  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.
