Dirac-Type Equations in a Gravitational Field,with Vector Wave Function

An analysis of the classical-quantum correspondence shows that it needs to identify a preferred class of coordinate systems, which defines a torsionless connection. One such class is that of the locally-geodesic systems, corresponding to the Levi-Civita connection. Another class, thus another connection, emerges if a preferred reference frame is available. From the classical Hamiltonian that rules geodesic motion, the correspondence yields two distinct Klein-Gordon equations and two distinct Dirac-type equations in a general metric, depending on the connection used. Each of these two equations is generally-covariant, transforms the wave function as a four-vector, and differs from the Fock-Weyl gravitational Dirac equation (DFW equation). One obeys the equivalence principle in an often-accepted sense, whereas the DFW equation obeys that principle only in an extended sense. Part of this work was done while the author was at Dipartimento di Fisica, Università di Bari and INFN Bari, Italy.     

GROUP OF TRANSFORMATIONS WITH RESPECT TO THE COUNTERPART OF RAPIDITY AND RELATED FIELD EQUATIONS

The Lorentz-group of transformations usually consists of linear transformations of the coordinates, keeping as invariant the norm of the four-vector in (Minkowski) space-time. Besides those linear transformations, one may construct different forms of nonlinear transformations of the coordinates keeping unchanged that respective invariant. In this paper we explore nonlinear transformations of second-order which have a natural interpretation within the framework of Yamaleev's concept of the counterpart of rapidity (co-rapidity). The purpose of developed concept is to show that the formulae for energy and momentum of the relativistic particle become regular near the zero-mass and speed of light states. Furthermore, in a covariant formulation, the co-rapidity is presented as a four-vector which admits an extension of the Lorentz-group of transformations. In this paper we additionally show, that in the same way as the rapidity is related to the electromagnetic field, the co-rapidity is related to the field of strengths, which are given by a four-vector. The corresponding equations of such a field are also constructed.     

A Generally Covariant Wave Equation for Grand Unified Field Theory

A generally covariant wave equation is derived geometrically for grand unified field theory. The equation states most generally that the covariant d'Alembertian acting on the vielbein vanishes for the four fields which are thought to exist in nature: gravitation, electromagnetism, weak field and strong field. The various known field equations are derived from the wave equation when the vielbein is the eigenfunction. When the wave equation is applied to gravitation the wave equation is the eigenequation of wave mechanics corresponding to Einstein's field equation in classical mechanics, the vielbein eigenfunction playing the role of the quantized gravitational field. The three Newton laws, Newton's law of universal gravitation, and the Poisson equation are recovered in the classical and nonrelativistic, weak-field limits of the quantized gravitational field. The single particle wave-equation and Klein-Gordon equations are recovered in the relativistic, weak-field limit of the wave equation when scalar components are considered of the vielbein eigenfunction of the quantized gravitational field. The Schrödinger equation is recovered in the non-relativistec, weak-field limit of the Klein-Gordon equation). The Dirac equation is recovered in this weak-field limit of the quantized gravitational field (the nonrelativistic limit of the relativistic, quantezed gravitational field when the vielbein plays the role of the spinor. The wave and field equations of O(3) electrodynamics are recovered when the vielbein becomes the relativistic dreibein (triad) eigenfunction whose three orthonormal space indices become identified with the three complex circular indices (1), (2), (3), and whose four spacetime indices are the indices of non-Euclidean spacetime (the base manifold). This dreibein is the potential dreibein of the O(3) electromagnetic field (an electromagnetic potential four-vector for each index (1), (2), (3)). The wave equation of the parity violating weak field is recovered when the orthonormal space indices of the relativistic dreibein eigenfunction are identified with the indices of the three massive weak field bosons. The wave equation of the strong field is recovered when the orthonormal space indices of the relativistic vielbein eigenfunction become the eight indices defined by the group generators of the SU (3) group.     

Einstein-aether theory with a Maxwell field: General formalism

We extend the Einstein-aether theory to include the Maxwell field in a nontrivial manner by taking into account its interaction with the time-like unit vector field characterizing the aether. We also include a generic matter term. We present a model with a Lagrangian that includes cross-terms linear and quadratic in the Maxwell tensor, linear and quadratic in the covariant derivative of the aether velocity four-vector, linear in its second covariant derivative and in the Riemann tensor. We decompose these terms with respect to the irreducible parts of the covariant derivative of the aether velocity, namely, the acceleration four-vector, the shear and vorticity tensors, and the expansion scalar. Furthermore, we discuss the influence of an aether non-uniform motion on the polarization and magnetization of the matter in such an aether environment, as well as on its dielectric and magnetic properties. The total self-consistent system of equations for the electromagnetic and the gravitational fields, and the dynamic equations for the unit vector aether field are obtained. Possible applications of this system are discussed. Based on the principles of effective field theories, we display in an appendix all the terms up to fourth order in derivative operators that can be considered in a Lagrangian that includes the metric, the electromagnetic and the aether fields.     

