Gauge theory and the Higgs mechanism based on differential geometry in the discrete space M4 x ZN.

Abstract

Weinberg-Salam theory and SU(5) grand unified theory (GUT) are reconstructed using the generalized differential calculus extended on the discrete space ${\mathit{M}}_{4}$\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{\mathit{N}}$. Our starting point is the generalized gauge field expressed by A(x,n)= ${\mathcal{J}}_{\mathit{i}}$ ${\mathrm{a}}_{\mathit{i}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$(x,n)${\mathit{scrda}}_{\mathit{i}}$(x,n), (n=1,2,...,N), where ${\mathit{a}}_{\mathit{i}}$(x,n) is the square matrix valued function defined on ${\mathit{M}}_{4}$\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{\mathit{N}}$ and scrd=d+ ${\mathcal{J}}_{\mathit{m}=1}^{\mathit{N}}$${\mathit{d}}_{\mathrm{\ensuremath{\chi}}\mathit{m}}$ is a generalized exterior derivative. We can construct the consistent algebra of ${\mathit{d}}_{\mathrm{\ensuremath{\chi}}\mathit{m}}$ which is an exterior derivative with respect to ${\mathit{Z}}_{\mathit{N}}$ and the spontaneous breakdown of gauge symmetry is coded in ${\mathit{d}}_{\mathrm{\ensuremath{\chi}}\mathit{m}}$. The unified picture of the gauge field and Higgs field as the generalized connection in noncommutative geometry is realized. Not only the Yang-Mills-Higgs Lagrangian but also the Dirac Lagrangian, invariant against the gauge transformation, is reproduced through the inner product between the differential forms. Three sheets (${\mathit{Z}}_{3}$) are necessary for Weinberg-Salam theory including strong interaction and the SU(5) GUT. Our formalism is applicable to a more realistic model such as the SO(10) unification model.