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.

/lexer

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).

quark-quark_scattering

SmallBookPile

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.

images6-ezgif com-resize

Matrix Scheme

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

quantum-chromodynamics-1-320

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)

UF1

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

UF2

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)

gifman

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)

0_kGdCmWqcFG_s8fIq

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

960x0

$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).

Hadron_colors svg

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

The-PMNS-Neutrino-Mixing-Matrix-The-non-diagonal-structure-and-the-smallness-of-the-U-e3 images (8) 16-0054-07 hr-web images (12) 1-neutrino-oscillation-l

Prime Identity

We are going to assign prime identity as a standard model that attempts to stimulate a quantum field model called eQuantum for the four (4) known fundamental forces.

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.

/lexer

This presentation was inspired by theoretical works from Hideki Yukawa who in 1935 had predicted the existence of mesons as the carrier particles of strong nuclear force.

Addition Zones

Here we would like to recompile the way we take on getting the arithmetic expresion of an individual unit expression (identity) such as a taxicab number below.

Note

It is a taxicab number, and is variously known as Ramanujan’s number and the Ramanujan-Hardy number, after an anecdote of the British mathematician GH Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital (Wikipedia).

These three (3) number are twin primes. We called the pairs as True Prime Pairs. Our scenario is mapping the distribution out of these pairs by taking the symmetrical behaviour of 36 as the smallest power (greater than 1) which is not a prime power.

Tip

The smallest square number expressible as the sum of four (4) consecutive primes in two ways (5 + 7 + 11 + 13 and 17 + 19) which are also two (2) couples of prime twins! (Prime Curios!).

$True Prime Pairs:
 (5,7), (11,13), (17,19)
 
 layer|  i  |   f
 -----+-----+---------
      |  1  | 5
   1  +-----+
      |  2  | 7
 -----+-----+---  } 36 » 6®
      |  3  | 11
   2  +-----+
      |  4  | 13
 -----+-----+---------
      |  5  | 17
   3  +-----+     } 36 » 6®
      |  6  | 19
 -----+-----+---------
  

Thus in short this is all about the method that we called as the 19 vs 18 Scenario of mapping the quantum way within a huge of primes objects (5 to 19) by lexering (11) the ungrammared feed (7) and parsering (13) across syntax (17).

Φ(1,2,3) = Φ(6,12,18) = Φ(13,37,61)

$True Prime Pairs:
(5,7), (11,13), (17,19)
 
layer | node | sub |  i  |  f
------+------+-----+----------
      |      |     |  1  | 
      |      |  1  +-----+          
      |  1   |     |  2  | (5)
      |      |-----+-----+
      |      |     |  3  |
  1   +------+  2  +-----+----
      |      |     |  4  |
      |      +-----+-----+
      |  2   |     |  5  | (7)
      |      |  3  +-----+
      |      |     |  6  |
------+------+-----+-----+------      } (36)
      |      |     |  7  |
      |      |  4  +-----+
      |  3   |     |  8  | (11)
      |      +-----+-----+
      |      |     |  9  |
  2   +------|  5  +-----+-----
      |      |     |  10 |
      |      |-----+-----+
      |  4   |     |  11 | (13)
      |      |  6  +-----+
      |      |     |  12 |
------+------+-----+-----+------------------
      |      |     |  13 |
      |      |  7  +-----+
      |  5   |     |  14 | (17)
      |      |-----+-----+
      |      |     |  15 |
  3   +------+  8  +-----+-----       } (36)
      |      |     |  16 |
      |      |-----+-----+
      |  6   |     |  17 | (19)
      |      |  9  +-----+
      |      |     |  18 |
------|------|-----+-----+------
  

The main background is that, as you may aware, the prime number theorem describes the asymptotic distribution of prime numbers which is still a major problem in mathematic.

Multiplication Zones

Instead of a proved formula we came to a unique expression called zeta function. This expression first appeared in a paper in 1737 entitled Variae observationes circa series infinitas.

Tip

This expression states that the sum of the zeta function is equal to the product of the reciprocal of one minus the reciprocal of primes to the powers. But what has this got to do with the primes? The answer is in the following product taken over the primes p (discovered by Leonhard Euler):

zeta function

This issue is actually come from Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that is considered to be the most important of unsolved problems in pure mathematics.

Note

In addition to the trivial roots, there also exist complex roots for real t. We find that the he first ten (10) non-trivial roots of the Riemann zeta function is occured when the values of t below 50. A plot of the values of ζ(1/2 + it) for t ranging from –50 to +50 is shown below. The roots occur each time the locus passes through the origin. (mathpages).

Meanwhile obtaining the non complex numbers it is easier to look at a graph like the one below which shows Li(x) (blue), R(x) (black), π(x) (red) and x/ln x (green); and then proclaim "R(x) is the best estimate of π(x)." Indeed it is for that range, but as we mentioned above, Li(x)-π(x) changes sign infinitely often, and near where it does, Li(x) would be the best value.

And we can see in the same way that the function Li(x)-(1/2)Li(x1/2) is ‘on the average' a better approximation than Li(x) to π(x); but no importance can be attached to the latter terms in Riemann's formula even by repeated averaging.

Exponentiation Zones

The problem is that the contributions from the non-trivial zeros at times swamps that of any but the main terms in these expansions.

