Localized solutions of the Dirac–Maxwell equations

Abstract

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.

Algebraic solution for the vector potential in the Dirac equation

“…In [9] it was noted that M had rank 3 and determinant zero, with a rank 1 right null space, and therefore could not be inverted to obtain a unique solution for the potential. Yet Radford [4] did just this, albeit in the bispinor representation. That work exploited the fact that A µ is real 2 , which was not used in [9].…”

“…This is the result previously reported by Eliezer [9] (who attributed to Dirac the antisymmetry argument using Cγ 5 = α x α z in the standard representation). The consistency conditions (6), (7), and (8) are equivalent to Radford's [4] 'reality' conditions.…”

“…It is not necessary to work explicity in 8 real components: regardless of which basis we use, the columns of M and the rows of G are just spinors in a vector space over R, and the matrix multiplication is just the calculation of inner products using (4). All that we require for the inversion is to find spinors φ ν where (…”

“…Since the mathematical foundations of the latter remain unclear, the Maxwell-Dirac equations continue to be of interest [1,2,3]. 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]).…”

“…In that paper, it was noted that the determinant of the matrix of coefficients of the vector potential A µ in the Dirac equation actually vanishes, and that therefore a umique algebraic inversion was not possible. The aim of the present paper is to reconcile [4] and [9], and to emphasise the legitimacy of the algebraic Ansatz, despite the negative conclusions of [9], by a careful analysis of the nature of the Dirac equation regarded as a linear system [10] for A µ . The main result is that the Dirac equation is indeed invertible if a real solution for the vector potential is required, and moreover that the treatment entrains an additional set of polynomial constraints on the wavefunction and its partial derivatives which must be carried forward in any further analysis.…”