Covariant representation theory of the Poincar\'e algebra and some of its extensions

TL;DR

This paper develops a covariant representation theory of the Poincaré algebra applicable in all dimensions above four, enabling systematic construction of polarization states and derivation of supersymmetry Ward identities for scattering amplitudes.

Contribution

It introduces a covariant framework extending Poincaré representation theory to higher dimensions, facilitating amplitude calculations and supersymmetry analysis.

Findings

Constructed covariant polarization states for all particles in higher dimensions.

Derived supersymmetry Ward identities in any dimension.

Presented higher-dimensional analogs of amplitude vanishing results.

Abstract

There has been substantial calculational progress in the last few years for gauge theory amplitudes which involve massless four dimensional particles. One of the central ingredients in this has been the ability to keep precise track of the Poincare algebra quantum numbers of the particles involved. Technically, this is most easily done using the well-known four dimensional spinor helicity method. In this article a natural generalization to all dimensions higher than four is obtained based on a covariant version of the representation theory of the Poincare algebra. Covariant expressions for all possible polarization states, both bosonic and fermionic, are constructed. For the fermionic states the analysis leads directly to pure spinors. The natural extension to the representation theory of the on-shell supersymmetry algebra results in an elementary derivation of the supersymmetry Ward identities for scattering amplitudes with massless or massive legs in any integer dimension from four onwards. As a proof-of-concept application a higher dimensional analog of the vanishing helicity-equal amplitudes in four dimensions is presented in (super) Yang-Mills theory, Einstein (super-)gravity and superstring theory in a flat background.