# Chromodynamics (lexer)

This section serve to study the internal (color) rotations of the gluon fields associated with the coloured quarks in quantum chromodynamics of colours of the gluon.

Tip

This section is referring to wiki page-24 of main section-2 that is inherited from the spin section-131 by prime spin-33 and span- with the partitions as below.

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A gauge colour rotation is a spacetime-dependent SU(3) group element. They span the Lie algebra of the SU(3) group in the defining representation.

## Feynman diagram

Note

In this Feynman diagram, an electron (e−) and a positron (e+) annihilate, producing a photon (γ, represented by the blue sine wave) that becomes a quark–antiquark pair (quark q, antiquark q̄), after which the antiquark radiates a gluon (g, represented by the green helix).

So basically there is a basic transformation between addition of `3 + 4 = 7` in to their multiplication of `3 x 4 = 12` while the 7 vs 12 will be treated as exponentiation.

## Matrix Scheme

Quarks have three colors. Color is to the strong interaction as electric charge is to the electromagnetic interaction.

``````red   anti-red,   red   anti-blue,   red   anti-green,
blue  anti-red,   blue  anti-blue,   blue  anti-green,
green anti-red,   green anti-blue,   green anti-green.
``````

This exponentiation takes important roles since by the multiplication zones the MEC30 forms a matrix of `8 x 8 = 64 = 8²` where the power of 2 stands as exponent

Note

During the last few years of the 12th century, Fibonacci undertook a series of travels around the Mediterranean. At this time, the world’s most prominent mathematicians were Arabs, and he spent much time studying with them. His work, whose title translates as the Book of Calculation, was extremely influential in that it popularized the use of the Arabic numerals in Europe, thereby revolutionizing arithmetic and allowing scientific experiment and discovery to progress more quickly. (Famous Mathematicians)

Since the first member is 30 then the form is initiated by a matrix of `5 x 6 = 30` which has to be transformed first to `6 x 6 = 36 = 6²` prior to the above MEC30's square.

Note

A square system of coupled nonlinear equations can be solved iteratively by Newton’s method. This method uses the Jacobian matrix of the system of equations. (Wikipedia)

Note

Fermions and bosons—fermions have quantum spin = 1/2.

• The elementary fermions are leptons and quarks.
• There are three generations of leptons: electron, muon, and tau, with electric charge −1, and their neutrinos with no electric charge.
• There are three generations of quarks: (u, d); (c, s); and (t, b).

The (u, c, t) quarks have electric charge 2/3 while the (d, s, b) quarks have electric charge −1/3. (IntechOpen)

Interactions in quantum chromodynamics are strong, so perturbation theory does not work. Therefore, Feynman diagrams used for quantum electrodynamics cannot be used.

Bosons have quantum spin = 1: photon, quantum of the electromagnetic field; gluon, quantum of the strong field; and W and Z, weak field quanta, which we do not need.

Note

An animation of color confinement, a property of the strong interaction. If energy is supplied to the quarks as shown, the gluon tube connecting quarks elongates until it reaches a point where it “snaps” and the energy added to the system results in the formation of a quark–antiquark pair. Thus single quarks are never seen in isolation. (Wikipedia)

``````  Fermion  | spinors | charged | neutrinos |   quark   | components | parameter
Field   |   (s)   |   (c)   |    (n)    | (q=s.c.n) |  Σ(c+n+q   | (complex)
===========+=========+=========+===========+===========+============+===========
bispinor-1 |    2    |    3    |     3     |    18     |     24     |   19+i5
-----------+---------+---------+-----------+-----------+------------+-----------
bispinor-2 |    2    |    3    |     3     |    18     |     24     |   17+i7 👈
===========+=========+=========+===========+===========+============+===========
bispinor-3 |    2    |    3    |     3     |    18     |     24     |   11+i13
-----------+---------+---------+-----------+-----------+------------+-----------
bispinor-4 |    2    |    3    |     3     |    18     |     24     |   19+i5
===========+=========+=========+===========+===========+============+===========
Total |    8    |   12    |    12     |    72     |     96     |   66+i30
``````

## Interactions

The subclasses of partitions systemically develops characters similar to the distribution of prime numbers.

Note

Unlike the strong force, the residual strong force diminishes with distance, and does so rapidly. The decrease is approximately as a negative exponential power of distance, though there is no simple expression known for this; see Yukawa potential. The rapid decrease with distance of the attractive residual force and the less rapid decrease of the repulsive electromagnetic force acting between protons within a nucleus, causes the instability of larger atomic nuclei, such as all those with atomic numbers larger than 82 (the element lead). (Wikipedia)

Note

Feynman diagram for the same process as in the animation, with the individual quark constituents shown, to illustrate how the fundamental strong interaction gives rise to the nuclear force. Straight lines are quarks, while multi-colored loops are gluons (the carriers of the fundamental force). Other gluons, which bind together the proton, neutron, and pion “in-flight”, are not shown. The π⁰ pion contains an anti-quark, shown to travel in the opposite direction, as per the Feynman–Stueckelberg interpretation. (Wikipedia)

Note

The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3×3 traceless Hermitian matrices used in the study of the strong interaction in particle physics. They span the Lie algebra of the SU(3) group in the defining representation.

• These matrices are traceless, Hermitian, and obey the extra trace orthonormality relation (so they can generate unitary matrix group elements of SU(3) through exponentiation[1]). These properties were chosen by Gell-Mann because they then naturally generalize the Pauli matrices for SU(2) to SU(3), which formed the basis for Gell-Mann’s quark model.[2] Gell-Mann’s generalization further extends to general SU(n). For their connection to the standard basis of Lie algebras, see the Weyl–Cartan basis.
• Since the eight matrices and the identity are a complete trace-orthogonal set spanning all 3×3 matrices, it is straightforward to find two Fierz completeness relations, (Li & Cheng, 4.134), analogous to that satisfied by the Pauli matrices. Namely, using the dot to sum over the eight matrices and using Greek indices for their row/column indices
• A particular choice of matrices is called a group representation, because any element of SU(3) can be written in the form using the Einstein notation, where the eight are real numbers and a sum over the index j is implied. Given one representation, an equivalent one may be obtained by an arbitrary unitary similarity transformation, since that leaves the commutator unchanged.
• The matrices can be realized as a representation of the infinitesimal generators of the special unitary group called SU(3). The Lie algebra of this group (a real Lie algebra in fact) has dimension eight and therefore it has some set with eight linearly independent generators, which can be written as g_{i}, with i taking values from 1 to 8

These matrices serve to study the internal (color) rotations of the gluon fields associated with the coloured quarks of quantum chromodynamics (cf. colours of the gluon). A gauge colour rotation is a spacetime-dependent SU(3) group element where summation over the eight indices (8) is implied. Wikipedia)

``````\$True Prime Pairs:
(5,7), (11,13), (17,19)

|    168    |    618    |
-----+-----+-----+-----+-----+                                             ---
19¨ |  3¨ |  4¨ |  6¨ |  6¨ | 4¤  ----->  assigned to "id:30"             19¨
-----+-----+-----+-----+-----+                                             ---
17¨ |  5¨ |  3¨ |  ❓ |  ❓ | 4¤ ✔️ --->  assigned to "id:31"              |
+-----+-----+-----+-----+                                              |
{12¨}|  .. |  .. |  2¤ (M & F)     ----->  assigned to "id:32"              |
+-----+-----+-----+                                                    |
11¨ |  .. |  .. |  .. | 3¤ ---->  Np(33)  assigned to "id:33"  ----->  👉 77¨
-----+-----+-----+-----+-----+                                              |
19¨ |  .. |  .. |  .. |  .. | 4¤  ----->  assigned to "id:34"              |
+-----+-----+-----+-----+                                              |
{18¨}|  .. |  .. |  .. | 3¤        ----->  assigned to "id:35"              |
+-----+-----+-----+-----+-----+-----+-----+-----+-----+               ---
43¨ |  .. |  .. |  .. |  .. |  .. |  .. |  .. |  .. |  .. | 9¤ (C1 & C2)  43¨
-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+               ---
139¨ |  1     2     3  |  4     5     6  |  7     8     9  |
Δ                 Δ                 Δ
``````

From the 50 we gonna split the 15 by bilateral 9 sums resulting 2 times 15+9=24 which is 48. So the total of involved objects is 50+48=98.

Note

Consider the evidence: scattering experiments strongly suggest a meson to be composed of a quark anti-quark pair and a baryon to be composed of three quarks. The famous 3R experiment also suggests that whatever force binds the quarks together has 3 types of charge (called the 3 colors).

• Now, into the realm of theory: we are looking for an internal symmetry having a 3-dimensional representation which can give rise to a neutral combination of 3 particles (otherwise no color-neutral baryons).
• The simplest such statement is that a linear combination of each type of charge (red + green + blue) must be neutral, and following William of Occam we believe that the simplest theory describing all the facts must be the correct one.
• We now postulate that the particles carrying this force, called gluons, must occur in color anti-color units (i.e. nine of them).
• BUT, red + blue + green is neutral, which means that the linear combination red anti-red + blue anti-blue + green anti-green must be non-interacting, since otherwise the colorless baryons would be able to emit these gluons and interact with each other via the strong force—contrary to the evidence. So, there can only be EIGHT gluons.

