Reconstruction of Weinberg-Salam theory in noncommutative geometry on ${\mathit{M}}_4$×${\mathit{Z}}_2$

Weinberg-Salam theory is reconstructed using the generalized differential calculus extended on the discrete space ${\mathit{M}}_4$×${\mathit{Z}}_2$. According to Chamseddine and co-workers, we introduce the field ${\mathit{a}}_{\mathit{i}}$(x,y) which is the square-matrix-valued function defined on ${\mathit{M}}_4$×${\mathit{Z}}_2$. The generalized gauge field is expressed as A(x,y)= ${\mathit{tsum}}_{\mathit{i}}$ ${\mathrm{a}}_{\mathit{i}}^{\mathrm{°}}$(x,y)${\mathit{scrda}}_{\mathit{i}}$(x,y), where scrd=d+${\mathit{d}}_{\mathrm{\chi }}$ is a generalized exterior derivative. We can construct the consistent algebra of ${\mathit{d}}_{\mathrm{\chi }}$ which is an exterior derivative with respect to ${\mathit{Z}}_2$. The spontaneous breakdown of gauge symmetry is coded in ${\mathit{d}}_{\mathrm{\chi }}$ with the introduction of the symmetry-breaking function M(y). The gauge field ${\mathit{A}}_{\mathrm{\mu }}$(x,y) and Higgs field Φ(x,y) are written in terms of ${\mathit{a}}_{\mathit{i}}$(x,y) and M(y), which may indicate ${\mathit{a}}_{\mathit{i}}$(x,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.