Electromagnetic Field Theory Without Divergence Problems 2. A Least Invasively Quantized Theory

Classical electrodynamics based on the Maxwell–Born–Infeld field equations coupled with a Hamilton–Jacobi law of point charge motion is partially quantized. The Hamilton–Jacobi phase function is supplemented by a dynamical amplitude field on configuration space. Both together combine into a single complex wave function satisfying a relativistic Klein–Gordon equation that is self-consistently coupled to the evolution equations for the point charges and the electromagnetic fields. Radiation-free stationary states exist. The hydrogen spectrum is discussed in some detail. Upper bounds for Born's “aether constant” are obtained. In the limit of small velocities of and negligible radiation from the point charges, the model reduces to Schrödinger's equation with Coulomb Hamiltonian, coupled with the de Broglie–Bohm guiding equation.     

Mapping nonlinear gravity into General Relativity with nonlinear electrodynamics

We show that families of nonlinear gravity theories formulated in a metric-affine approach and coupled to a nonlinear theory of electrodynamics can be mapped into general relativity (GR) coupled to another nonlinear theory of electrodynamics. This allows to generate solutions of the former from those of the latter using purely algebraic transformations. This correspondence is explicitly illustrated with the Eddington-inspired Born–Infeld theory of gravity, for which we consider a family of nonlinear electrodynamics and show that, under the map, preserve their algebraic structure. For the particular case of Maxwell electrodynamics coupled to Born–Infeld gravity we find, via this correspondence, a Born–Infeld-type nonlinear electrodynamics on the GR side. Solving the spherically symmetric electrovacuum case for the latter, we show how the map provides directly the right solutions for the former. This procedure opens a new door to explore astrophysical and cosmological scenarios in nonlinear gravity theories by exploiting the full power of the analytical and numerical methods developed within the framework of GR.     

Born–Infeld–Chern–Simons theory: Hamiltonian embedding,duality and bosonization

In this paper we study in detail the equivalence of the recently introduced Born–Infeld self-dual model to the Abelian Born–Infeld–Chern–Simons model in 2+1 dimensions. We first apply the improved Batalin, Fradkin and Tyutin scheme, to embed the Born–Infeld self-dual model to a gauge system and show that the embedded model is equivalent to Abelian Born–Infeld–Chern–Simons theory. Next, using Buscher's duality procedure, we demonstrate this equivalence in a covariant Lagrangian formulation and also derive the mapping between the n-point correlators of the (dual) field strength in Born–Infeld–Chern–Simons theory and of basic field in Born–Infeld self-dual model. Using this equivalence, the bosonization of a massive Dirac theory with a non-polynomial Thirring type current–current coupling, to leading order in (inverse) fermion mass is also discussed. We also rederive it using a master Lagrangian. Finally, the operator equivalence between the fermionic current and (dual) field strength of Born–Infeld–Chern–Simons theory is deduced at the level of correlators and using this the current–current commutators are obtained.     

The Born–Infeld sphaleron

We study the SU(2) electroweak model in which the standard Yang–Mills coupling is supplemented by a Born–Infeld term. The deformation of the sphaleron and bisphaleron solutions due to the Born–Infeld term is investigated and new branches of solutions are exhibited. Especially, we find a new branch of solutions connecting the Born–Infeld sphaleron to the first solution of the Kerner–Gal'tsov series.     

Invariant length scale in relativistic kinematics: lessons from Dirichlet branes

Dirac–Born–Infeld theory is shown to possess a hidden invariance associated with its maximal electric field strength. The local Lorentz symmetry O(1,n) on a Dirichlet-n-brane is thereby enhanced to an O(1,n)×O(1,n) gauge group, encoding both an invariant velocity and acceleration (or length) scale. The presence of this enlarged gauge group predicts consequences for the kinematics of observers on Dirichlet branes, with admissible accelerations being bounded from above. An important lesson is that the introduction of a fundamental length scale into relativistic kinematics does not enforce a deformation of Lorentz boosts, as one might assume naively. The exhibited structures further show that Moffat's non-symmetric gravitational theory qualifies as a candidate for a consistent Born–Infeld type gravity with regulated solutions.     

