The full classical Dirac-Maxwell equations are considered in a somewhat novel form and under various simplifying assumptions. A reduction of the equations is performed in the case when the Dirac field is static. A further reduction of the equations is made under the assumption of spherical symmetry. These static spherically symmetric equations are examined in some detail and a numerical solution presented. Some surprising results emerge from this investigation:• Spherical symmetry necessitates the existence of a magnetic monopole.• There exists a uniquely defined solution, determined only by the demand that the solution be analytic at infinity.• The equations describe highly compact objects with an inner onion like shell structure.

A reduction of the Dirac-Maxwell equations in the case of static cylindrical symmetry is performed. The behaviour of the resulting system of o.d.e.s is examined analytically and numerical solutions presented. There are two classes of solutions.• The first type of solution is a Dirac field surrounding a charged "wire." The Dirac field is highly localised, being concentrated in cylindrical shells about the wire. A comparison with the usual linearized theory demonstrates that this localisation is entirely due to the non-linearities in the equations which result from the inclusion of the "self-field".• The second class of solutions have the electrostatic potential finite along the axis of symmetry but unbounded at large distances from the axis.

We study the full Maxwell-Dirac equations: Dirac field with minimally coupled electromagnetic field and Maxwell field with Dirac current as source. Our particular interest is the static case in which the Dirac current is purely time-like -the "electron" is at rest in some Lorentz frame. In this case we prove two theorems under rather general assumptions. Firstly, that if the system is also stationary (time independent in some gauge) then the system as a whole must have vanishing total charge, i.e. it must be electrically neutral. In fact, the theorem only requires that the system be asymptotically stationary and static. Secondly, we show, in the axially symmetric case, that if there are external Coulomb fields then these must necessarily be magnetically charged -all Coulomb external sources are electrically charged magnetic monopoles.

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The Maxwell-Dirac equations are the equations for electronic matter, the "classical" theory underlying QED. The system combines the Dirac equations with the Maxwell equations sourced by the Dirac current.A stationary Maxwell-Dirac system has ψ = e −iEt φ, with φ independent of t. The system is said to be isolated if the dependent variables obey quite weak regularity and decay conditions. In this paper we prove the following results for isolated, stationary Maxwell-Dirac systems,• there are no embedded eigenvalues in the essential spectrum, i.e. −m ≤ E ≤ m;• if |E| < m then the Dirac field decays exponentially as |x| → ∞;• if |E| = m then the system is "asymptotically" static and decays exponentially if the total charge is non-zero.