Maxwell-Dirac Equations Research Articles

Four component electromagnetic fields and electrodynamics

Dirac-Maxwell equations with magnetic monopoles are generalized to electromagnetic fields by introducing fourth components to the fields and their solutions are obtained. The formalism is presented into tensor, dyonic as well as quaternionic forms and conservation theorems for the field energy and momenta are obtained involving the new contribution from the mutual interaction of the fields and currents. The generation of the standard modeste, tm andtem ofem waves is also obtained in the formalism.

General solution of the classical Schwinger model

The general solution of the massless Maxwell–Dirac equations in 1+1 dimensions, the classical version of the Schwinger model, is found. This solution serves as a good introduction in an intermediate course of mathematical physics to the study of concentrated propagating waves.

Nonperturbative quantum electrodynamics: The Lamb shift

The nonlinear integro-differential equation, obtained from the coupled Maxwell-Dirac equations by eliminating the potential Aμ, is solved by iteration rather than perturbation. The energy shift is complex, the imaginary part giving the spontaneous emission. Both self-energy and vacuum polarization terms are obtained. All results, including renormalization terms, are finite.

Global solutions for the coupled maxwell-dirac equations

This letter is a result announcement. The evolution group of coupled Maxwell-Dirac equations is intertwined with the evolution group of the free equations by a time-independent mapping on a domain which contains more than the free electromagnetic field. As a consequence, there exist global solutions with a nonzero electron field.

Weyl and conformal covariant field theories

It is shown that, by imposing covariance with respect to the 11-parameter Weyl group, most of the known renormalizable field theories, and only these, are obtained and that they admit, locally in the Minkowski spaceM3,1, a conformal group as a higher symmetry group. Globally conformal covariant Lagrangian field theories are first defined in a pseudo-Euclidean spaceM4,2, where the conformal group acts linearly. Subsequently, a compactified Minkowski spaceMc3,1 is defined where the ordinary Minkowski space is densely imbedded. Conformally invariant Lagrangians and corresponding quasi-invariant massive-field equations inM4,2,Mc3,1,M3,1 are considered and the transformation properties of the mass terms discussed. Conformal quasi-invariant field equations for spinor fields interacting with (conformal) scalar and vector fields are discussed. The spinor fields always appear in the equations of motion as doublets of Dirac spinors, both having canonical dimensions. The simplest form of minimally coupled spinor-vector field equations appears to be apt to represent intermediate boson weak interactions rather than electromagnetic ones, since the interaction always induces transitions among the elements of the spinor doublets. To obtain the familiar field equations of electrodynamics, one may introduce appropriate conformally covariant factors in the spinor-vector Lagrangian density. By imposing two conformally invariant gauge conditions on the conformal six-vector field, one obtains conformally covariant Dirac-Maxwell field equations inM3,1 for a conformal doublet of equally charged fermions (reminiscent of μ, e). By not imposing any gauge condition on the conformal vector field, two new Lorentz scalar fields,A+,A−, coupled to the spinor doublet, are predicted by the conformal theory. Their properties and possible role in mass generation are briefly discussed.

Global Solutions of the Cauchy Problem for the (Classical) Coupled Maxwell-Dirac and Other Nonlinear Dirac Equations in One Space Dimension

The existence of global solutions is proved for the Maxwell-Dirac equations, for the Thirring model (Dirac equation with vector self-interaction), for the Klein-Gordon-Dirac equations and for two Dirac equations coupled through a vector-vector interaction (Federbusch model) in one space dimension. The proof is based on charge conservation, and depends on an “a priori” estimate of ${\left \| {} \right \|_\infty }$ for the Dirac field. This estimate is obtained only on the basis of algebraical properties of the nonlinear term, and allows us to simplify the proofs of global existence. We obtain it by computing ${\partial _0}{j^1} + {\partial _1}{j^0}$, with ${\partial _0}{j^0} + {\partial _1}{j^1} = 0$ being the continuity equation which expresses charge conservation. We also prove global existence for the Thirring and Federbusch models coupled in standard form with the electromagnetic field.

Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac and other nonlinear Dirac equations in one space dimension

The existence of global solutions is proved for the Maxwell-Dirac equations, for the Thirring model (Dirac equation with vector self-interaction), for the Klein-Gordon-Dirac equations and for two Dirac equations coupled through a vector-vector interaction (Federbusch model) in one space dimension. The proof is based on charge conservation, and depends on an “a priori” estimate of ‖ ‖ ∞ {\left \| {} \right \|_\infty } for the Dirac field. This estimate is obtained only on the basis of algebraical properties of the nonlinear term, and allows us to simplify the proofs of global existence. We obtain it by computing ∂ 0 j 1 + ∂ 1 j 0 {\partial _0}{j^1} + {\partial _1}{j^0} , with ∂ 0 j 0 + ∂ 1 j 1 = 0 {\partial _0}{j^0} + {\partial _1}{j^1} = 0 being the continuity equation which expresses charge conservation. We also prove global existence for the Thirring and Federbusch models coupled in standard form with the electromagnetic field.

