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

TL;DR

This paper develops Lorentz-invariant polynomial invariants for multiple Dirac spinors to characterize their entanglement, extending previous methods and analyzing their behavior under local unitary transformations.

Contribution

It introduces a generalized method for constructing Lorentz-invariant polynomials for multiple Dirac particles, identifying new invariants of degrees 2 and 4 for three and four spinors.

Findings

Identified 67 degree 4 polynomials for three spinors.

Constructed 16 degree 2 and 26 degree 4 polynomials for four spinors.

Described how to extend polynomial construction to five or more spinors.

Abstract

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 method is a generalization of the method used in [Phys. Rev. A {\bf 105}, 032402 (2022), arXiv:2103.07784] for the case of two Dirac particles. All polynomials constructed by this method are identically zero for product states. The behaviour of the polynomials under local unitary evolution that acts unitarily on any subspace defined by fixed particle momenta is described. By design all of the polynomials have invariant absolute values on this kind of subspaces if the evolution is locally generated by 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 or additional couplings. Polynomials of degree 2 and 4 are derived for the cases of three and four Dirac spinors. For three spinors no non-zero degree 2 polynomials are found but 67 linearly independent polynomials of degree 4 are identified. For four spinors 16 linearly independent polynomials of degree 2 are constructed as well as 26 polynomials of degree 4 selected from a much larger number. The relations of these polynomials to the polynomial spin entanglement invariants of three and four non-relativistic spin-$\frac{1}{2}$ particles are described. Moreover, it is described how degree 4 polynomials for five spinors can be constructed and how degree 2 polynomials can be constructed for any even number of spinors.