Comments on the radial plane waves

TL;DR

This paper discusses the properties of radial plane waves, highlighting that their orthogonality breaks down in infinite volume, but they remain useful for radial path integrals despite being overcomplete.

Contribution

It clarifies the orthogonality properties of radial plane waves and demonstrates their continued usefulness in constructing radial path integrals.

Findings

Orthogonality of radial plane waves breaks in infinite volume.

Radial plane waves become overcomplete.

They remain useful for radial path integrals.

Abstract

The orthogonality of the radial plane waves, introduced by Fujikawa, turns out to be broken for the case of infinite volume. We will find, though they become overcomplete, the concept of the radial plane waves remains useful for constructing radial path integrals.