  The left-right symmetric gauge model with the symmetry of $SU(3)_c\times
SU(2)_L\times SU(2)_R\times U(1)$ is reconstructed in a new scheme of the
noncommutative differential geometry (NCG) on the discrete space $M_4\times
Z_4$ which is a product space of Minkowski space and four points space. The
characteristic point of this new scheme is to take the fermion field to be a
vector in a 24-dimensional space which contains all leptons and quarks.
Corresponding to this specification, all gauge and Higgs boson fields are
represented in $24\times 24$ matrix forms. We incorporate two Higgs doublet
bosons $h$ and $SU(2)_R$ adjoint Higgs $\xi_R$ which are as usual transformed
as $(2,2^\ast,0)$ and $(1,3,-2)$ under $SU(2)_L\times SU(2)_R\times U(1)$,
respectively. Owing to the revise of the algebraic rules in a new NCG, we can
obtain the necessary potential and interacting terms between these Higgs bosons
which are responsible for giving masses to the particles included. Among the
Higgs doublet bosons, one CP-even scalar boson survives in the weak energy
scale and other four scalar bosons acquire the mass of the $SU(2)_R\times U(1)$
breaking scale, which is similar to the situation in the standard model.
$\xi_R$ is responsible to spontaneously break $SU(2)\ma{R} \times U(1)$ down to
$U(1)\ma{Y}$ and so well explains the seesaw mechanism. Up and down quarks have
the different masses through the vacuum expectation value of $h$.

High Energy Physics - Theory