All coordinates transformations that separate the center of mass kinetic energy, their group structure and geometry

TL;DR

This paper characterizes the most general coordinate transformations that exactly separate center of mass kinetic energy in quantum many-body systems, revealing a new group structure and geometric properties of these transformations.

Contribution

It identifies a broad class of coordinate transformations with free parameters, generalizing Jacobi coordinates, and uncovers a hidden additive group structure linked to their geometry.

Findings

Generalized coordinate transformations with free parameters.

Discovery of a new additive group structure.

Connections between transformation geometry and group properties.

Abstract

The most general coordinates transformations that allow for the exact separation of the kinetic energy operator of a quantum many-body system into total center of mass kinetic energy and internal kinetic energy are found and discussed. We find i) that the suitable transformations, depending on the number of particles, have a certain number of free parameters and this allows for the generalization of the Jacobi coordinates to a much larger class of coordinates with the same properties and ii) that there is a new, uncommon, additive group structure hidden in the transformation matrices that is connected to certain geometric properties of the set of coordinates.