On the possibility of differential-algebraic elimination of the spinor field from the Maxwell--Dirac electrodynamics

TL;DR

This paper explores the potential to eliminate the spinor field from Maxwell--Dirac equations, showing that the electromagnetic field can evolve independently without the spinor components.

Contribution

It demonstrates, through linearization and rank analysis, that the spinor field can be algebraically eliminated, leading to a formulation of the electromagnetic evolution independently.

Findings

Spinor components are uniquely determined by electromagnetic fields.

Fourth-order derivatives of the electromagnetic potential are determined by lower derivatives.

The electromagnetic field can evolve independently, enabling a Cauchy problem formulation.

Abstract

We investigate whether the spinor field can be differential-algebraically eliminated from the Maxwell--Dirac equations in a particular gauge. To this end, we construct a generic truncated power-series solution and linearize the prolonged system of the Maxwell--Dirac equations about this solution. We then analyze the ranks of the coefficient matrices associated with the linearized system. Our results indicate that, generically, the spinor components are uniquely determined by the electromagnetic field and its derivatives. Furthermore, the fourth-order time derivatives of the components of the electromagnetic four-potential are uniquely determined by derivatives of the lower order with respect to time. These findings strongly suggest that the spinor field can be differential-algebraically eliminated. Furthermore, the resulting equations describe independent evolution of the electromagnetic field, that is, a Cauchy problem can be formulated in terms of the electromagnetic variables alone.