Low degree Lorentz invariant polynomials as potential entanglement invariants for multiple Dirac spinors

A system of multiple spacelike separated Dirac particles is considered and a method for constructing polynomial invariants under the spinor representations of the local proper orthochronous Lorentz groups is described. The polynomials constructed by this method are identically zero for product states. The behaviour of the polynomials under local unitary evolutions that act unitarily on any subspace defined by fixed particle momenta is considered. By design all of the polynomials have invariant absolute values on such subspaces if the evolutions are generated by local zero-mass Dirac Hamiltonians. Depending on construction some polynomials have invariant absolute values also for the case of nonzero-mass or additional couplings. Because of these properties the polynomials are considered potential candidates for describing the spinor entanglement of multiple Dirac particles, with either zero or arbitrary mass. Polynomials of degree 2 and 4 are derived for the cases of three and four Dirac spinors. The relations of these polynomials to the polynomial spin entanglement invariants of three and four non-relativistic spin-$\frac{1}{2}$ particles are described.