Amplituhedra, and Beyond

TL;DR

This review introduces geometric methods, especially amplituhedra and positive geometries, for understanding scattering amplitudes in quantum field theories, highlighting recent advances and open problems in the field.

Contribution

It provides a comprehensive overview of the role of amplituhedra and positive geometries in quantum field theory, summarizing recent developments and future challenges.

Findings

Introduction of amplituhedra as geometric encodings of scattering amplitudes

Overview of positive geometries and their physical significance

Discussion of recent progress and open problems in the field

Abstract

This review is a primer on recently established geometric methods for observables in quantum field theories. The main emphasis is on amplituhedra, i.e. geometries encoding scattering amplitudes for a variety of theories. These pertain to a broader family of geometries called positive geometries, whose basics we review. We also describe other members of this family that are associated with different physical quantities and briefly consider the most recent developments related to positive geometries. Finally, we discuss the main open problems in the field. This is a Topical Review invited by Journal of Physics A: Mathematical and Theoretical.