Lagrangian Formulation of Connes' Gauge Theory

TL;DR

This paper derives Connes' gauge theory from a Lagrangian invariance principle, discusses spontaneous symmetry breaking with multiple fermion generations, and extends the bosonic Lagrangian with additional parameters.

Contribution

It provides a Lagrangian formulation of Connes' gauge theory based on invariance, highlighting symmetry breaking conditions and generalizing the bosonic sector.

Findings

Gauge invariance leads to Connes' gauge field formulation.

Spontaneous symmetry breaking requires multiple fermion generations.

The bosonic Lagrangian includes two additional parameters.

Abstract

It is shown that Connes' generalized gauge field in non-commutative geometry is derived by simply requiring that Dirac lagrangian be invariant under local transformations of the unitary elements of the algebra, which define the gauge group. The spontaneous breakdown of the gauge symmetry is guaranteed provided the chiral fermions exist in more than one generations as first observed by Connes-Lott. It is also pointed out that the most general gauge invariant lagrangian in the bosonic sector has two more parameters than in the original Connes-Lott scheme.