Lorentz- and permutation-invariants of particles. (22nd March 2021)

• Record Type: Journal Article
• Authors: Gripaios, Ben; Haddadin, Ward; Lester, Christopher G
• Is Part Of: Journal of physics. Volume 54:Number 15(2021)
• Publication Date: 2021-03-22
• Subjects: mathematical physics -- invariant theory -- Hironaka decomposition -- minimal algebra generators -- invariant polynomial generators; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/abe58c
• Languages: English
• ISSNs: 1751-8113

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.