Relativistic generalization and extension to the non-Abelian gauge theory of Feynman's proof of the Maxwell equations

R. P. Feynman showed F. J. Dyson a proof of the Lorentz force law and the homogeneous Maxwell equations, which the obtained starting from Newton's law of motion and the commutation relations between position and velocity for a single nonrelativistic particle. We formulate both a special relativistic and a general relativistic versions of Feynman's derivation. Especially in the general relativistic version we prove that the only possible fields that can consistently act on a quantum mechanical particle are scalar, gauge, and gravitational fields. We also extend Feynman's scheme to the case of non-Abelian gauge theory in the special relativistic context.     

An unexpected result in Einstein–Maxwell electrostatics

I examine a known exact static solution of the Einstein–Maxwell equations representing the exterior field of two charged masses. I find a property totally unexpected according to classical electrostatics: the electric field does not vanish between two like charges. The point where it does vanish (electrically neutral point) is found in the general case.     

Minimal Coupling and Feynman's Proof

The non quantum relativistic version of theproof of Feynman for the Maxwell equations is discussedin a framework with a minimum number of hypothesesrequired. From the present point of view it is clear that the classical equations of motioncorresponding to the gauge field interactions can bededuced from the minimal coupling rule, and we claimhere resides the essence of the proof ofFeynman.     

Geonic black holes and remnants in Eddington-inspired Born–Infeld gravity

We show that electrically charged solutions within the Eddington-inspired Born–Infeld theory of gravity replace the central singularity by a wormhole supported by the electric field. As a result, the total energy associated with the electric field is finite and similar to that found in the Born–Infeld electromagnetic theory. When a certain charge-to-mass ratio is satisfied, in the lowest part of the mass and charge spectrum the event horizon disappears, yielding stable remnants. We argue that quantum effects in the matter sector can lower the mass of these remnants from the Planck scale down to the TeV scale.     

Classical Electrodynamics of Point Charges

A simple mathematical procedure is introduced which allows redefining in an exact way divergent integrals and limits that appear in the basic equations of classical electrodynamics with point charges. In this way all divergences are at once removed without affecting the locality and the relativistic covariance of the theory, and with no need for mass renormalization. The procedure is first used to obtain a finite expression for the electromagnetic energy-momentum of the system. We show that the relativistic Lorentz-Dirac equation can be deduced from the conservation of this electromagnetic energy-momentum plus the usual mechanical term. Then we derive a finite lagrangian, which depends on the particle variables and on the actual electromagnetic potentials at a given time. From this lagrangian the equations of motion of both particles and fields can be derived via Hamilton's variational principle. The hamiltonian formulation of the theory can be obtained in a straightforward way. This leads to an interesting comparison between the resulting divergence-free expression of the hamiltonian functional and the standard renormalization rules for perturbative quantum electrodynamics.     

Shock wave polarizations and optical metrics in the Born and the Born–Infeld electrodynamics

We analyze the behavior of shock waves in nonlinear theories of electrodynamics. For this, by use of generalized Hadamard step functions of increasing order, the electromagnetic potential is developed in a series expansion near the shock wave front. This brings about a corresponding expansion of the respective electromagnetic field equations which allows for deriving relations that determine the jump coefficients in the expansion series of the potential. We compute the components of a suitable gauge-normalized version of the jump coefficients given for a prescribed tetrad compatible with the shock front foliation. The solution of the first-order jump relations shows that, in contrast to linear Maxwell’s electrodynamics, in general the propagation of shock waves in nonlinear theories is governed by optical metrics and polarization conditions describing the propagation of two differently polarized waves (leading to a possible appearance of birefringence). In detail, shock waves are analyzed in the Born and Born–Infeld theories verifying that the Born–Infeld model exhibits no birefringence and the Born model does. The obtained results are compared to those ones found in literature. New results for the polarization of the two different waves are derived for Born-type electrodynamics.     

Can the macroscopic Maxwell equations be obtained from the microscopic Maxwell-Lorentz equations by performing averages?

