# The Nakano–Nishijima–Gell-Mann Formula from Discrete Galois Fields

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## Abstract

**:**

## 1. Introduction

## 2. Galois Field

## 3. Lorentz Group in a Galois Field: The Coish Group

## 4. Gauge Transformation in Galois Field

## 5. Classification of Hadrons

## 6. Quark Confinement

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of the discrete multivaluedness with phase factor $\omega $. Each set of ${\mathbb{F}}_{{p}^{2}}$ constitutes a cyclotomic field. When $p=4n-1$, the field has $p+1$ elements. A Lorentz group in a Galois field has $p+1$ multivaluedness such as ${\omega}^{\alpha}a,\phantom{\rule{4pt}{0ex}}\alpha =0,1,\cdots ,p$. Regarding $\omega $ as a phase factor, it constitute a global gauge transformation in a Galois field.

**Figure 2.**Classification of hadrons with Equation (23). Our model take the n and I axis instead of Y and ${I}_{z}$. We did not use $\mathrm{SU}\left(3\right)$ symmetry but assume the same quark constituent as in the standard model of particle physics. Charge numbers such as $Q=2/3=2\xb7{3}^{p-2}\mathrm{mod}p$ are shown in Figure 3. Note that this figure superimposes baryons, anti-bayons, and mesons, which differ by baryon numbers. So, K

^{0}and $\Xi $

^{0}, for instance, are distinguished.

**Figure 3.**The values of $1/3={3}^{p-2}\phantom{\rule{4.pt}{0ex}}\mathrm{mod}\phantom{\rule{4.pt}{0ex}}p=(p+1)/3$ and $2/3=2\xb7{3}^{p-2}\phantom{\rule{4.pt}{0ex}}\mathrm{mod}\phantom{\rule{4.pt}{0ex}}p=2(p+1)/3$ are shown for the primes of the form $p=3n-1$. Fractional numbers in a Galois field are large numbers proportional to p.

**Figure 4.**Electromagnetic self-energy ${E}_{\mathrm{em}}={k}_{e}\frac{{Q}^{2}}{2a}{c}^{2}$ where Q is the charge number in Galois field, ${k}_{e}$ is the Coulomb constant, a is the classical electron radius, and c is the speed of light. The energy is independent of p for integral charge but increase for fractional charge. The GUT energy is shown as a rough standard.

**Table 1.**Quantum numbers of mesons in a Galois field. Q is the charge number of baryon and $j,k$ are total isospins of quarks. n is derived from Equation (23).

Meson | Q | n | I | j | k | Representation |
---|---|---|---|---|---|---|

${K}^{+}$ | 1 | 0 | 1/2 | 1/2 | 0 | ${\parallel a\parallel}^{0}{a}^{\left(1\right)}\otimes {\overline{a}}^{\left(0\right)}$ |

${K}^{0}$ | 0 | −1/2 | 1/2 | 1/2 | 0 | ${\parallel a\parallel}^{-\frac{1}{2}}{a}^{\left(1\right)}\otimes {\overline{a}}^{\left(0\right)}$ |

${\pi}^{+}$ | 1 | 1/2 | 0 | 1/2 | 1/2 | ${\parallel a\parallel}^{\frac{1}{2}}{a}^{\left(1\right)}\otimes {\overline{a}}^{\left(1\right)}$ |

${\pi}^{0}$ | 0 | 0 | 0 | 1/2 | 1/2 | ${\parallel a\parallel}^{0}{a}^{\left(1\right)}\otimes {\overline{a}}^{\left(1\right)}$ |

${\pi}^{-}$ | −1 | −1/2 | 0 | 1/2 | 1/2 | ${\parallel a\parallel}^{-\frac{1}{2}}{a}^{\left(1\right)}\otimes {\overline{a}}^{\left(1\right)}$ |

$\overline{{K}^{0}}$ | 0 | 1/2 | −1/2 | 0 | 1/2 | ${\parallel a\parallel}^{\frac{1}{2}}{a}^{\left(0\right)}\otimes {\overline{a}}^{\left(1\right)}$ |

${K}^{-}$ | −1 | 0 | −1/2 | 0 | 1/2 | ${\parallel a\parallel}^{0}{a}^{\left(0\right)}\otimes {\overline{a}}^{\left(1\right)}$ |

**Table 2.**Quantum numbers of baryons in a Galois field. Q is the charge number of baryon and $i,j,k$ are total isospins of quark. n is derived from Equation (23).

Baryon | Q | n | I | i | j | k | Representation |
---|---|---|---|---|---|---|---|

p | 1 | −1 | 3/2 | 1/2 | 1/2 | 1/2 | ${\parallel a\parallel}^{-1}{a}^{\left(1\right)}\otimes {a}^{\left(1\right)}\otimes {a}^{\left(1\right)}$ |

n | 0 | −3/2 | 3/2 | 1/2 | 1/2 | 1/2 | ${\parallel a\parallel}^{-\frac{3}{2}}{a}^{\left(1\right)}\otimes {a}^{\left(1\right)}\otimes {a}^{\left(1\right)}$ |

${\Sigma}^{+}$ | 1 | −1/2 | 1 | 1/2 | 1/2 | 0 | ${\parallel a\parallel}^{-\frac{1}{2}}{a}^{\left(1\right)}\otimes {a}^{\left(1\right)}\otimes {a}^{\left(0\right)}$ |

${\Sigma}^{0}$ | 0 | −1 | 1 | 1/2 | 1/2 | 0 | ${\parallel a\parallel}^{-1}{a}^{\left(1\right)}\otimes {a}^{\left(1\right)}\otimes {a}^{\left(0\right)}$ |

${\Sigma}^{-}$ | −1 | −3/2 | 1 | 1/2 | 1/2 | 0 | ${\parallel a\parallel}^{-\frac{3}{2}}{a}^{\left(1\right)}\otimes {a}^{\left(1\right)}\otimes {a}^{\left(0\right)}$ |

${\Xi}^{0}$ | 0 | −1/2 | 1/2 | 1/2 | 0 | 0 | ${\parallel a\parallel}^{-\frac{1}{2}}{a}^{\left(1\right)}\otimes {a}^{\left(0\right)}\otimes {a}^{\left(0\right)}$ |

${\Xi}^{-}$ | −1 | −1 | 1/2 | 1/2 | 0 | 0 | ${\parallel a\parallel}^{-1}{a}^{\left(1\right)}\otimes {a}^{\left(0\right)}\otimes {a}^{\left(0\right)}$ |

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**MDPI and ACS Style**

Nakatsugawa, K.; Ohaga, M.; Fujii, T.; Matsuyama, T.; Tanda, S.
The Nakano–Nishijima–Gell-Mann Formula from Discrete Galois Fields. *Symmetry* **2020**, *12*, 1603.
https://doi.org/10.3390/sym12101603

**AMA Style**

Nakatsugawa K, Ohaga M, Fujii T, Matsuyama T, Tanda S.
The Nakano–Nishijima–Gell-Mann Formula from Discrete Galois Fields. *Symmetry*. 2020; 12(10):1603.
https://doi.org/10.3390/sym12101603

**Chicago/Turabian Style**

Nakatsugawa, Keiji, Motoo Ohaga, Toshiyuki Fujii, Toyoki Matsuyama, and Satoshi Tanda.
2020. "The Nakano–Nishijima–Gell-Mann Formula from Discrete Galois Fields" *Symmetry* 12, no. 10: 1603.
https://doi.org/10.3390/sym12101603