This is just Occam’s razor again: a hypothetical particle that can’t interact with anything, and therefore can’t be detected, doesn’t exist. The simplest theory describing the above is the SU(3) one with the gluons as the basis states of the Lie algebra. That is, gluons transform in the adjoint representation of SU(3), which is 8-dimensional. (Physics FAQ)

Please note that we are not talking about the number of 19 which is the 8th prime. Here we are talking about 19th as sequence follow backward position of 19 as per the scheme below where the 19th prime which is 67 goes 15 from 66 to 51.

π(1000) = π(Φ x 618) = 168 = 100 + 68 = (50x2) + (66+2) = 102 + 66

``````\$True Prime Pairs:
(5,7), (11,13), (17,19)

|    168    |    618    |
-----+-----+-----+-----+-----+                                             ---
19¨ |  3¨ |  4¨ |  6¨ |  6¨ | 4¤  ----->  assigned to "id:30"             19¨
-----+-----+-----+-----+-👇--+                                             ---
17¨ |  5¨ |  3¨ |  ❓ |  7¨ | 4¤ ✔️ --->  assigned to "id:31"              |
+-----+-----+-----+-----+                                              |
{12¨}|  .. |  .. |  2¤ (M & F)     ----->  assigned to "id:32"              |
+-----+-----+-----+                                                    |
11¨ |  .. |  .. |  .. | 3¤ ---->  Np(33)  assigned to "id:33"  ----->  👉 77¨
-----+-----+-----+-----+-----+                                              |
19¨ |  .. |  .. |  .. |  .. | 4¤  ----->  assigned to "id:34"              |
+-----+-----+-----+-----+                                              |
{18¨}|  .. |  .. |  .. | 3¤        ----->  assigned to "id:35"              |
+-----+-----+-----+-----+-----+-----+-----+-----+-----+               ---
43¨ |  .. |  .. |  .. |  .. |  .. |  .. |  .. |  .. |  .. | 9¤ (C1 & C2)  43¨
-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+               ---
139¨ |  1     2     3  |  4     5     6  |  7     8     9  |
Δ                 Δ                 Δ
``````

In number theory, the partition functionp(n) represents the number of possible partitions of a non-negative integer n. Integers can be considered either in themselves or as solutions to equations (Diophantine geometry).

Note

Young diagrams associated to the partitions of the positive integers 1 through 8. They are arranged so that images under the reflection about the main diagonal of the square are conjugate partitions (Wikipedia).

Note

In mathematics, orthonormality typically implies a norm which has a value of unity (1). Gell-Mann matrices, however, are normalized to a value of 2.

• Thus, the trace of the pairwise product results in the ortho-normalization condition where delta is the Kronecker delta.
• This is so the embedded Pauli matrices corresponding to the three embedded subalgebras of SU(2) are conventionally normalized.
• In this three-dimensional matrix representation, the Cartan subalgebra is the set of linear combinations (with real coefficients) of the two matrices which commute with each other.

The SU(2) Casimirs of these subalgebras mutually commute. However, any unitary similarity transformation of these subalgebras will yield SU(2) subalgebras. There is an uncountable number of such transformations. (Wikipedia)

``````\$True Prime Pairs:
(5,7), (11,13), (17,19)

|    168    |    618    |
-----+-----+-----+-----+-----+                                             ---
19¨ |  3¨ |  4¨ |  6¨ |  6¨ | 4¤  ----->  assigned to "id:30"             19¨
-----+-----+-----+-👇--+-----+                                             ---
17¨ | {5¨}| {3¨}|  2¨ |  7¨ | 4¤ ✔️ --->  assigned to "id:31"              |
+-----+-----+-----+-----+                                              |
{12¨}|  .. |  .. |  2¤ (M & F)     ----->  assigned to "id:32"              |
+-----+-----+-----+                                                    |
11¨ |  .. |  .. |  .. | 3¤ ---->  Np(33)  assigned to "id:33"  ----->  👉 77¨
-----+-----+-----+-----+-----+                                              |
19¨ |  .. |  .. |  .. |  .. | 4¤  ----->  assigned to "id:34"              |
+-----+-----+-----+-----+                                              |
{18¨}|  .. |  .. |  .. | 3¤        ----->  assigned to "id:35"              |
+-----+-----+-----+-----+-----+-----+-----+-----+-----+               ---
43¨ |  .. |  .. |  .. |  .. |  .. |  .. |  .. |  .. |  .. | 9¤ (C1 & C2)  43¨
-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+               ---
139¨ |  1     2     3  |  4     5     6  |  7     8     9  |
Δ                 Δ                 Δ
``````