity equations can be reproduced from the T-duality covariant equations of motion of the double field theory by choosing a non-standard solution of the strong constraint. Then, by using the doubled formalism, we show the Weyl invariance of the bosonic string sigma model on a generalized gravity background. According to the dual-coordinate dependence of the dilaton, the Fradkin-Tseytlin term nicely removes the Weyl anomaly. This result seems likely to support that string theories can be consistently defined on arbitrary generalized supergravity backgrounds.

Microscopic theory of the refractive index

We examine the refractive index from the viewpoint of modern first-principles materials physics. We first argue that the standard formula, n2=ɛrμr, is generally in conflict with fundamental principles on the microscopic level. Instead, it turns out that an allegedly approximate relation, n2=ɛr, which is already being used for most practical purposes, can be justified theoretically at optical wavelengths. More generally, starting from the fundamental, Lorentz-covariant electromagnetic wave equation in materials as used in plasma physics, we rederive a well-known, three-dimensional form of the wave equation in materials and thereby clarify the connection between the covariant fundamental response tensor and the various Cartesian tensors used to describe optical properties. Finally, we prove a general theorem by which the fundamental, covariant wave equation can be reformulated concisely in terms of the microscopic dielectric tensor.

Kerr-Schild–Kundt metrics are universal

We define (non-Einsteinian) universal metrics as the metrics that solve the source-free covariant field equations of generic gravity theories. Here, extending the rather scarce family of universal metrics known in the literature, we show that the Kerr-Schild–Kundt class of metrics are universal. Besides being interesting on their own, these metrics can provide consistent backgrounds for quantum field theory at extremely high energies.

Analytical solutions in rotating linear dilaton black holes: Resonant frequencies, quantization, greybody factor, and Hawking radiation

Charged massive scalar fields are studied in the gravitational, electromagnetic, dilaton, and axion fields of rotating linear dilaton black holes. In this geometry, we separate the covariant Klein-Gordon equation into radial and angular parts and obtain the exact solutions of both the equations in terms of the confluent Heun functions. Using the radial solution, we study the problems of resonant frequencies, entropy/area quantization, and greybody factor. We also analyze the behavior of the wave solutions near the event horizon of the rotating linear dilaton black hole and derive its Hawking temperature via the Damour-Ruffini-Sannan method.

Anomalous magnetohydrodynamics in the extreme relativistic domain

The evolution equations of anomalous magnetohydrodynamics are derived in the extreme relativistic regime and contrasted with the treatment of hydromagnetic nonlinearities pioneered by Lichnerowicz in the absence of anomalous currents. In particular we explore the situation where the conventional vector currents are complemented by the axial-vector currents arising either from the pseudo-Nambu-Goldstone bosons of a spontaneously broken symmetry or because of finite fermionic density effects. After expanding the generally covariant equations in inverse powers of the conductivity, the relativistic analog of the magnetic diffusivity equation is derived in the presence of vortical and magnetic currents. While the anomalous contributions are generally suppressed by the diffusivity, they are shown to disappear in the perfectly conducting limit. When the flow is irrotational, boost invariant and with vanishing four-acceleration, the corresponding evolution equations are explicitly integrated so that the various physical regimes can be directly verified.

From smooth curves to universal metrics

A special class of metrics, called universal metrics, solve all gravity theories defined by covariant field equations purely based on the metric tensor. Since we currently lack the knowledge of what the full of quantum-corrected field equations of gravity are at a given microscopic length scale, these metrics are particularly important in understanding quantum fields in curved backgrounds in a consistent way. But, finding explicit universal metrics has been a hard problem as there does not seem to be a procedure for it. In this work, we overcome this difficulty and give a construction of universal metrics of d-dimensional spacetime from curves constrained to live in a (d-1)-dimensional Minkowski spacetime or a Euclidean space.

Pressure gradient errors in a covariant method of implementing the σ-coordinate: idealized experiments and geometric analysis

AbstractA new approach is proposed to use the covariant scalar equations of the σ-coordinate (the covariant method), in which the pressure gradient force (PGF) has only one term in each horizontal momentum equation, and the PGF errors are much reduced in the computational space. In addition, the validity of reducing the PGF errors by this covariant method in the computational and physical space over steep terrain is investigated. First, the authors implement a set of idealized experiments of increasing terrain slope to compare the PGF errors of the covariant method and those of the classic method in the computational space. The results demonstrate that the PGF errors of the covariant method are consistently much-reduced, compared to those of the classic method. More importantly, the steeper the terrain, the greater the reduction in the ratio of the PGF errors via the covariant method. Next, the authors use geometric analysis to further investigate the PGF errors in the physical space, and the results illu...

Dynamics of expansion of the Universe in the models with nonminimally coupled dark energy

We consider the dark energy model with barotropic equation of state, which interacts with dark matter through gravitation and another force, causing the energy-momentum exchange between them. Both components are described in approximation of ideal fluids, which are parametrized by density and equation of state parameters. Three types of interactions between dark components are considered: the interaction independent from their densities, the one proportional to density of dark energy and the one proportional to density of dark matter. The equations which describe the expansion dynamics of homogeneous and isotropic Universe and evolution of densities of both components for different values of interaction parameter are obtained on the bases of the general covariant conservation equations and Einstein's ones. For three kinds of interactions we show the existence of the range of values of parameters of dark energy for which the densities of dark components and their sum are negative. We find the conditions of positivity of density of dark energy and dark matter. The constraints on the value of parameter of interaction are derived. The dynamics of expansion of the Universe with these interactions of dark energy and dark matter is analysed.