Scalar Pre-potentials for Spinor and Tensor Fields on Spacetime

TL;DR

This paper reviews a technique for solving classical linear PDE systems in Minkowski spacetime using complex solutions and Hertz potentials, enabling the construction of regular, propagating solutions for various fields including Maxwell, Einstein, and Dirac spinors.

Contribution

It introduces a unified method employing complexified scalar Laplacian solutions and Clifford algebra to generate non-singular, chiral, pulse-like solutions for tensor and spinor fields, extending previous approaches.

Findings

Constructed non-singular pulse solutions for Maxwell and Einstein fields.

Extended the technique to generate solutions for the Dirac equation.

Demonstrated applications in modeling laser pulses and astrophysical jets.

Abstract

We review a technique for solving a class of classical linear partial differential systems of relevance to physics in Minkowski spacetime. All the equations are amenable to analysis in terms of complex solutions in the kernel of the scalar Laplacian and a complexified Hertz potential. The complexification prescription ensures the existence of regular physical solutions with chirality and propagating, non-singular, pulse-like characteristics that are bounded in all three spatial dimensions. The technique is applied to the source-free Maxwell, Bopp-Land\'e-Podolsky and linearised Einstein field systems, and particular solutions are used for constructing classical models describing single-cycle laser pulses and a mechanism is discussed for initiating astrophysical jets. Our article concludes with a brief introduction to spacetime Clifford algebra ideals that we use to represent spinor fields. We employ these to demonstrate how the same technique used for tensor fields enables one to construct new propagating, chiral, non-singular, pulse-like spinor solutions to the massless Dirac equation in Minkowski spacetime.