The Dirac equation as one fourth-order equation for one function -- a general, manifestly covariant form

TL;DR

This paper generalizes a previous result showing the Dirac equation can be expressed as a fourth-order equation for a single component, maintaining covariance and broadening applicability across different representations and components.

Contribution

The work extends earlier findings by deriving a general, covariant fourth-order form of the Dirac equation applicable to any gamma-matrix representation and component.

Findings

Equivalent fourth-order equation for one Dirac spinor component

Maintains manifest relativistic covariance

Applicable to arbitrary gamma-matrix representations

Abstract

Previously (A. Akhmeteli, J. Math. Phys., v. 52, p. 082303 (2011)), the Dirac equation in an arbitrary electromagnetic field was shown to be generally equivalent to a fourth-order equation for just one component of the four-component Dirac spinor function. This was done for a specific (chiral) representation of gamma-matrices and for a specific component. In the current work, the result is generalized for a general representation of gamma-matrices and a general component (satisfying some conditions). The resulting equivalent of the Dirac equation is also manifestly relativistically covariant and should be useful in applications of the Dirac equation.