Evolution of the concept of the curvature in the momentum space

TL;DR

This paper reviews the historical development of the concept of curvature in momentum space over 80 years, highlighting its evolution from general relativity analogies to noncommutative geometries, aiding quantum gravity understanding.

Contribution

It provides a comprehensive historical overview of the evolution of curvature in momentum space, connecting various theoretical proposals and motivations.

Findings

Evolution from general relativistic equations to noncommutative geometries

Motivations shifted from locality to non-commutativity

Highlights the role of curvature in quantum gravity theories

Abstract

We review the history of the curved momentum space from Max Born to the recent relative locality, \k{appa}-Poincare and noncommutative geometric proposals. We found that the concept of the curvature in the momentum space and motivations has been evolved during 80 years from the Max Born time. Motivations has been evolved from introducing general relativistic like equation for the momentum space to the relaxation of the concept of locality in special relativity and non-commutativity. This study can help us for a better understanding of this concept in quantum gravity.