Representation of quantum field theory in an extended spin space and fermion mass hierarchy

TL;DR

This paper develops a matrix space framework based on spin degrees of freedom to represent quantum field theory, deriving gauge-invariant interactions and naturally explaining fermion mass hierarchy in higher dimensions.

Contribution

It introduces a novel spin space matrix formalism that maintains Lorentz symmetry and derives gauge interactions and fermion mass hierarchy in higher-dimensional models.

Findings

Constructed scalar, fermion, and gauge fields within the spin space formalism.

Derived gauge-invariant, renormalizable Lagrangians consistent with standard-model connections.

Naturally obtained Higgs-like scalars producing fermion mass hierarchy in 7+1 dimensions.

Abstract

We consider a matrix space based on the spin degree of freedom, describing both a Hilbert state space, and its corresponding symmetry operators. Under the requirement that the Lorentz symmetry be kept, at given dimension, scalar symmetries, and their representations are determined. Symmetries are flavor or gauge-like, with fixed chirality. After spin 0, 1/2, and 1 fields are obtained in this space, we construct associated interactive gauge-invariant renormalizable terms, showing their equivalence to a Lagrangian formulation, using as example the previously studied (5+1)-dimensional case, with many standard-model connections. At 7+1 dimensions, a pair of Higgs-like scalar Lagrangian is obtained naturally producing mass hierarchy within a fermion flavor doublet.