Non-linear conformally invariant generalization of the Poisson equation to D>2 dimensions

TL;DR

This paper introduces a unique non-linear, conformally invariant scalar theory generalizing the Poisson equation to higher dimensions, providing exact solutions and insights into multi-charge systems.

Contribution

It presents the only second-order conformally invariant scalar theory with a specific non-linear dielectric response and derives exact solutions for multi-charge configurations.

Findings

Exact two-point-force expressions

Energy functions for multi-charge systems

Virial theorem for many-particle systems

Abstract

I propound a non-linear generalization of the Poisson equation describing a "medium" in D dimensions with a "dielectric constant" proportional to the field strength to the power D-2. It is the only conformally invariant scalar theory that is second order, and in which the scalar $phi$ couples to the sources $\rho$ via a $\phi\rho$ contact term. The symmetry is used to generate solutions for the field for some non-trivial configurations (e.g. for two oppositely charged points). Systems comprising N point charges afford further application of the symmetry. For these I derive e.g. exact expressions for the following quantities: the general two-point-charge force; the energy function and the forces in any three-body configuration with zero total charge; the few-body force for some special configurations; the virial theorem for an arbitrary, bound, many-particle system relating the time-average kinetic energy to the particle charges. Possible connections with an underlying conformal quantum field theory are mentioned.