Non-Commutative Differential Geometry on Discrete Space $M_4\times Z\ma{N}$ and Gauge Theory

TL;DR

This paper advances non-commutative differential geometry on discrete spaces to better explain gauge fields and Higgs particles as gauge connections, leading to specific mass relations and a unified geometric framework.

Contribution

It provides a consistent algebraic formulation of NCG on $M_4\times Z_N$, linking Higgs fields to gauge connections and deriving mass relations within a unified geometric approach.

Findings

Higgs field is a gauge connection on discrete space.

Derived a mass relation for Higgs and W bosons.

Confirmed Higgs kinetic and potential terms as curvatures.

Abstract

The algebra of non-commutative differential geometry (NCG) on the discrete space $M_4\times Z\ma{N}$ previously proposed by the present author is improved to give the consistent explanation of the generalized gauge field as the generalized connection on $M_4\times Z\ma{N}$. The nilpotency of the generalized exterior derivative ${\mbf d}$ is easily proved. The matrix formulation where the generalized gauge field is denoted in matrix form is shown to have the same content with the ordinary formulation using {\mbf d}, which helps us understand the implications of the algebraic rules of NCG on $M_4\times Z\ma{N}$. The Lagrangian of spontaneously broken gauge theory which has the extra restriction on the coupling constant of the Higgs potential is obtained by taking the inner product of the generalized field strength. The covariant derivative operating on the fermion field determines the parallel transformation on $M_4\times Z\ma{N}$, which confirms that the Higgs field is the connection on the discrete space. This implies that the Higgs particle is a gauge particle on the same footing as the weak bosons. Thus, it is expected that the mass relation $m\ma{H}=\frac{4}{\sqrt{3}}m\ma{W}\sin\theta\ma{W}$ proposed by the present author holds without any correction in the same way as the mass relation $m\ma{W}=m\ma{Z}\cos\theta\ma{W}$. The Higgs kinetic and potential terms are regarded as the curvatures on $M_4\times Z\ma{N}$.