A solvable algebra for massless fermions

TL;DR

The work identifies the stabiliser of massless fermions as the maximal solvable subgroup of the proper Lorentz group, the Borel subgroup $\mathrm{Bor}(1,3;p)$, reflecting a projective momentum space rather than a semisimple little group. It shows that massless states decompose into left- and right-handed sectors described by the Kronecker-sum $\mathrm{sol}_2^-\boxplus\mathrm{sol}_2^+$, and derives induced Lorentz transformations, Weyl equations, and a covariant representation framework via Chevalley bases and Kronecker-sum realizations. The paper also constructs explicit common eigenvectors, links to the $(k,l)$ representation family, and provides explicit forms for $J_i^\boxplus$ and $K_i^\boxplus$, demonstrating how massless chirality emerges from a four-generator solvable algebra. The results set the stage for connecting Weyl- and Dirac-type descriptions in the massless limit and offer a formal handle on spin-flip effects, with further work anticipated on their physical implications.

Abstract

We derive the stabiliser group of the four-vector, also known as Wigner's little group, in case of massless particle states, as the maximal solvable subgroup of the proper orthochronous Lorentz group of dimension four, known as the Borel subgroup. In the absence of mass, particle states are disentangled into left- and right-handed chiral states, governed by the maximal solvable subgroups ${\rm sol}_2^\pm$ of order two. Induced Lorentz transformations are constructed and applied to general representations of particle states. Finally, in our conclusions it is argued how the spin-flip contribution might be closely related to the occurrence of nonphysical spin operators.