The Nakano-Nishijima-Gell-Mann Formula From Discrete Galois Field

If the 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. Consequently, instead of the NNG formula, we obtained the new formula Q = 2(n+ I), 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.

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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 Pub., 1987
4) K. Nakatsugawa, M. Ohaga,T. Fujii,T. Matsuyama and S. Tanda, Symmetry 12 (2020) 1603.