Non-Commutative Differential Geometry and Standard Model

TL;DR

This paper extends non-commutative differential geometry to incorporate Sogami's idea into the Standard Model, enabling a gauge-invariant formulation with a generalized field strength and a novel matrix calculus approach.

Contribution

It introduces a new extension of non-commutative geometry that allows for a generalized gauge-invariant Lagrangian and reformulates the Standard Model with a novel matrix calculus method.

Findings

Generalized the field strength to include non-nilpotent exterior derivatives.

Reformulated the Standard Model with left- and right-handed fermions on different sheets.

Developed a matrix calculus approach independent of discrete space-time.

Abstract

We incorporate Sogami's idea in the standard model into our previous formulation of non-commutative differential geometry by extending the action of the extra exterior derivative operator on spinors defined over the discrete space-time; four dimensinal Minkovski space multiplyed by two point discrete space. The extension consists in making it possible to require that the operator become nilpotent when acting on the spinors. It is shown that the generalized field strength leads to the most general, gauge-invariant Yang-Mills-Higgs Lagrangian even if the extra exterior derivative operator is not nilpotent, while the fermionic part remains intact. The proof is given for a single Higgs model. The method is applied to reformulate the standard model by putting left-handed fermion doublets on the upper sheet and right-handed fermion singlets on the lower sheet with generation mixing among quarks being taken into account. We also present a matrix calculus of the method without referring to the discrete space-time.