Causal green function in relativistic quantum mechanics

The causal Green function or Feynman propagator for the free-field Klein-Gordon equation and related singular functions, defined as distributions, are related to the causal time-boundary data. Probability densities and amplitudes are defined in terms of the solutions of the Klein-Gordon equation for a complex scalar field interacting with an electromagnetic field. The convergence of the perturbation expansion of the solution of the Klein-Gordon equation for a charged scalar particle in an external field is shown for well-behaved electromagnetic potentials. Other relativistic wave equations are discussed briefly.     

Invariance of the Massless Field Equations under Changes of the Metric

It is shown that the Maxwell equations with sources, expressed in terms of the covariant tensor field F_ijand the current density four-vector Jⁱ, are invariant under the change of the metric g_ijby g_ij = g_ij+ l_il_jif l_iis a principal null direction of F_ijand that an analogous result holds in the case of the massless Klein-Gordon equation if l_iis null and orthogonal to the gradient of the field and in the case of the null dust equations if l_iis parallel to the dust four-velocity. An elementary proof of the following generalization of the Xanthopoulos theorem is also given: Let (g_ij, F_ij) be an exact solution of the Einstein-Maxwell equations and let l_ibe a principal null direction of F_ij, then (g_ij+ l_il_j, F_ij) is also an exact solution of the Einstein-Maxwell equations if and only if (l_il_j, 0) satisfies the Einstein-Maxwell equations linearized about the background solution (g_ij, F_ij). Furthermore, analogous theorems, where the source of the gravitational field is a massless Klein-Gordon field or null dust, are presented.     

Konstruktion definiter Ausdrücke für die Teilchendichte des Klein-Gordon-Feldes

An expression for a charge density of the Klein-Gordon field is constructed which is the 0-component of a four-vector as well as positive-definite if one confines oneself to positive frequency solutions only. The latter is not the case for the usual density as was shown in a preceding paper.     

Quadratic algebras and noncommutative integration of Klein-Gordon equations in non-steckel Riemann spaces

The method of noncommutative integration of linear partial differential equations is used to solve the Klein-Gordon equations in Riemann space, in the case when the set of noncommutating symmetry operators of this equation for a quadratic algebra consists of one second-order operator and several first-order operators. Solutions that do not permit variable separation are presented.Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 83–87, May, 1995.     

Thomas Precession and the Bargmann-Michel-Telegdi Equation

A direct method showing the Thomas precession for an evolution of any vector quantity (a spatial part of a four-vector) is proposed. A useful application of this method is a possibility to trace correctly the presence of the Thomas precession in the Bargmann-Michel-Telegdi equation. It is pointed out that the Thomas precession is not incorporated in the kinematical term of the Bargmann-Michel-Telegdi equation, as it is commonly believed. When the Bargmann-Michel-Telegdi equation is interpreted in curved spacetimes, this term is shown to be equivalent to the affine connection term in the covariant derivative of the spin four-vector evolving in a gravitational field. It then contributes to the geodetic precession. The described problem is an interesting and unexpected example showing that approximate methods used in special relativity, in this case to identify the Thomas precession, can distort the true meaning of physical laws.     

Modified Scattering for the Boson Star Equation

We consider the question of scattering for the boson star equation in three space dimensions. This is a semi-relativistic Klein-Gordon equation with a cubic nonlinearity of Hartree type. We combine weighted estimates, obtained by exploiting a special null structure present in the equation, and a refined asymptotic analysis performed in Fourier space, to obtain global solutions evolving from small and localized Cauchy data. We describe the behavior of such solutions at infinity by identifying a suitable nonlinear asymptotic correction to scattering. As a byproduct of the weighted energy estimates alone, we also obtain global existence and (linear) scattering for solutions of semi-relativistic Hartree equations with potentials decaying faster than Coulomb.     

Complex energies in relativistic quantum theory

A new four-component spin-1/2 wave equation for ordinary mass is discussed. It is shown that this equation has a conserved current not easily identified with a transition probability, only pure imaginary energy states, and is covariant. A tachyon-like Klein-Gordon equation is satisfied by this equation, but rest states are explicitly constructed.     

Further Evidence against the Possibility of Faster-than-Light Particles

Arguments are added to the growing number of arguments against the possibility of faster-than-light particles. Causal loops with faster-than-light particles are discussed in the framework of non-quantum special relativity. Thought experiments with steady currents of faster-than-light particles show that the reinterpretation principle is incompatible with the causality principle. If both principles are used consequently paradoxial situations arise. These are visualized utilizing Brehme diagrams. The final conclusion is that particles cannot travel on a macroscopic scale faster than light if space and time are homogeneous and if the relativity and causality principles are valid.     

