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

TL;DR

This work tackles the question of whether the spinor field in Maxwell-Dirac electrodynamics can be eliminated via differential-algebraic methods. It adopts a strategy of constructing a generic truncated power-series solution, prolonging the equations, and linearizing about the solution to analyze the dependence of spinor components on the electromagnetic field and its derivatives, using a fixed gauge and a rational approximation $e^2=221/2410$. Through rank analyses of the linearized coefficient matrices, the study finds that the spinor variations are locally uniquely determined by the electromagnetic field (e.g., a rank drop of 1 in the test for a particular component), suggesting that differential-algebraic elimination is plausible in a local sense, though not proven globally. The results are computational and indicate significant but not insurmountable barriers to full elimination with current methods, offering conceptual significance for reduced Maxwell-Dirac models and highlighting the need for higher-order prolongations for derivatives of spinor components.

Abstract

We investigate whether the spinor field can be eliminated from the Maxwell-Dirac equations by differential algebraic methods. A generic truncated power series solution is constructed, the prolonged system of the Maxwell-Dirac electrodynamics is linearized about the solution, and the ranks of the associated coefficient matrices are computed. The results indicate that, generically, the spinor components are uniquely determined by the electromagnetic field and its derivatives. This strongly suggests that differential-algebraic elimination of the spinor field is possible.