SS symmetry
ArticleThe Nakano–Nishijima–Gell-Mann Formula from
Discrete Galois Fields
Keiji Nakatsugawa 1,2, *, Motoo Ohaga 1,2 , Toshiyuki Fujii 2,3 , Toyoki Matsuyama 2,4
and Satoshi Tanda 1,2, *
1
2
3
4

*

Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan;
ohaga_mm@eis.hokudai.ac.jp
Center of Education and Research for Topological Science and Technology, Hokkaido University,
Sapporo 060-8628, Japan; tfujii@asahikawa-med.ac.jp (T.F.); matsuyat@nara-edu.ac.jp (T.M.)
Department of Physics, Asahikawa Medical University, Asahikawa 078-8510, Japan;
Department of Physics, Nara University of Education, Takabatake-cho, Nara 630-8528, Japan
Correspondence: keiji_nakatsugawa@eng.hokudai.ac.jp (K.N.); tanda@eng.hokudai.ac.jp (S.T.)

Received: 12 August 2020; Accepted: 22 September 2020; Published: 26 September 2020


Abstract: The well known Nakano–Nishijima–Gell-Mann (NNG) formula relates certain quantum
numbers of elementary particles to their charge number. This equation, which phenomenologicallyintroduces the quantum numbers Iz (isospin), S (strangeness), etc., is constructed using group theory
with real numbers R. But, using a discrete Galois field F p instead of R 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 bycoprimeness between half-integer (1/2) of isospin and number of composite particles (e.g., three).
Keywords: quantum theory; finite mathematics; Nakano–Nishijima–Gell-Mann formula; fractional
quantum number; quark confinement

1. Introduction
The standard model of particle physics has been established based on physics with real
numbers (R). However, problems such as the problem of infinite in gravity are yet unsolved.
To avoid such problems, quantum gravity theories like superstring theory or loop quantum gravity
are developing, but neither of those theories has been completed. On the other hand, the assumptionthat all of the physics in the universe is constructed solely of R has not been proved. For instance,
Yukawa considered extended wave function of elementary particles using “elementary domains” [1]and Snyder [2] and Finkelstein [3] simply considered quantization of space-time. However,these approaches are incomplete and a unified formulation is unknown. Here, we assert that adifferent mathematical structure, namely a world with a finite [4] and discrete space-time [5–14],
is needed for the following reasons.
The description of a discrete space-time would require a discrete mathematical structure. Unitarytransformation from quantum mechanics and Lorentz (Möbius) transformation from relativity requires
this structure to be a number field, where addition, subtraction, multiplication, and division can be
defined. The set of integers Z is not a field because division is not defined. For instance, fractional
numbers such as 1/3 are not defined. What we would consider in this paper is called a Galois field(F p = Z/pZ), that is, the set of integers where two numbers differing by a multiple of a prime number
p are equivalent [5–14]. F p has a finite, periodic, and discrete structure. Mathematical and physicaljustification to consider this field are as follows.

Symmetry 2020, 12, 1603; doi:10.3390/sym12101603

www.mdpi.com/journal/symmetry

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Mathematically, there is another system called p-adic numbers Q p (p : prime) that is equivalent toR [15]. Q p is regarded as the limit of p-adic integers Z/pn Z with n → ∞.
n→∞Z/pZ, Z/p2 Z, Z/p3 Z, · · · , Z/pn Z −→ Q p .
Moreover, In adele ring theory, R is regarded as the limit of Q p with p → ∞ [16].
p→∞

Q2 , Q3 , Q5 , · · · , Q p −→ R.
So, from the viewpoint of number theory, the geometry constructed by the real numbers R is only anapproximation of a finite and discrete geometry [11,17]. Physics over R may be obtained from physics
over F p under the appropriate limit.
Physically, Galois fields can appear naturally in our universe from multiple perspectives. First,
recent observation of microwave background radiation suggests that the world is finite and periodic(I120 : Poincaré Dodecahedron) [4]. Second, if space-time has the Planck length as a characteristic
scale, then Lorentz transformation must be discrete [18]. Theoretically, Jarnefelt estimated a huge81
but finite number of world coordinates as 1010 [5,7,19]. Third, the theory of quantum gravity needs
to be expanded because of the difficulty of renormalizability. This problem would not appear in F p
in the first place [5,6,13,14], so the expansion is expected to be achieved by using a discrete system.
In standard particle physics theory, the existence of antiparticles depends on additional assumptions,while in the Galois field such strange assumptions are not necessary [13]. Fourth, the field theory of
Galois field was investigated by Nambu in the context of recurrence time [6]. Recently, the discretenessof time is also discussed the context of time crystals [20–24]. So, if space-time is both finite and discrete,
we have to re-examine the laws of physics [6].
In this article, we re-examine a previous model and classify hadrons by using a discrete Galois
field F p . When we reconstruct new theories with a Galois field, these new theories must satisfy
fundamental conservation laws, namely those related to unitarity, Lorentz invariance, and gauge
invariance. Consequently, instead of the Nakano–Nishijima–Gell-Mann (NNG) formula (which is
proposed using continuous coordinate R), we obtained the alternate formula Q = 2(n + I ), where Q ischarge number, n is multivaluedness in Galois field, and I is total isospin. This formula is derived from
Meson (two quarks) and Baryon (three quarks) representations in a unified way. The use of isospin
was not investigated in detail by previous authors [5,7–12,19,25] who introduced finite geometry into
the field theory. Since the NNG formula also applies to individual quarks, we apply the new formula
to up-quark and down-quark, which inevitably lead to fractional numbers. Fractional numbers in
Galois fields are, however, very large numbers. Therefore, isolated quarks with fractional charge canhave a very large energy. We surmise that quark confinement ascribes to this fractionality in Galois field.
These results may be a starting point to develop a theory without problems of infinity.
2. Galois Field
Here, we introduce the Galois fields F p and F p2 . A number field is a mathematical structure where
addition, subtraction, multiplication and division are defined. Let Z/mZ be the set of integers where
two numbers differing by an integral multiple of m ∈ Z are regarded equivalent. A well-known result
from number theory is that Z/mZ is a number field if and only if m is a prime number. This finite fieldis called a Galois field which is denoted by F p :F p = {0, 1, 2, . . . , p − 1}.

(1)

A Galois field is not an ordered field. But, as was made clear in Reference [25], F p can be partiallyordered if p has the formkp = 8x ∏ qi − 1
i =1

(2)

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where x is an odd integer and ∏i qi is the product of the first k odd primes. So, the usual notion ofmagnitude (large and small) and sign (positive and negative) hold for the first N ∼ qk ∼ lnp elements.
Therefore, we consider a prime of the form in Equation (2) which is assumed to be large enough toconstruct a geometry containing all coordinates in the universe.
Quantum mechanics requires complex numbers. To consider complex numbers in a Galois field,
it is natural to take an analogue of the ordinary extension of real numbers to complex numbers. That is
to say, the set of “complex” numbersz = x + iy,( x, y ∈ F p )(3)

i 2 = −1

(4)

is a “complex” Galois field which is denoted by F p2 . For a prime p of the form p = 4n − 1 (n ∈ Z),
complex conjugation is given by i p = −i = i∗ and z p = x − iy = z∗ since we have

( x + iy) p = x p − iy p = x − iy.

(5)

The first equality holds because p is a character of the field, that is p · a = 0 for all a ∈ F p ; The second
equality follows from Fermat’s little theorem a p ≡ a mod p. Note that |z|2 can become negative for the
non-ordered part of F p .
3. Lorentz Group in a Galois Field: The Coish Group
Lorentz transformation can be recovered with the Galois field introduced in the previous
section [5,8,11,12]. Coordinates in Minkowski space-time are closely related to spinor structure throughthe Pauli matrix representation
X=

x0 + x3
x1 + ix2

x1 − ix2
x0 − x3!

= x0 1 + x1 σ1 + x2 σ2 + x3 σ3(6)where x0 , x1 , x2 , x3 ∈ F p and
σ1 =

0
1

!
1
,
0

0
i

σ2 =

!
−i
,
0

σ3 =

1
0

0
−1

!

are Pauli matrices. (See Reference [26] for the case of xµ ∈ R.) The metric form is given as thedeterminant of X:k X k = x02 − x12 − x22 − x32 .
(7)
Let a be a 2 × 2 matrix with coefficients in F p2 . The linear transformation
0

X = a† Xa(8)

is a proper Lorentz transformation if it leaves the metric form Equation (7) invariant, that is,
if k ak∗ k ak = k ak p+1 = 1 holds. Such matrices form the orthogonal group SL(2, F p2 ). In finite
geometry, however, there also exist transformations with k ak p+1 = −1 which reverse the sign of themetric [5].
In addition, suppose that matrices a1 and a2 lead to the same Lorentz transformation.
Then U ≡ a1 a2−1 transforms X asU † XU = X.
(9)
Since the set of X contains the identity matrix, Equation (9) leads to U † U = UU † = 1. So, U is a unitarymatrix of the form
U = λ1.
(10)

Symmetry 2020, 12, 1603

4 of 11Given that U is unitary, λ satisfies
λ∗ λ = 1.