On the Maxwell-Dirac equations with zero magnetic field and their solution in two space dimensions

Under the assumption of a vanishing magnetic field (curl A = 0), a transformation of variables is exhibited which uncouples the Maxwell-Dirac equations. It is then shown that the Cauchy problem in two space dimensions has a unique solution for C ∞ data with compact support.

The electromagnetic string with spin

The relations betweeb a field F μν M satisfying the usual Maxwell equations and a field F μν D satisfying the symmetric Maxwell-Dirac equations, and the singular potential solving both of these is given. The action principle is formulated in both forms and the reality of the string is shown. A string with spin is constructed by placing electric charges at its end-points. The motion and interactions of the string, the relation between flux and angular momentum quantization and the passage to two-body Hamiltonians are examined.

Properties of the Solutions of the Cauchy Problem for the (Classical) Coupled Maxwell-Dirac Equations in One Space Dimension

Properties of the Solutions of the Cauchy Problem for the (Classical) Coupled Maxwell-Dirac Equations in One Space Dimension

Properties of the solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac equations in one space dimension

Solutions of the Maxwell-Dirac equations coupled through the standard electromagnetic interaction are shown to blow up at each spatial point for large times. This is used to show that these solutions do not tend asymptotically to free solutions. In addition it is used to prove that these equations do not admit a nontrivial stationary solution.

Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac equations in one space dimension

The existence of global solutions of the Cauchy problem is proved for the Maxwell-Dirac equations coupled through the standard electromagnetic interaction. The proof depends on the conservation of charge and an a priori estimate on the electromagnetic potential. The technique also applies to the Dirac-Klein-Gordon equations with Yukawa coupling.

On the Cauchy Problem for the Coupled Maxwell-Dirac Equations

The Cauchy problem for the coupled Maxwell-Dirac equations is solved within an arbitrary bounded region of space-time. An integral part of the proof is that the Cauchy problem for a cut-off version of these equations has a global solution. The analysis requires that the size of the Cauchy data or the coupling constant be suitably restricted.

Parametric dispersion relations for electron Feynman amplitudes

By extending the manifestly covariant form of the Feynman notation to the mass coordinate and its Fourier transform, the proper time coordinate (J. Schwinger, Phys. Rev. Rev. 112, 331(1961)) are derived explicitly and alternatively for the case of Dirac-Maxwell equations. An interpretation Further data for a study of multipion resonances in pi /sup +/ + d reactions observed in the Lawrence Radiation Laboratory 72-in. bubble chamber exposed to a 1.23 Bev/c pion beam from the Bevatron indicates the existence of a second neutral 3-pion resonaddee- with a mass of - approximately 550 Mev. The

The renormalization of Dirac-Maxwell equations

A previous hypothesis, that the free energy projection of the interaction term of the Dirac-Maxwell equations combines with the bare electron to form the true experimental electron and that such reformulated Dirac-Maxwell equations in terms of the experimental electron are already renormalized, is shown to be true by constructing an explicit analytic representation for the free energy operator, but it is found that a convergent quantum electrodynamics is a necessary prerequisite for its validity. Such quantum electrodynamics can be constructed with the help of the Feynman cut off or with a non local interaction term in which the structure of the Dirac-Maxwell equations is maintained. The additional result that the charge renormalization is unity is obtained.

Renormalized Dirac Maxwell equations

The Dirac Maxwell equations are reformulated by eliminating that part of the interaction term which contributes only to the mass and charge of the bare electron and then by replacing the bare mass and charge of the electron by its experimental mass and charge. Then it is shown that a renormalization scheme is contained within them. Two schemes for renormalization are described and it is shown that the first scheme, although not the same, follows the pattern of Dyson’s renormalization program.

A simple non-local quantum electrodynamics

The Dirac-Maxwell equations are modified, according to Wataghin, by displacing the fields at the point of their interaction by introducing displacement parameters. When these parameters, which are taken to be small, tend to zero the original Dirac-Maxwell equations are obtained. The original charge renormalization program, the theory of Feynman « cut-off » and Yukawa’s non-local theory of particles with internal structure and the consequent convergent quantum electrodynamics can be considered as special cases of these modified DiracMaxwell equations. By combining the methods of these processes a convergent quantum electrodynamics, in which the original Feynman graph integrals are modified by the multiplication of a simple modifying factor, is obtained. This is illustrated by the calculations of second order Feynman graphs. It is then suggested that these equations can be considered to be free from the need of charge renormalization and the actual charge renormalization is due to the use of the Born approximation.