A Field-Theoretic Approach to Connes' Gauge Theory on $M_4\times Z_2$

TL;DR

This paper reformulates Connes' gauge theory on a product space, revealing nonuniqueness in the field strength and exploring implications for the Higgs potential and charge quantization.

Contribution

It introduces a new field strength definition in Connes' gauge theory, demonstrating its impact on the Higgs potential and clarifying the geometric origins of charge quantization.

Findings

The field strength in Connes' gauge theory is not unique.

A new field strength leads to a generation-number independent Higgs potential.

The nonuniqueness is linked to differential geometry extensions on $M_4\times Z_2$.

Abstract

Connes' gauge theory on $M_4\times Z_2$ is reformulated in the Lagrangian level. It is pointed out that the field strength in Connes' gauge theory is not unique. We explicitly construct a field strength different from Connes' one and prove that our definition leads to the generation-number independent Higgs potential. It is also shown that the nonuniqueness is related to the assumption that two different extensions of the differential geometry are possible when the extra one-form basis $\chi$ is introduced to define the differential geometry on $M_4\times Z_2$. Our reformulation is applied to the standard model based on Connes' color-flavor algebra. A connection between the unimodularity condition and the electric charge quantization is then discussed in the presence or absence of $\nu_R$.