Warning

A. E. Ingham says it this way: Considerable importance was attached formerly to a function suggested by Riemann as an approximation to π(x)… This function represents π(x) with astonishing accuracy for all values of x for which π(x) has been calculated, but we now see that its superiority over Li(x) is illusory… and for special values of x (as large as we please) the one approximation will deviate as widely as the other from the true value (primes.utm.edu).

Moreover in it was verified numerically, in a rigorous way using interval arithmetic, that The Riemann hypothesis is true up to 3 · 10^12. That is, all zeroes β+iγ of the Riemann zeta-function with 0<γ≤3⋅1012 have β=1/2.

Danger

We have Λ ≤ 0.2. The next entry is conditional on taking H a little higher than 10*13, which of course, is not achieved by Theorem 1. This would enable one to prove Λ < 0.19. Given that our value of H falls between the entries in this table, it is possible that some extra decimals could be wrought out of the calculation. We have not pursued this (arXiv:2004.09765).

This Euler formula represents the distribution of a group of numbers that are positioned at regular intervals on a straight line to each other. Riemann later extended the definition of zeta(s) to all complex numbers (except the simple pole at s=1 with residue one). Euler's product still holds if the real part of s is greater than one. Riemann derived the functional equation of zeta function.

Danger

The Riemann zeta function has the trivial zeros at -2, -4, -6, … (the poles of gamma(s/2)). Using the Euler product (with the functional equation) it is easy to show that all the other zeros are in the critical strip of non-real complex numbers with 0 < Re(s) < 1, and that they are symmetric about the critical line Re(s)=1/2. The unproved Riemann hypothesis is that all of the nontrivial zeros are actually on the critical line (primes.utm.edu).

If both of the above statements are true then mathematically this Riemann Hypothesis is proven to be incorrect because it only applies to certain cases or limitations. So first of all the basis of the Riemann Hypothesis has to be considered.

Warning

The solution is not only to prove Re(z)= 1/2 but also to calculate ways for the imaginary part of the complex root of ζ(z)=0 and also to solve the functional equations. (Riemann Zeta - pdf)

On the other hand, the possibility of obtaining the function of the distribution of prime numbers shall go backwards since it needs significant studies to be traced.

Or may be start again from the Euleur Function.

Identition Zones

Freeman Dyson discovered an intriguing connection between quantum physics and Montgomery's pair correlation conjecture about the zeros of the zeta function which dealts with the distribution of primes.

Note

The Mathematical Elementary Cell 30 (MEC30) standard unites the mathematical and physical results of 1972 by the mathematician Hugh Montgomery and the physicist Freeman Dyson and thus reproduces energy distribution in systems as a path plan more accurately than a measurement. (Google Patent DE102011101032A9)

The path plan assume that a symmetric distribution of prime numbers with equal axial lengths from a middle zero axis = 15 is able to determine the distribution of primes in a given number space. This assumption finally bring us to the equation of Euler's identity.

Note

Euler’s identity is considered to be an exemplar of deep mathematical beauty as it shows a profound connection between the most fundamental numbers. Three (3) of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation (Wikipedia).

The finiteness position of Euler's identity by the said MEC30 opens up the possibility of accurately representing the self-similarity based on the distribution of True Prime Pairs so that all number would belongs together with their own identitities.

Tip

There are 28 octonion Fano plane triangles that correspond directly to the 28 Trott quartic curve bitangents.

These bitangents are directly related to the Legendre functions used in the Shroedinger spherical harmonic electron orbital probability densities.
Shown below is a graphic of these overlaid onto the n=5, l=2, m=1 element, which is assigned to gold (Au).
When using an algorithm based on the E8 positive algebra root assignments, the “flipped” Fano plane has E8 algebra root number 79 (the atomic number of Au) and split real even group number of 228 (in Clifford/Pascal triangle order).
This matrix is shown to be useful in providing direct relationships between E8 and the lower dimensional Dynkin and Coxeter-Dynkin geometries contained within it, geometries that are visualized in the form of real and virtual 3 dimensional objects.
A direct linkage between E8, the folding matrix, fundamental physics particles in an extended Standard Model Gravi GUT, quaternions, and octonions is introduced, and its importance is investigated and described.
E8 and its 4D children, the 600-cell and 120-cell (pages on which I have some work, amongst others) and its grandkids (2 of the 3D 5 Platonic Solids, one of which is the 3D version of the 2D Pentagon) are all related to the Fibonacci numbers and the Golden Ratio.
And finally, the {7, 8} dimensions in physics can be identified with quark color, as {7} preserves the blue quark positions, while {8} moves the dual concentric rings of quarks while preserving their relative positions within the rings. It is interesting t note that the dimensions {6, 7, 8} are appropriately labeled {r, g, b} in SRE coordinates, since in this projection the SRE math coordinates are located at the afforementioned 6 triple overlap points at center of the quark’s {r, g, ¯ g, b, ¯ ¯b} concentric rings (the intersection of the gluons triality lines)

So that kind of explains why most of my 2D art, 3D objects and sculptures (e.g. furniture like the dodecahedron table below), and 4D youtube animations all use the Golden Ratio theme. (E8 to H4 folding matrix - pdf)

Nothing is going to be easly about the nature of prime numbers but they demonstrably congruent to something organized. Let's discuss starting with the addition zones.

Reference: