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We clarify the structure of the Bazhanov-Baxter model of the 3-dim N-state integrable model. There are two essential points, i) the cubic symmetries, and ii) the spherical trigonometry parametrization, to understand the structure of this model. We propose two approaches to find a candidate as a solution of the tetrahedron equation, and we find a new solution., 23 pages, Latex

We propose a canonical procedure to quantize fields with interaction on discrete space-time. The time evolution operator that reproduces the field equation is represented by using canonical variables. The generator of the operator is a conserved quantity, but its existence is not obvious. It is possible to calculate the S-matrix perturbatively. Our quantization gives the same results as those given by the path integral quantization.

We have examined solutions of tetrahedron equations from the elliptic free fermion model by using Korepanov mechanism based on tetrahedral Zamolodchikov algebras. As a byproduct, we have found a new integrable 2-dim. lattice model. We have also studied the relation between tetrahedral Zamolodchikov algebras and tetrahedron equations., 10 pages, Latex

The problem whether there exist conserved quantities in the theory on discrete spacetime is studied. This problem is not obvious in the theory on discrete spacetime, because Noether's theorem is not applicable. The same problem arises even in the classical dynamics on discrete time. Some examples are shown in the classical dynamics with conserved quantities. The equation of motion is interpreted as an iterative map, which does not behave chaotically if there exists a conserved quantity. By suitable discretization the equation obtains a conserved quantity. In conclusion, an example of classical field theory with a conserved quantity is shown on four-dimensional discrete spacetime.