Green Function of the Poisson Equation: D=2,3,4

TL;DR

This paper investigates the Green function of the Poisson equation across two, three, and four dimensions, revealing dimensional analogies and symmetries, with particular focus on the 2D case and SO(4) symmetry in 4D.

Contribution

It provides a detailed analysis of the Green function in multiple dimensions, highlighting new insights into the 4D case and its relation to lower dimensions.

Findings

In 2D, the scale L interacts with the zero magnetic quantum number component.

In 3D, standard Green function expressions are reviewed for comparison.

In 4D, analogies with 3D spherical harmonics and SO(4) symmetry are established.

Abstract

We study the Green function of the Poisson equation in two, three and four dimensions. The solution g of the equation nabla^2 g(x - x') = delta^(D)(x - x'), where x and x are D-dimensional position vectors, is customarily expanded into radial and angular coordinates. For the two-dimensional case (D=2), we find a subtle interplay of the necessarily introduced scale L with the radial component of zero magnetic quantum number. For D=3, the well-known expressions are briefly recalled; this is done in order to highlight the analogy with the four-dimensional case, where we uncover analogies of the four-dimensional spherical harmonics with the familiar three-dimensional case. Remarks on the SO(4) symmetry of the hydrogen atom complete the investigations.