The Nakano-Nishijima-Gell-Mann Formula From Galois Fields

If world has a finite compact space (I₁₂₀: Poincare Dodecahedron) [1] and discrete coordinates [2,3], what happens? In this case, the problem of infinities in gravity and in the standard model might be avoided. To avoid this problem, quantum gravity theories such as the superstring theory or the loop quantum gravity are developing, but neither of those theories have been completed. We reconstruct the Nakano-Nishijima-Gell-Mann (NNG) formula by using a discrete Galois field without using continuous coordinate. When we reconstruct new theories with a Galois field, these new theories must satisfy fundamental conservation law related to unitary, Lorentz, and gauge invariance.
Here, we reexamine previous model [2] using isospin I_z. Consequently, instead of the NNG formula, we obtained the alternate formula Q = 2(n + I_z), where Q is charge number and n is multi-valuedness in Galois field. These results may be a starting point to develop a theory without many problems of infinity.

1) J.-P. Luminet et al.: Nature 425 (2003) 593.
2) H. R. Coish: Phys.Rev. 114 (1959) 383.
3) Y. Nambu, Field Theory of Galois Fields, In I. A. Batalin (ed), Quantum Field Theory and Quantum Statistics, Vol. 1, p. 625. IOP Publishing, 1987, also in Broken Symmetry, World Scientific Pub. Co. Inc., 1995.