Reconstruction of Weinberg-Salam theory in noncommutative geometry on M4 x Z2.

Abstract

Weinberg-Salam theory is reconstructed using the generalized differential calculus extended on the discrete space [ital M][sub 4][times][ital Z][sub 2]. According to Chamseddine and co-workers, we introduce the field [ital a][sub [ital i]]([ital x],[ital y]) which is the square-matrix-valued function defined on [ital M][sub 4][times][ital Z][sub 2]. The generalized gauge field is expressed as [ital A]([ital x],[ital y])= [ital tsum][sub [ital i]] a[sub [ital i]][sup [degree]]([ital x],[ital y])[ital scrda][sub [ital i]]([ital x],[ital y]), where [ital scrd]=[ital d]+[ital d][sub [chi]] is a generalized exterior derivative. We can construct the consistent algebra of [ital d][sub [chi]] which is an exterior derivative with respect to [ital Z][sub 2]. The spontaneous breakdown of gauge symmetry is coded in [ital d][sub [chi]] with the introduction of the symmetry-breaking function [ital M]([ital y]). The gauge field [ital A][sub [mu]]([ital x],[ital y]) and Higgs field [Phi]([ital x],[ital y]) are written in terms of [ital a][sub [ital i]]([ital x],[ital y]) and [ital M]([ital y]), which may indicate [ital a][sub [ital i]]([ital x],[ital y]) is a more fundamental object. Not only the Yang-Mills-Higgs Lagrangian but also the Dirac Lagrangian is reproduced through the inner product between the differential forms in a completely gauge-invariant way.