Dirac Equation in Space–Time with Torsion

The Dirac equation in a curved space–time endowed with compatible affine connection is reconsidered. After a detailed decomposition of the total action, the equation is obtained by varying with respect to the Dirac spinor and the torsion field. The result is a known Dirac-like equation with constraints that can be interpreted as the equation of a self-interacting spin 1/2 particle in curved space–time. The scheme is then translated into the language of the 2-spinor formalism of curved space–time based on the choice of a null tetrad frame. The spinorial equation so obtained coincides with the standard one in case of no torsion, while in general it remains a nonlinear equation describing a self-interacting spin 1/2 particle. The nonlinearity is produced by the interaction of the particle with its own current that remains conserved as in the free torsion case.     

Effect of Torsion in Dirac Equation for Coulomb Potential in Robertson–Walker Space–Time

The Dirac equation with Coulomb-like potential and self-interaction term, that originates from torsion, is studied in the Robertson–Walker space–time. It is shown that the angular dependence of the equation can be separated also in presence of nonlinear terms. Under reasonable physical assumptions, the time dependence is also separated. An extended perturbative calculation can then be applied qualitatively. The conclusion is that the perturbation of the energy levels of the system, as consequence of the self-interacting term, is not relevant on physical grounds.     

Dirac Equation with Central Potential: Discrete Spectrum of the Hydrogen Atom in the Robertson–Walker Space–Time

The formulation of the Dirac equation withelectromagnetic field for a general space–time isspecialized to the Robertson–Walker metric. For aclass of physically meaningful electromagneticpotentials the angular part of the wave function separates asin the free-field case. The scheme is explicitly studiedfor a Coulomb potential. By using a realisticapproximation method one recovers the discrete energy levels of the hydrogen atom in Minkowski space.In case of static space–time, the result is exactfor zero curvature, while it is approximate for nonzerocurvature. The very good order of accuracy of the result is established by a comparison withsimilar qualitative and perturbative results.     

Radiation Reaction of the Classical Off-Shell Relativistic Charged Particle

It has been shown by Gupta and Padmanabhan that the radiation reaction force of the Abraham–Lorentz–Dirac equation can be obtained by a coordinate transformation from the inertial frame of an accelerating charged particle to that of the laboratory. We show that the problem may be formulated in a flat space of five dimensions, with five corresponding gauge fields in the framework of the classical version of a fully gauge covariant form of the Stueckelberg–Feynman–Schwinger covariant mechanics (the zero mode fields of the 0, 1, 2, 3 components correspond to the Maxwell fields). Without additional constraints, the particles and fields are not confined to their mass shells. We show that in the mass-shell limit, the generalized Lorentz force obtained by means of the retarded Green's functions for the five dimensional field equations provides the classical Abraham–Lorentz–Dirac radiation reaction terms (with renormalized mass and charge). We also obtain general coupled equations for the orbit and the off-shell dynamical mass during the evolution as well as an autonomous non-linear equation of third order for the off-shell mass. The theory does not admit radiation if the particle does not move off-shell. The structure of the equations implies that mass-shell deviation is bounded when the external field is removed.     

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.     

Field Equations of Arbitrary Spin in Space-time with Torsion

A two spinor lagrangian formulation of field equations for massive particle of arbitrary spin is proposed in a curved space-time with torsion. The interaction between fields and torsion is expressed by generalizing the situation of the Dirac equation. The resulting field equations are different (except for the spin-1/2 case) from those obtained by promoting the covariant derivatives of the torsion free equations to include torsion. The non linearity of the equations, that is induced by torsion, can be interpreted as a self-interaction of the particle. The spin-1 and spin-3/2 cases are studied with some details by translating into tensor form. There result the Proca and Rarita-Schwinger field equations with torsion, respectively. PACS numbers: 03.65.Pm; 04.20.Cv; 04.20.Fy.     

Kinetic equation for a system of supersymmetric Dirac particles in electromagnetic and gravitational fields

Using the algebra of anticommuting variables, we formulate the Liouville equation for a system of supersymmetric Dirac particles acted upon by gravitational and electromagnetic fields. A macroscopic equation is obtained for the spin tensor. A generalization of our equations is made to the case of space-time with torsion. Exact solutions are found for a Boltzmann gas and for a collisionless neutral gas in the field of plane gravitational waves.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 16–21, January, 1985.The author thanks his colleagues in the Dept. of Relativity Theory and Gravitation at Kazansk University for discussions of the results.     

Nongeneric SUSY in Spinning NUT–Kerr–Newman Space–Time

The supersymmetric extension of theNUT–Kerr–Newman (NUT–KN)space–time is investigated. Along with fourstandard supersymmetries, this type of space–timeadmits fermionic symmetry generated by the square root of the bosonic constant of motion exceptthe Hamiltonian. Such a new supersymmetry corresponds tothe Killing–Yano tensor, which plays an importantrole in solving various field equations in thisspace–time.     

A Modified Theory of Gravity with Torsion and Its Applications to Cosmology and Particle Physics

In this paper we consider the most general least-order derivative theory of gravity in which not only curvature but also torsion is explicitly present in the Lagrangian, and where all independent fields have their own coupling constant: we will apply this theory to the case of ELKO fields, which is the acronym of the German Eigenspinoren des LadungsKonjugationsOperators designating eigenspinors of the charge conjugation operator, and thus they are a Majorana-like special type of spinors; and to the Dirac fields, the most general type of spinors. We shall see that because torsion has a coupling constant that is still undetermined, the ELKO and Dirac field equations are endowed with self-interactions whose coupling constant is undetermined: we discuss different applications according to the value of the coupling constants and the different properties that consequently follow. We highlight that in this approach, the ELKO and Dirac field’s self-interactions depend on the coupling constant as a parameter that may even make these non-linearities manifest at subatomic scales.     

A Discussion on Dirac Field Theory, No-Go Theorems and Renormalizability

We study Dirac field equations coupled to electrodynamics with metric and torsion fields: we discuss how special spinorial solutions are incompatible with torsion; eventually these results will be used to sketch a discussion on the problem of renormalizability of point-like particles.     

The incompressible Navier–Stokes equations from black hole membrane dynamics

We consider the dynamics of a d+1 space–time dimensional membrane defined by the event horizon of a black brane in (d+2)-dimensional asymptotically Anti-de Sitter space–time and show that it is described by the d-dimensional incompressible Navier–Stokes equations of non-relativistic fluids. The fluid velocity corresponds to the normal to the horizon while the rate of change in the fluid energy is equal to minus the rate of change in the horizon cross-sectional area. The analysis is performed in the Membrane Paradigm approach to black holes and it holds for a general non-singular null hypersurface, provided a large scale hydrodynamic limit exists. Thus we find, for instance, that the dynamics of the Rindler acceleration horizon is also described by the incompressible Navier–Stokes equations. The result resembles the relation between the Burgers and KPZ equations and we discuss its implications.     

Properties of exact solutions for a massive classical Yang-Mills field

Exact solutions for a massive Yang-Mills field are found and solutions of classical Wong equations and quantum Dirac equations are discussed for the field configurations obtained. A procedure for constructing constant fields is given and transition to solutions of the Yang-Mills equations in the case of a massless field is discussed.Translated from Izvestriya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 96–100, May, 1986.In conclusion, the authors express their gratitude to V. Ch. Zhukovskii and V. R. Khalilov for valuable remarks and discussions.     

Yang-Mills fields permitted by abelian complete sets of first-order symmetry operators of the Dirac equation

Nonequivalent complete sets of first-order symmetry operators of the Dirac free equation determine the Yang-Mills field, permitting complete variable separation in the Dirac equations with an external Yang-Mills field. Typical representatives of the classes of permissible fields are considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 30–34, October, 1989.     

