Internal spaces of fermion and boson fields, described with the superposition of odd and even products of $γ^{a}$, enable understanding of all the second-quantised fields in an equivalent way

TL;DR

This paper advocates a unified description of second-quantised fermion and boson fields using Clifford-algebra-based internal spaces, where fermions are represented by odd basis vectors and bosons by even basis vectors. By embedding the internal space in even dimensions such as $d=(13+1)$ and constraining observable momenta to $d=(3+1)$, the framework naturally yields fermion families and two orthogonal boson groups corresponding to gauge and scalar sectors, with the vacuum defined as a quantum vacuum rather than a Dirac sea. The authors formulate a simple Lorentz-invariant action that combines these internal-basis vectors with ordinary spacetime, and illustrate the construction via toy models in $d=(5+1)$ and the full $d=(13+1)$ case, showing how known fermions and gauge fields may emerge after symmetry breaking. Key open questions concern renormalisability, anomalies, and the precise mechanism of symmetry breaking, as well as potential predictions for Higgs-like scalars, gravitons, and additional families or dark matter. Overall, the work provides a cohesive algebraic route to describe spins, charges, and family structure in a single framework, motivating further exploration of its phenomenological and theoretical consequences.

Abstract

Using the odd and even ``basis vectors'', which are the superposition of odd and even products of $γ^a$'s, to describe the internal spaces of the second quantised fermion and boson fields, respectively, offers in even-dimensional spaces, like it is $d=(13+1)$, the unique description of all the properties of the observed fermion fields (quarks and leptons and antiquarks and antileptons appearing in families) and boson fields (gravitons, photons, weak bosons, gluons and scalars) in a unique way, providing that all the fields have non zero momenta only in $d =(3+1)$ of the ordinary space-time, bosons have the space index $α$ (which is for tensors and vectors $μ=(0,1,2,3)$ and for scalars $σ\ge 5$). In any even-dimensional space, there is the same number of internal states of fermions appearing in families and their Hermitian conjugate partners as it is of the two orthogonal groups of boson fields having the Hermitian conjugate partners within the same group. A simple action for massless fermion and boson fields describes all the fields uniquely. The paper overviews the theory, presents new achievements and discusses the open problems of this theory.