BRST invariant Lagrangian of spontaneously broken gauge theories in noncommutative geometry

TL;DR

This paper develops a BRST invariant Lagrangian for spontaneously broken gauge theories within noncommutative geometry, using a generalized differential geometric approach on discrete spaces, and applies it to the SU(2)×U(1) model.

Contribution

It introduces a characteristic formulation for gauge theories in noncommutative geometry that simplifies the derivation of BRST invariant Lagrangians and demonstrates its application to the electroweak model.

Findings

Successfully derived BRST invariant Lagrangian in NCG framework.

Applied the formulation to SU(2)×U(1) gauge theory.

Showed the approach aligns with previous matrix derivative methods.

Abstract

The quantization of spontaneously broken gauge theories in noncommutative geometry(NCG) has been sought for some time, because quantization is crucial for making the NCG approach a reliable and physically acceptable theory. Lee, Hwang and Ne'eman recently succeeded in realizing the BRST quantization of gauge theories in NCG in the matrix derivative approach proposed by Coquereaux et al. The present author has proposed a characteristic formulation to reconstruct a gauge theory in NCG on the discrete space $M_4\times Z_{_N}$. Since this formulation is a generalization of the differential geometry on the ordinary manifold to that on the discrete manifold, it is more familiar than other approaches. In this paper, we show that within our formulation we can obtain the BRST invariant Lagrangian in the same way as Lee, Hwang and Ne'eman and apply it to the SU(2)$\times$U(1) gauge theory.