Maxwell Dirac Research Articles

An ongoing big bang model in the special relativistic Maxwell–Dirac equations

An exact, analytical solution of the combined Maxwell–Dirac equation is presented. The solution is regular inside a wedgelike domain of the flat space–time. On the boundary of that domain, the solution exhibits singular behavior.

General solutions of Maxwell–Dirac equations in 1+1-dimensional space–time and a spatially confined solution

The most general solution of the system of massless Maxwell–Dirac equations in 1+1-dimensional space–time (or classical Schwinger theory) is obtained in terms of four arbitrary functions and one arbitrary constant. A particular example is furnished for which the Dirac wave function vanishes completely outside a finite spatial range. Maxwell–Dirac equations for a nonzero mass parameter are reduced to a single, real, fourth-order, nonlinear partial differential equation. A particular class of solutions for this complicated equation is provided.

A class of exact plane wave solutions of the Maxwell–Dirac equations

Exact solutions of Maxwell–Dirac equations are investigated for which the Dirac field is of the type ψ(x)=α(p)eipλxλ. In the subclass where the mass parameter m≠0, there exists no nontrivial solution of the problem. In the subclass where the mass parameter m=0, there exist infinitely many solutions inherent with arbitrary functions. Furthermore, every solution for m=0 must have a null four-current vector field associated with it.

Analysis in space-time bundles IV. Natural bundles deforming into and composed of the same invariant factors as the spin and form bundles

We study two interesting new bundles over the universal cosmos M̃ (or maximal isotropic space-time), which may be physically applicable. The treatment is from a homogeneous vector bundle point of view and uses the notation and some of the results of the treatment in Papers I–III ( S. M. Paneitz and I. E. Segal, J. Funct. Anal. 47 (1982), 78–142; 49 (1982), 335–414; 54 (1983) , 18–22)) of conventional bundles over M̃. The “spannor” bundle deforms into essentially the usual spinor bundle as a conformally invariant parameter that may be interpreted as the space curvature becomes arbitrarily small. From a Minkowski space standpoint, however, the spannors involve a nontrivial action of space-time translations that deforms into a trivial action in the spinor limit and also have more complex transformation properties under discrete symmetries. Also studied are the “plyors,” consisting of the dual to the bundle product of the spannors with themselves. Composition series for the spannor and plyor section spaces are treated, relative to the conformal group, and irreducible subquotients are identified with certain that occur in conventional bundles. In particular, factors corresponding to the Maxwell and massless Dirac equations, and which may represent certain of the observed elementary particles, are determined. A gauge and conformally invariant nonlinear coupling between spannors and plyors, constituting essentially a generalization of that used in quantum electrodynamics, is developed, and an associated invariant nonlinear partial differential equation is derived. Covariant and causal quantization for spannors (as fermions) and plyors (as bosons) is formulated algebraically. The present treatment is basically mathematical, but physical motivations and possible interpretations are briefly noted.

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.

Conservation laws for the classical Maxwell–Dirac and Klein–Gordon–Dirac equations

Solutions to the classical coupled Maxwell–Dirac and Klein–Gordon–Dirac equations in a space–time of dimension four are considered. These equations are invariant under the 15-dimensional conformal group, in the case of zero mass. The resulting conservation laws are explicitly exhibited in terms of the Cauchy data at a fixed time in a form suitable for analysis by the techniques of partial differential equations.

Local-Lagrangian Quantum Field Theory of Electric and Magnetic Charges

We present a local Lagrangian density, depending on a pair of four-potentials $A$ and $B$, and charged fields ${\ensuremath{\psi}}_{n}$ with electric and magnetic charges ${e}_{n}$ and ${g}_{n}$. The resulting local Lagrangian field equations are equivalent to Maxwell's and Dirac's equations. The Lagrangian depends on a fixed four-vector, so manifest isotropy is lost and is regained only for quantized values of (${e}_{n}{g}_{m}\ensuremath{-}{g}_{n}{e}_{m}$). This condition results from the requirement that the representation of the Poincar\'e Lie algebra which results from Poincar\'e invariance, integrate to a representation of the finite Poincar\'e group. The finite Lorentz transformation laws of $A$, $B$, and ${\ensuremath{\psi}}_{n}$ are presented here for the first time. The familiar apparatus of Lagrangian field theory is applied to yield directly the canonical commutation relations, the energy-momentum tensor, and Feynman's rules.

Sur une methode quaternionique de separation des variables

It is shown how the use of quaternions allows a direct integration by separation of variables of systems of partial differential equations. It follows that the resolution of Maxwell's or Dirac's equations is considerably simplified and made uniform. As an example we apply the method to Dirac's hydrogenoïd atom and to the corresponding selection rules.

MAXWELL'S EQUATIONS AND MATRIX ELEMENTS IN QUANTUM ELECTRODYNAMICS

Matrix elements of second-order processes in quantum electrodynamics are obtainable directly from the use of Maxwell's and Dirac's equations.

A New Approach to Kinematic Cosmology—(B)

Maxwell's equations, Lorentz's equations of motion, and Dirac's equations are investigated on a cosmological background. Maxwell's equations are conform invariant and the equations of motion can easily be made conform invariant. Then, without introducing any new assumptions, Dirac's equations are seen to be conform invariant. In open universes the solutions of Maxwell's and Dirac's equations are the same as in flat Minkowski space. The behavior of these equations is different in closed universes by reasons of topology. In closed universes both Maxwell's and Dirac's equations provide new eigenvalue problems. Dirac's equations exhibit a marked difference between an elliptic and a spherical closed universe.