Quantization in Cartesian coordinates and the Hofer metric

TL;DR

This paper explores why Cartesian coordinates are uniquely suited for quantum commutation relations, attributing this to the existence and uniqueness of the Hofer metric on the space of canonical transformations.

Contribution

It links the special role of Cartesian coordinates in quantum mechanics to the mathematical properties of the Hofer metric, providing a new geometric perspective.

Findings

Cartesian coordinates simplify canonical commutation relations

The Hofer metric's uniqueness explains coordinate preference

Provides a geometric foundation for quantization in phase space

Abstract

P.A.M. Dirac had stated that the Cartesian coordinates are uniquely suited for expressing the canonical commutation relations in a simple form. By contrast, expressing these commutation relations in any other coordinate system is more complicated and less obvious. The question that we address in this work, is the reason why this is true. We claim that this unique role of the Cartesian coordinates is a result of the existence and uniqueness of the Hofer metric on the space of canonical transformations of the phase space of the system getting quantized.