Causal interpretation of the modified Klein-Gordon equation

A consistent causal interpretation of the Klein-Gordon equation treated as a field equation has been developed, and leads to a model of entities described by the Klein-Gordon equation, i.e., spinless, massive bosons, as objectively existing fields. The question arises, however, as to whether a causal interpretation based on a particle ontology of the Klein-Gordon equation is also possible. Our purpose in this article will be to indicate, by making what we believe is a best possible attempt at developing a particle interpretation of the Klein-Gordon equation, that such an interpretation is untenable. To resolve the nonpositive-definite probability density difficulties with the Klein-Gordon equation, we modify this equation by the introduction of an evolution parameter. We base our subsequent considerations on this modified Klein-Gordon equation. Partly to motivate the need for a relativistic causal interpretation and partly to give emphasis to aspects of the causal interpretation often overlooked, we begin our article with a brief historical survey of the causal interpretation.Other work commitments prevented publication of this article in the special issue ofFoundations of Physics in honor of Prof. J. P. Vigier. I would nevertheless like to dedicate this work to Prof. Vigier in recognition of this untiring contributions to the causal interpretation in particular and to the foundations of physics in general. I take this opportunity to thank Prof. Vigier for his help during my Royal Society fellowship spent at the Institut Henri Poincaré in the academic year 1988–1989.     

Quadratic algebras applied to noncommutative integration of the Klein-Gordon equation: Four-dimensional quadratic algebras containing three-dimensional nilpotent lie algebras

The study is continued on noncommutative integration of linear partial differential equations [1] in application to the exact integration of quantum-mechanical equations in a Riemann space. That method gives solutions to the Klein-Gordon equation when the set of noncommutative symmetry operations for that equation forms a quadratic algebra consisting of one second-order operator and of first-order operators forming a Lie algebra. The paper is a continuation of [2], where a single nontrivial example is used to demonstrate noncommutative integration of the Klein-Gordon equation in a Riemann space not permitting variable separation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 89–94, March, 1995.     

Punctured Haag Duality in Locally Covariant Quantum Field Theories

We investigate a new property of nets of local algebras over 4-dimensional globally hyperbolic spacetimes, called punctured Haag duality. This property consists in the usual Haag duality for the restriction of the net to the causal complement of a point p of the spacetime. Punctured Haag duality implies Haag duality and local definiteness. Our main result is that, if we deal with a locally covariant quantum field theory in the sense of Brunetti, Fredenhagen and Verch, then also the converse holds. The free Klein-Gordon field provides an example in which this property is verified.     

Solution of a linearized kinetic model for an ultrarelativistic gas

A linearized model of the Boltzmann equation for a relativistic gas is shown to be reducible, in the ultrarelativistic limit and for (1 + 1)-dimensional problems, to a system of three uncoupled transport equations, one of which is well known. A general method for solving these equations is recalled, with a few new details, and applied to the solution of two boundary value problems. The first of these describes the propagation of an impulsive change in a half space and is shown to give an explicit example of the recently proved result that no signal can propagate with speed larger than the speed of light, according to the relativistic Boltzmann equation. The second problem deals with steady oscillations in a half space and illustrates the meaning of certain recent results concerning the dispersion relation for linear waves in relativistic gas.     

Matched solutions of nonlinear forced equations and Rossby vortices

An analytical method for solving nonlinear equations with local forcing is proposed. It is shown in an example that a nonlinear forced equation may have many solutions, which generally do not turn to the solution of a linear equation in the limit of the nonlinear term becoming small. The solution of the Korteweg-de Vries (KdV) equation with forcing is applied to the problem of topographic Rossby vortices in shear flow. Solutions of other nonlinear equations with forcing are also obtained.     

Klein-Gordon equations for energy-momentum of the relativistic particle in rapidity space

The notion of four-rapidity is defined as a four-vector with one time-like and three space-like coordinates. It is proved, the energy and momentum defined in the space of four-rapidity obey Klein-Gordon equations constrained by the classical trajectory of a relativistic particle. It is shown, for small values of a proper mass influence of the constraint is weakened and the classical motion gains features of a wave motion.     

Coupling of Tachyons to Electromagnetism

We consider a fourth-order equation implying abradyonic and a tachyonic mode of propagation for ascalar field. The electromagnetic field is introducedvia the gauge covariant derivative. We show that the interacting fourth-order equation isequivalent to a second-order Klein-Gordon equation alsominimally coupled, with the tachyon living in closedloops connected only to photon lines. The equivalence shows that the fourth-order theory isrenormalizable and unitary.     

Spinless Duffin-Kemmer-Petiau Oscillator in a Galilean Non-commutative Phase Space

We examine Galilei-invariant linear wave equations in a non-commutative phase space. Specifically, we establish and solve the Galilean covariant Duffin-Kemmer-Petiau equation for spin-0 fields in a harmonic oscillator potential. We obtain these wave equations with a Galilean covariant approach, based on a (4+1)-dimensional manifold with light-cone coordinates followed by a reduction to a (3+1)-dimensional spacetime. We find the exact wave functions and their energy levels, and we examine the effects of non-commutativity.