Lorentz- and permutation-invariants of particles

Abstract

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 × n matrix of dot products. We extend the first theorem to include the action of an arbitrary permutation group P ⊂ 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 → 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 formulæ for the computation of the relevant Hilbert series for d ⩽ 4.