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

• Main Authors: Keiji Nakatsugawa, Motoo Ohaga, Toshiyuki Fujii, Toyoki Matsuyama, Satoshi Tanda
• Format: Article
• Language: English
• Published: MDPI AG 2020-09-01
• Series: Symmetry
• Subjects: quantum theory; finite mathematics; Nakano–Nishijima–Gell-Mann formula; fractional quantum number; quark confinement
• Online Access: https://www.mdpi.com/2073-8994/12/10/1603

The well known Nakano–Nishijima–Gell-Mann (NNG) formula relates certain quantum numbers of elementary particles to their charge number. This equation, which phenomenologically introduces the quantum numbers <inline-formula><math display="inline"><semantics><msub><mi>I</mi><mi>z</mi></msub></semantics></math></inline-formula> (isospin), <i>S</i> (strangeness), etc., is constructed using group theory with real numbers <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">R</mi></semantics></math></inline-formula>. But, using a discrete Galois field <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">F</mi><mi>p</mi></msub></semantics></math></inline-formula> instead of <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">R</mi></semantics></math></inline-formula> and assuring the fundamental invariance laws such as unitarity, Lorentz invariance, and gauge invariance, we derive the NNG formula deductively from Meson (two quarks) and Baryon (three quarks) representations in a unified way. Moreover, we show that quark confinement ascribes to the inevitable fractionality caused by coprimeness between half-integer (1/2) of isospin and number of composite particles (e.g., three).