Complex-Distance Potential Theory and Hyperbolic Equations

TL;DR

This paper extends potential theory into complex space, establishing a novel link between elliptic and hyperbolic equations through complexified distance functions, with applications to physics and electromagnetic wave analysis.

Contribution

It introduces a complexified potential theory that connects elliptic and hyperbolic equations without requiring analyticity of initial data.

Findings

Complex potential theory models extended particles in physics.

New connection between elliptic and hyperbolic equations in complex spacetime.

Application to Dirac and Maxwell equations with electromagnetic wavelets.

Abstract

An extension of potential theory in R^n is obtained by continuing the Euclidean distance function holomorphically to C^n. The resulting Newtonian potential is generated by an extended source distribution D(z) in C^n whose restriction to R^n is the delta function. This provides a natural model for extended particles in physics. In C^n, interpreted as complex spacetime, D(z) acts as a propagator generating solutions of the wave equation from their initial values. This gives a new connection between elliptic and hyperbolic equations that does not assume analyticity of the Cauchy data. Generalized to Clifford analysis, it induces a similar connection between solutions of elliptic and hyperbolic Dirac equations. There is a natural application to the time-dependent, inhomogeneous Dirac and Maxwell equations, and the `electromagnetic wavelets' introduced previously are an example.