Lorentz- and permutation-invariants of particles

Abstract

Two theorems of Weyl tell us that the algebra of Lorentz- (and parity-) invariant polynomials in the momenta of $n$ particles are generated by the dot products and that the redundancies which arise when $n$ exceeds the spacetime dimension $d$ are generated by the $(d+1)$-minors of the $n \times n$ matrix of dot products. We extend the first theorem to include the action of an arbitrary permutation group $P \subset S_n$ on the particles, to take account of the quantum-field-theoretic fact that particles can be indistinguishable. Doing so provides a convenient set of variables for describing scattering processes involving identical particles, such as $pp \to jjj$, for which we provide an explicit minimal set of Lorentz- and permutation-invariant generators. Additionally, we use the Cohen-Macaulay structure of the Lorentz-invariant algebra to provide a more direct characterisation in terms of a Hironaka decomposition. Among the benefits of this approach is that it can be generalized straightforwardly to when parity is not a symmetry and to cases where a permutation group acts on the particles. In the first non-trivial case, $n=d+1$, we give a homogeneous system of parameters that is valid for the action of an arbitrary permutation symmetry and make a conjecture for the full Hironaka decomposition in the case without permutation symmetry. An appendix gives formul\ae\ for the computation of the relevant Hilbert series for $d \leq 4$.