(11)

In a Galois field, such λ become discrete and multivalued. That is, for a prime p of the form p = 4n − 1(such as in Equation (2)) we have
λ = ω α , a1 = ω α a2 .
(12)
where α = 0, 1, · · · , p. So, the the orthogonal group in a Galois field has p + 1 multivaluedness aswith a cyclotomic field, and ω acts as a phase factor (Figure 1). In this way, we obtain the full Coishgroup. The spinor group, which is a subgroup of the rotational group, has the same multivaluedness.
This multivaluedness will plays an essential role for the classification of hadrons.

Figure 1. Schematic representation of the discrete multivaluedness with phase factor ω. Each set of
F p2 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 ω α a, α = 0, 1, · · · , p. Regarding ω as a phase factor,
it constitute a global gauge transformation in a Galois field.

4. Gauge Transformation in Galois Field
In this section, the charge number Q is derived with gauge transformation in Galois field inaccordance with the derivation given in Reference [5]. Because of homomorphism, an orthogonalgroup leads a spinor group. The irreducible representation of the orthogonal group must be a matrix
with elements of F p2 as its coefficients. The spinor group is a subgroup of of the two dimensional
general linear homogeneous Galois group GLH (2, p2 ) over the complex finite field F p2 . The irreducible
representations of GLH (2, p2 ) are given by Brauer and Nesbitt [27] and extended by Coish [5] ask akn a(2j) ⊗ ā(2k) ,

(13)

where
j, k = 0, 1/2, 1, 3/2, · · · ( p − 1)/2,(14)

n = 0, ±1/2, ±1, · · · ± ( p + 1/2), p + 1.

(15)

Here, a is a spinor matrix which transforms a spinor
ψ1
ψ2

!

→

ψ10
ψ20

!

=a

ψ1
ψ2

!
.

(16)

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5 of 11k akn is interpreted as the multivaluedness of the representation. Next, consider the polynomialjf (ψ) =

∑ f m ψ1

j+m

m=− j

j−m

ψ2

.

(17)

n
o
j+m j−m
The set ψ1 ψ2
can be regarded as the linearly independent set of basis for a (2j + 1)-dimensional
vector space. The action of a on f (ψ) is given byf ( aψ) =

j

j

m=− j

m=− j

(2j) j+n j−n
ψ2 .

∑ f m (ψ10 ) j+m (ψ20 ) j−m = ∑ f m amn ψ1
j+m

(18)

j−m

So, a(2j) is a matrix transforming the monomials ψ1 ψ2 . In particular, the phase factor ω αtransforms a → ω α a and a(2j) → ω 2jα a(2j) . This phase factor corresponds to a gauge transformation,
and Equation (13) is transformed as follows

k akn a(2j) ⊗ ā(2k) −→ ω Qα k akn a(2j) ⊗ ā(2k) ,(19)

Q = 2( n + j − k ).

(20)

where
Therefore, we naturally derive Equation (20) as the charge number reflecting discreteness of gaugetransformation in Galois field.
5. Classification of Hadrons
Now, we can classify Hadrons in a Galois field. Coish interpreted j and k as spins and represented
baryons and mesons with Equations (13) and (20). However, his model does not agree with thestandard model of particle physics. So, we give a different interpretation. Instead of spins, we interpretj and k as isospin. More precisely, to encompass the multiplet structure of hadrons, it would beappropriate to interpret j, k as the total isospin I instead of Iz . Mesons, which are composed of particles
and anti-particles, are transformed by the representation( j,−k,n)Dmeason ≡ k akn a(2j) ⊗ ā(2k) ,(21)

but it is now interpreted as a representation of GLH (2, p2 ) × GLH (2, p2 )∗ . Baryons, which arecomposed of three quarks, are transformed by another representation(i,j,k,n)Dbaryon ≡ k akn a(2i) ⊗ a(2j) ⊗ a(2k)(22)

of GLH (2, p2 ) × GLH (2, p2 ) × GLH (2, p2 ). Here, the contribution of strangeness appears as amodification of the multivaluedness n.
Mesons and baryons can be classified according to the charge number
Q = 2( n + I ),(23)

where I = j − k for mesons and I = i + j + k for baryons.
Tables 1 and 2, respectively, show quantum numbers and representations of mesons and baryons
in a Galois field. Here, we did not use SU(3) symmetry but assume the known quark contents.

