Deriving spin-1 quartic interaction vertices from closure of the Poincar\'e algebra

TL;DR

This paper derives the quartic interaction vertex in pure Yang-Mills theory using Poincaré algebra closure, highlighting the necessity of the Jacobi identity for structure constants and showing spin generator corrections vanish at this order.

Contribution

It provides a derivation of the quartic vertex from algebraic principles and clarifies the role of the Jacobi identity in Yang-Mills interactions.

Findings

Derivation of the quartic vertex from Poincaré algebra closure

Proof that structure constants must satisfy the Jacobi identity

Demonstration that spin generator corrections vanish at this order

Abstract

We derive the quartic interaction vertex of pure Yang-Mills theory by demanding closure of the light-cone Poincar\'e algebra in four-dimensional Minkowski spacetime. This calculation explicitly shows why structure constants must satisfy the Jacobi identity. We then prove that corrections to the spin generator, for spin one at this order, vanish. We comment briefly on higher spin fields in this context.