It is shown that the usual procedures of obtaining the macroscopic Maxwell equations from the microscopic Maxwell-Lorentz equations by performing averages contain an arbitrary choice of gauge. By a suitable different choice of the gauge the so-obtained Maxwell equations can be cast back to the form of the starting Maxwell-Lorentz equations. Therefore one cannot consider the Maxwell equations to be obtainable from the Maxwell-Lorentz equations by simply performing averages. The implication of this result is that besides the electromagnetic fields produced by the moving electric charges, as given by the Maxwell-Lorentz equations, there may be some other agents that cannot be identified as some kind of motion of the electric charges and that participate in the production of the electromagnetic fields.     

Generalized Born–Infeld actions and projective cubic curves

We investigate supersymmetric Born–Infeld Lagrangians with a second non–linearly realized supersymmetry. The resulting non–linear structure is more complex than the square root present in the standard Born–Infeld action, and nonetheless the quadratic constraints determining these models can be solved exactly in all cases containing three vector multiplets. The corresponding models are classified by cubic holomorphic prepotentials. Their symmetry structures are associated to projective cubic varieties.     

Eikonal approximation to 5D wave equations as geodesic motion in a curved 4D spacetime

We first derive the relation between the eikonal approximation to the Maxwell wave equations in an inhomogeneous anisotropic medium and geodesic motion in a three dimensional Riemannian manifold using a method which identifies the symplectic structure of the corresponding mechanics. We then apply an analogous method to the five dimensional generalization of Maxwell theory required by the gauge invariance of Stueckelbergs covariant classical and quantum dynamics to demonstrate, in the eikonal approximation, the existence of geodesic motion for the flow of mass in a four dimensional pseudo-Riemannian manifold. No motion of the medium is required. These results provide a foundation for the geometrical optics of the five dimensional radiation theory and establish a model in which there is mass flow along geodesics. Finally, we discuss the interesting case of relativistic quantum theory in an anisotropic medium as well. In this case the eikonal approximation to the relativistic quantum mechanical current coincides with the geodesic flow governed by the pseudo-Riemannian metric obtained from the eikonal approximation to solutions of the Stueckelberg-Schrödinger equation. This construction provides a model for an underlying quantum mechanical structure for classical dynamical motion along geodesics on a pseudo-Riemannian manifold. The locally symplectic structure which emerges is that of Stueckelbergs covariant mechanics on this manifold.This revised version was published online in April 2005. The publishing date was inserted.     

Determination of the Electromagnetic Lagrangian from a System of Poisson Brackets

The Lagrangian and Hamiltonian formulations of electromagnetism are reviewed and the Maxwell equations are obtained from the Hamiltonian for a system of many electric charges. It is shown that three of the equations which were obtained from the Hamiltonian, namely the Lorentz force law and two Maxwell equations, can be obtained as well from a set of postulated Poisson brackets. It is shown how the results derived from these brackets can be used to reconstruct the original Lagrangian for the theory aided by some reasoning based on physical concepts.     

Hydrodynamic analogies between the equations of classical hydrodynamics and electrodynamics in electrochemistry

It is shown that the use of equations of hydrodynamics of an incompressible and compressible fluid gives similar results for a number of experimental data from the field of classical electrodynamics used in electrochemistry. The analogue of electric current in wires is a stream that creates around itself a flow of a fluid. The analogue of electric field is the acceleration of a flow, whereas the analogue of magnetic induction is the frequency of a rotational motion of the fluid. Ampere’s law in hydrodynamics describes the interaction of flows with real bodies in terms of the Zhukovsky equation. The power laws in the fluid are similar, with some distinctions, to Maxwell equations. The expansion of the equations of conservation of momentum and mass in a series in perturbations leads to wave equations also similar to the Maxwell equations for the propagation of electromagnetic waves.     