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

Q

n

I

j

k

Representation

K+

1

0

1/2

1/2

0

k ak0 a(1) ⊗ ā(0)

K0

0

−1/2

1/2

1/2

0

k ak− 2 a(1) ⊗ ā(0)

π+

1

1/2

0

1/2

1/2

k ak 2 a(1) ⊗ ā(1)

π0

0

0

0

1/2

1/2

k ak0 a(1) ⊗ ā(1)

π−

−1

−1/2

0

1/2

1/2

k ak− 2 a(1) ⊗ ā(1)

K¯0

0

1/2

−1/2

0

1/2

k ak 2 a(0) ⊗ ā(1)

K−

−1

0

−1/2

0

1/2

k ak0 a(0) ⊗ ā(1)

1

1

1

1

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

k a k −1 a (1) ⊗ a (1) ⊗ a (1)

n

0

−3/2

3/2

1/2

1/2

1/2

k a k − 2 a (1) ⊗ a (1) ⊗ a (1)

Σ+

1

−1/2

1

1/2

1/2

0

k a k − 2 a (1) ⊗ a (1) ⊗ a (0)

Σ0

0

−1

1

1/2

1/2

0

k a k −1 a (1) ⊗ a (1) ⊗ a (0)

Σ−

−1

−3/2

1

1/2

1/2

0

k a k − 2 a (1) ⊗ a (1) ⊗ a (0)

Ξ0

0

−1/2

1/2

1/2

0

0

k a k − 2 a (1) ⊗ a (0) ⊗ a (0)

Ξ−

−1

−1

1/2

1/2

0

0

k a k −1 a (1) ⊗ a (0) ⊗ a (0)

3

1

3

1

We derived Equation (23) equivalent to the NNG formula
Q = Iz +

Y
,
2

(24)

where Y is the hypercharge. Figure 2 shows the classification of hadrons with n and I. Particles andanti-particles are related by CP conjugation (that is, point symmetry around the origin n = I = 0) as inReference [5]. Different values of I indicate the multiplet structure of hadrons. By comparison with the
NNG formula, n includes both the isospin Iz and the hypercharge Y. In this way, Galois fields can beused to organize the quantum number of hadrons. Similarly, as it is well known, the quantum numberof leptons are related with an equation similar to the NNG formula; the origins of the weak isospin
and weak hypercharge, which are properties of leptons introduced phenomenologically, may also be
explained by the Galois field.

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Figure 2. Classification of hadrons with Equation (23). Our model take the n and I axis instead of Y
and Iz . We did not use SU(3) symmetry but assume the same quark constituent as in the standard
model of particle physics. Charge numbers such as Q = 2/3 = 2 · 3 p−2 mod p are shown in Figure 3.
Note that this figure superimposes baryons, anti-bayons, and mesons, which differ by baryon numbers.
So, K0 and Ξ0 , for instance, are distinguished.

More generally, we can give a unified representation for arbitrary composit particles. First, we
can give a representation of a single particle as k akn a(2j) . Likewise, a representation of particles with
N (= 2, 3, 4, 5 or higher) constituents is possible withNO

k aknl a(2jl )θνl

(25)

l =1

where jl are half-integers, νl = 0 or 1, θ0 is the identity map a(2j)θ0 = a(2j) , and θ1 maps a(2j) to the1conjugate group a(2j)θ1 = ā(2j) with coefficients α p = a∗ (c.f. References [5,27]). Then, the charge
number is given by Q = ∑lN=1 2(nl ± jl ) where plus and minus correspond to νl = 0 and νl = 1,
respectively. This includes a representation of mesons, baryons, pentaquark, and so on, in a unified way.

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6. Quark Confinement
Finally, we provide a possible approach toward quark confinement which ascribes to fractionality
in Galois field.
Suppose for the moment that up-quark and down-quark can be described by the following
irreducible representation of GLH (2, p2 ):a u = k a k n u a (1) ,a d = k a k n d a (1) .

