The Dirac–Maxwell equations with cylindrical symmetry

Abstract

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.

On the connection between the solutions to the Dirac and Weyl equations and the corresponding electromagnetic four-potentials

“…In this work the Dirac equation is expressed in a 2-spinor form, which allows it to be (covariantly) solved for the electromagnetic 4-potential, in terms of the wave function and its derivatives. This approach subsequently led to some physically interesting results, see also [14], [15], [16] (for a review see [17]). In [18] it was demonstrated that the Dirac equation is indeed algebraically invertible if a real solution for the vector potential is required.…”

The stationary Maxwell–Dirac equations

“…These solutions do, however, exhibit interesting non-linear behaviour which would not have been apparent through perturbation expansions. The particular solutions found in [8] and [9] exhibit just this sort of behaviour -localisation and charge screening. See also Das [10] and the more recent work of Finster, Smoller and Yau [11].…”

Algebraic solution for the vector potential in the Dirac equation

“…Gaussian elimination in this 8 × 4 real system then leads to a solution for A µ and also implies a set of four additional linearly independent constraints, linear in ∂ µ ψ, which can be identified in this case with bilinear identities proposed by Radford [4] and Booth and Radford [5] using van der Waerden 2-spinor notation. Either approach ultimately derives from the structure of the Dirac algebra and the symmetry properties of the γ matrices.…”

“…Radford [4] and Booth and Radford [5] used van der Waerden notation in order to derive a complex form of the vector potential subject to additional reality conditions. Here the 2 spinor version is reached via the Weyl representation of the Dirac algebra (see for example [11]), wherein…”

“…Recently Radford [4] handled the Maxwell-Dirac equations firstly by solving the Dirac equation for the electromagnetic potential in terms of the wavefunction and its derivatives, and then substituting this solution in the Maxwell equations. This approach subsequently led to some physically interesting results [5,6,7] (for a review see [8]). …”