On the uniqueness of solutions to gauge covariant Poisson equations with compact Lie algebras

TL;DR

This paper proves the uniqueness of solutions to gauge covariant Poisson equations with compact Lie algebra values, showing solutions are uniquely determined by their behavior at infinity under general smoothness conditions.

Contribution

It establishes the uniqueness of gauge covariant Laplace equation solutions in various dimensions, extending to cases with merely compact Lie algebras and analyzing solutions with specific growth at infinity.

Findings

Solutions vanish at infinity are unique and identically zero.

Solutions with certain growth conditions are covariant constants.

Uniqueness holds across different spatial dimensions.

Abstract

It is shown, under rather general smoothness conditions on the gauge potential, which takes values in an arbitrary semi-simple compact Lie algebra ${\bf g}$, that if a (${\bf g}$-valued) solution to the gauge covariant Laplace equation exists, which vanishes at spatial infinity, in the cases of 1,2,3,... space dimensions, then the solution is identically zero. This result is also valid if the Lie algebra is merely compact. Consequently, a solution to the gauge covariant Poisson equation is uniquely determined by its asymptotic radial limit at spatial infinity. In the cases of one or two space dimensions a related result is proved, namely that if a solution to the gauge covariant Laplace equation exists, which is unbounded at spatial infinity, but with a certain dimension-dependent condition on the asymptotic growth of its norm, then the solution in question is a covariant constant.