BRST invariant lagrangian of spontaneously broken gauge theory in noncommutative differential geometry

Quantization of spontaneously broken gauge theory in noncommutative geometry(NCG) has been urged to be confirmed for long time because it is crucial for NCG approach to become reliable and physical theory. Lee, Hwang and Ne'eman succeeded very recently BRST quantization of gauge theory in NCG in the matrix derivative approach due to Coquereaux et al. Morita and the present author has also proposed another formalism to reconstruct the gauge theory in NCG on the discrete space M_4\times Z_{_N}. Our formalism is a generalization of the differential geometry on the ordinary manifold into the discrete one, then it is more familiar than that of other approaches. We show also in our formalism to be able to obtain the BRST invariant Lagrangian in the same way as Lee, Hwang and Ne'eman and apply it to the SU(2)\timesU(1) gauge theory