Second-Order Fermions

TL;DR

This paper explores a second-order formulation of fermions using two-component spinors, simplifying calculations and providing new insights into gauge theories, the Standard Model, and potential unification with gravity.

Contribution

It develops a detailed second-order formalism for fermions in quantum field theories, including consistency checks and applications to unification scenarios.

Findings

Simplified fermion propagators and Feynman rules.

Confirmed anomaly cancellation and unitarity in the new formalism.

Explored gauge groups for embedding the Standard Model and unification with gravity.

Abstract

It has been proposed several times in the past that one can obtain an equivalent, but in many aspects simpler description of fermions by first reformulating their first-order (Dirac) Lagrangian in terms of two-component spinors, and then integrating out the spinors of one chirality ($e.g.$ primed or dotted). The resulting new Lagrangian is second-order in derivatives, and contains two-component spinors of only one chirality. The new second-order formulation simplifies the fermion Feynman rules of the theory considerably, $e.g.$ the propagator becomes a multiple of an identity matrix in the field space. The aim of this thesis is to work out the details of this formulation for theories such as Quantum Electrodynamics, and the Standard Model of elementary particles. After having developed the tools necessary to establish the second-order formalism as an equivalent approach to spinor field theories, we proceed with some important consistency checks that the new formulation is required to pass, namely the presence or absence of anomalies in their perturbative and non-perturbative description, and the unitarity of the S-Matrix derived from their Lagrangian. Another aspect which is studied is unification, where we seek novel gauge-groups that can be used to embed all of the Standard Model content: forces and fermionic representations. Finally, we will explore the possibility to unify gravity and the Standard Model when the former is seen as a diffeomorphism invariant gauge-theory.