Geometrization of the physics with teleparallelism. II. Towards a fully geometric Dirac equation

In an accompanying paper (I), it is shown that the basic equations of the theory of Lorentzian connections with teleparallelism (TP) acquire standard forms of physical field equations upon removal of the constraints represented by the Bianchi identities. A classical physical theory results that supersedes general relativity and Maxwell-Lorentz electrodynamics if the connection is viewed as Finslerian. The theory also encompasses a short-range, strong, classical interaction. It has, however, an open end, since the source side of the torsion field equation is not geometric.In this paper, Kaehler's partial geometrization of the Dirac equation is taken as a starting point for the development of fully geometric Dirac equations via the correspondence principle given in I. For this purpose, Kaehler's calculus (where the spinors are differential forms) is generalized so that it also applies when the torsion is not zero. The point is then made that the forms can take values in tangent Clifford algebras rather than in tensor algebras. The basic Eigenschaft of the Kaehler calculus also is examined from the physical perspective of dimensional analysis.Geometric Dirac equations of great structural simplicity are finally inferred from the standard Dirac equation by using the aforementioned correspondence principle. The realm of application of the Dirac theory is thus enriched in principle, though only at an abstract level at this point: the standard spinors, which are scalar-valued forms in the Kaehler version of that theory, become Clifford-valued. In addition, the geometrization of the Dirac equation implies a geometrization of the Dirac current. When this current is replaced in the field equations for the torsion, the theory of Paper I becomes fully geometric.     

Three-Dimensional Numerical Analysis for Millimeter Wave TWTs

Based on Macro–particle model, the electron beam can be subdivided into 3D Macro–particles to calculate space–charge forces and 3D large–signal working equations are obtained. Finally, the numerical results for a uniform magnetic focusing field are given.     

Quantum cellular automaton theory of light

We present a quantum theory of light based on the recent derivation of Weyl and Dirac quantum fields from general principles ruling the interactions of a countable set of abstract quantum systems, without using space–time and mechanics (D’Ariano and Perinotti, 2014). In a Planckian interpretation of the discreteness, the usual quantum field theory corresponds to the so-called relativistic regime of small wave-vectors. Within the present framework the photon is a composite particle made of an entangled pair of free Weyl Fermions, and the usual Bosonic statistics is recovered in the low photon density limit, whereas the Maxwell equations describe the relativistic regime. We derive the main phenomenological features of the theory in the ultra-relativistic regime, consisting in a dispersive propagation in vacuum, and in the occurrence of a small longitudinal polarization, along with a saturation effect originated by the Fermionic nature of the photon. We then discuss whether all these effects can be experimentally tested, and observe that only the dispersive effects are accessible to the current technology via observations of gamma-ray bursts.     

Possible experimental manifestations of the torsion field

The question whether it is possible in principle to obtain experimental evidence of the existence of torsion fields is discussed. Torsion is introduced as an element of the universal gravitational interaction complementary to the metric. An equation is written which is an analog of the Pauli equation in the torsion and electromagnetic external fields. The equations of motion of a weakly relativistic particle in an external torsion field are obtained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 5–12, March, 1992.     

Dirac Field of Kaluza–Klein Theory in Weitzenböck Space

The Dirac equation in five-dimensional Weitzenbo;auck space is dervied. The effectof spin–spin interaction induced by torsion is revealed by use of the Diracequation in the weak-field situation. A comparison is made of the Dirac equationof Kaluza–Klein theory in three types of spaces. It is concluded that, from thepoint of view of simplicity, the Weitzenböck space is the most suitable one forestablishing Kaluza–Klein theory.     

Is the universe homogeneous and isotropic in the time when quark-gluon plasma exists?

In this study, quark and strange quark matter which exist in the first seconds of the early Universe have been studied in the context of general relativity to be able to obtain space–time geometry of first seconds of the early Universe. For this purpose, Einstein’s field equations for quark and strange quark matter in the non static spherically symmetric space–time have been solved by using experimental result that anisotropy parameter of quark matter is very small. We have concluded from obtained solutions that the space–time structure of first seconds of the Early Universe is homogeneous and isotropic. Also we have concluded that the color interactions of the quarks may be origin of primordial magnetic field in the early universe.