(26)

We expect that proton and neutron can be expressed as a p = au ⊗ au ⊗ ad and an = au ⊗ ad ⊗ ad ,respectively. Then, the quantum numbers of proton and neutron can be written in terms of theirconstituent quarks as follows:n p = 2nu + nd ,(27)

nn = nu + 2nd ,(28)

I p = In = I.

(29)

Substitute this decomposition into Equation (23) and we obtain
Q p = 2(2nu + nd + I ),Qn = 2(nu + 2nd + I ).

(30)

2Qn − Q p − 2I
.
6

(31)

Then, we can solve for nu and ndnu =2Q p − Qn − 2I
,
6

nd =

Taking the values Q p = 1, Qn = 0 and I = 3/2 as in Table 2 we obtain
1
nu = − ,62
nd = − ,3(32)which are not half-integers as in (15). Therefore, irreducible representations of quarks as inEquation (26) is not possible.
Now, suppose that we can nevertheless write the corresponding charge numbers as elements ofF p . Then, we obtain Qu = 2/3 and Qd = −1/3. Counterintuitively, these fractional quantum numbers
in F p are very large, which can be shown as follows. The modular inverses in of an integer a in F p isdefined bya−1 a ≡ 1 mod p.
(33)
On the other hand, from Fermat’s little theorem
a p−1 ≡ 1 mod p(34)

a−1 ≡ a p−2 mod p.

(35)

we obtain
The prime given by Equation (2) contains the first odd prime q1 = 3, so it has the form p = 3n − 1 forsome n ∈ N. In this case, the fractional number 1/3 in F p takes the value 3−1 = ( p + 1)/3 (see Figure 3).
Therefore, fractional numbers in Galois field are large numbers of order p. Consequently, isolated quark
can have, for instance a huge electromagnetic self-energy (Figure 4). In this case, quark confinement in
F p may be a consequence of energy minimization.

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Figure 3. The values of 1/3 = 3 p−2 mod p = ( p + 1)/3 and 2/3 = 2 · 3 p−2 mod p = 2( p + 1)/3 areshown for the primes of the form p = 3n − 1. Fractional numbers in a Galois field are large numbers
proportional to p.

2

2
Figure 4. Electromagnetic self-energy Eem = k e Q
2a c 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 arough standard.

7. Discussion and Conclusions
Finally, we summarize and discuss our results. The NNG formula is a phenomenological equation.
By using a Galois field, the NNG Formula (24) is rewritten as (23). The new quantum number n, which
includes isospin Iz and hypercharge Y, arises from the multivaluedness of a linear representation of
the Galois field. We also discussed quark confinement which ascribes to fractionality of Galois field.
After all, these results may indicate finiteness and discreteness of the world. While such a method
is unconventional, it is not inherently inconsistent under one assumption, namely that the geometry
constructed by the real numbers R is only an approximation of a finite and discrete geometry [11,17].

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As future developments, we expect that composite particles such as pentaquarks [28–30] canbe explained by generalizing Equation (25) to any number of particles and combining the quantum
numbers n and I to a single “grand spin” as discussed in [30]. We expect that the inclusion of otherflavors such as c (charm), t (top) and b (bottom) would require extension of our result to GLH (3, p2 )(analogous to SU(3)), GLH (4, p2 ), and so on, which goes in parallel to the extension of SU(2) to SU(3) or
higher. Representation of particles in Figure 2 can be modified accordingly. This extention can be done
based on previous studies [31] and using a method equivalent to the “enlarging lemma” [32]. Recently,
with the development of quantum information, Galois fields have begun to be used to constructspace-time models as error-correcting codes [33]. Further application of Galois fields is expected in thefoundation of quantum physics [34] and time crystal with discrete space-time symmetry [20–24].
Author Contributions: Conceptualization, S.T., K.N., T.F. and T.M.; methodology, M.O., K.N., T.F. and S.T.;
validation, all authors; formal analysis, K.N., M.O., T.F. and S.T.; investigation, all authors; writing, K.N., M.O.,
and S.T.; visualization, K.N., M.O. and S.T.; supervision, S.T.; project administration, S.T. All authors have readand agreed to the published version of the manuscript.
Funding: This research received no external funding.
Acknowledgments: We thank Kohkichi Konno, Masanori Yamanaka, Thomas Zeugmann, Tatsuhiko N. Ikeda,
Masanori Morishita and Ryotaro Okazaki for stimulating and valuable discussions.
Conflicts of Interest: The authors declare no conflict of interest.

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