Relativistic partial waves for celestial amplitudes

TL;DR

This paper develops a formalism for expanding four-point celestial amplitudes of massless particles into relativistic partial waves, establishing their orthogonality, completeness, and connection to Minkowski space formulations.

Contribution

It introduces a relativistic partial wave expansion for celestial amplitudes, including eigenfunctions, orthogonality, and completeness, with applications to various particle types and string theory.

Findings

Derived eigenfunctions for celestial Poincaré Casimir operators.

Verified orthogonality and completeness of the relativistic partial waves.

Demonstrated expansions for scalars, gluons, gravitons, and superstring gluons.

Abstract

The formalism of relativistic partial wave expansion is developed for four-point celestial amplitudes of massless external particles. In particular, relativistic partial waves are found as eigenfunctions to the product representation of celestial Poincar\'e Casimir operators with appropriate eigenvalues. The requirement of hermiticity of Casimir operators is used to fix the corresponding integral inner product, and orthogonality of the obtained relativistic partial waves is verified explicitly. The completeness relation, as well as the relativistic partial wave expansion follow. Example celestial amplitudes of scalars, gluons, gravitons and open superstring gluons are expanded on the basis of relativistic partial waves for demonstration. A connection with the formulation of relativistic partial waves in the bulk of Minkowski space is made in appendices.