The Ampere-Neumann Electrodynamics of Metallic Conductors

This is a review of the old electrodynamics which prevailed during the first half of the 165-year history of electromagnetism. Amperes principal achievement was the deduction of his empirical force law from experiments with several current balances. Faraday then discovered electro-magnetic induction. This prompted F. E. Neumann to work out a quantitative explanation of induction based on Amperes force law. It involved the concept of the electrodynamic potential which, as we know now, is the same entity as magnetic energy. With the newtonian principle of virtual work, Neumann found his potential yielded the correct mechanical forces on metallic current circuits. Neumanns theory contains a physical quantity which today is called the magnetic vector potential and treated as a mathematical contrivance. Neumanns mutual inductance formula has become a powerful tool of inductance calculations. Maxwell made a major contribution to the Ampere-Neumann electrodynamics by developing the mean-geometric-distance method for calculating the inductance of conductors of finite cross-sections. This became particularly useful after Sommerfeld solved Neumanns double integral for parallel, straight wires. Maxwell built all of Neumanns mathematical theory into his field equations but the lingo changed. Electrodynamic potential became kinetic energy of the field; conductor element interactions became flux linkage; and so on. Maxwells equations do not contain a magnetic force law. He believed both Amperes law and the law currently in use, which was first suggested by Grassmann in 1845, were compatible with field theory. Lorentz later found that the motion of charges in vacuum obeyed only Grassmanns law and not Amperes. From then onward the old electrodynamics fell into disuse and field theory has reigned supremely ever since. Recent developments have shown the conflict between Amperes and Grassmanns law to be related to the nature, of the electric current. Conduction currents in metals obey Amperes law and convection currents in vacuum obey Grassmanns law. Both laws agree on the reaction forces between closed metallic circuits, because the relativistic contribution from the Grassmann law then integrates to zero. This fact appears to have mislead Lorentz in believing that the drifting electron in vacuum is magnetically equivalent to the current element of metals. An examination of the long debate concerning the validity of Newtons third law of motion in electromagnetism proves the Ampere-Neumann electro-dynamics to be valid for metallic circuits while the theory of special relativity and field momentum conservation are required for convecting charges in vacuum. This conclusion is strongly supported by experimental evidence. It demands a change in the concept of the metallic current element.     

A modification of Einstein–Schrödinger theory that contains both general relativity and electrodynamics

We modify the Einstein–Schrödinger theory to include a cosmological constant Λ _z which multiplies the symmetric metric, and we show how the theory can be easily coupled to additional fields. The cosmological constant Λ _z is assumed to be nearly cancelled by Schrödinger’s cosmological constant Λ _b which multiplies the nonsymmetric fundamental tensor, such that the total Λ =  Λ _z +  Λ _b matches measurement. The resulting theory becomes exactly Einstein–Maxwell theory in the limit as |Λ _z | → ∞. For |Λ _z | ~ 1/(Planck length)² the field equations match the ordinary Einstein and Maxwell equations except for extra terms which are < 10^?16 of the usual terms for worst-case field strengths and rates-of-change accessible to measurement. Additional fields can be included in the Lagrangian, and these fields may couple to the symmetric metric and the electromagnetic vector potential, just as in Einstein–Maxwell theory. The ordinary Lorentz force equation is obtained by taking the divergence of the Einstein equations when sources are included. The Einstein–Infeld–Hoffmann (EIH) equations of motion match the equations of motion for Einstein–Maxwell theory to Newtonian/Coulombian order, which proves the existence of a Lorentz force without requiring sources. This fixes a problem of the original Einstein–Schrödinger theory, which failed to predict a Lorentz force. An exact charged solution matches the Reissner–Nordström solution except for additional terms which are ~10^?66 of the usual terms for worst-case radii accessible to measurement. An exact electromagnetic plane-wave solution is identical to its counterpart in Einstein–Maxwell theory.     

Wave-Corpuscle Mechanics for Electric Charges

It is well known that the concept of a point charge interacting with the electromagnetic (EM) field has a problem. To address that problem we introduce the concept of wave-corpuscle to describe spinless elementary charges interacting with the classical EM field. Every charge interacts only with the EM field and is described by a complex valued wave function over the 4-dimensional space time continuum. A system of many charges interacting with the EM field is defined by a local, gauge and Lorentz invariant Lagrangian with a key ingredient—a nonlinear self-interaction term providing for a cohesive force assigned to every charge. An ideal wave-corpuscle is an exact solution to the Euler-Lagrange equations describing both free and accelerated motions. It carries explicitly features of a point charge and the de Broglie wave. Our analysis shows that a system of well separated charges moving with nonrelativistic velocities are represented accurately as wave-corpuscles governed by the Newton equations of motion for point charges interacting with the Lorentz forces. In this regime the nonlinearities are “stealthy” and don’t show explicitly anywhere, but they provide for the binding forces that keep localized every individual charge. The theory can also be applied to closely interacting charges as in hydrogen atom where it produces discrete energy spectrum.