Non-commutative differential geometry on discrete space M(4)xZ(N) and gauge theory

The algebra of non-commutative differential geometry (NCG) on the discrete space M(4)xZ(N) previously proposed by the present author is improved to give a consistent explanation of the generalized gauge held as a generalized connection on M(4) x Z(N). The nilpotency of the generalized exterior derivative d is easily proved. The matrix formulation where the generalized gauge held is denoted in matrix form is shown to have the same content with the ordinary formulation using d, which helps us understand the implications of the algebraic rules of NCG on M(4) x Z(N). The Lagrangian of the spontaneously broken gauge theory which has the extra restriction on the coupling constant of the Higgs potential is obtained by taking the inner product of the generalized field strength. The covariant derivative operating on the fermion field determines the parallel transformation on M(4)xZ(N), which confirms that the Higgs held is the connection on the discrete space. This implies that the Higgs particle is a gauge particle on the same footing as the weak bosons. The Higgs kinetic and potential terms are regarded as the curvatures on M4xZ(N).