A note on hidden classes in spinor classification

TL;DR

This work extends the standard Lounesto spinor classification by exploring alternative spinor duals beyond the Dirac adjoint, motivated by theories beyond the Standard Model. It analyzes the admissible subclasses under the Fierz-Pauli-Kofink identities and a dual-algebra constraint, constructing potential duals Delta from space-time multivectors and discrete symmetries. The study identifies a range of mathematically allowed subclasses, with class 1 offering the richest extension, but many configurations are forbidden, and physical realization depends on the chosen dual, sometimes implying Lorentz-violating spin sums or additional quantum labels. Overall, the paper provides a rigorous algebraic framework for probing beyond-SM spinor structures and clarifies when new duals yield well-defined quantum fields, guiding future theoretical developments.

Abstract

The Lounesto classification is a well-established scheme for categorizing spinors based on their physical content, which are determined by their associated bilinear forms. It consists of six disjoint classes encompassing the known spinors within the context of the standard model of high-energy physics. However, advancements in theories beyond the standard model have opened the door to potential new spinorial adjoint structures, leading to new unforeseen classess. These developments indicate the potential for extending the standard Lounesto classification. In this paper, we explore all possible subclasses that could extend the Lounesto scheme. We highlight the most relevant subclasses by imposing constraints to their corresponding dual structures, thus broadening our understanding of spinor and its applications in theoretical physics.