Multiplication Zones (18-30)

Multiplication is the form of expression set equal to the inverse function of symmetrical exponentation which stand as multiplicative identity reflects a point across the origin.

Tip

This section is referring to wiki page-9 of gist section-5 that is inherited from the gist section-19 by prime spin- and span- with the partitions as below.

  1. Symmetrical Breaking (spin 1)
  2. The Angular Momentum (spin 2)
  3. Entrypoint of Momentum (spin 3)
  4. The Mapping of Spacetime (spin 4)
  5. Similar Order of Magnitude (spin 5)
  6. The Search for The Graviton (spin 6)
  7. Elementary Retracements (spin 7)
  8. The Recycling Momentum (spin 8)
  9. Exchange Entrypoint (spin 9)
  10. The Mapping Order (spin 10)
  11. Magnitude Order (spin 11)

The multiplication zones is a symmetric matrix representing the multilinear relationship of a stretching and shearing within the plane of the base unit.

Standard Model

In particle physics, study of the symmetry and its breaking play very important role in order to get useful information about the nature.

Note

Similarly the Standard Model incorporates three generations of quarks, so its fermionic content can be summarized. In addition, the Standard Model involves gauge bosons (photons for the electromagnetic interaction, W and Z for the weak interaction, and eight (8) gluons for the strong interaction), plus the (scalar) Higgs particle. This is what all known matter in the Universe consists of. (Netrinos)

Multiplets-of-the-1-2-spin-baryon-in-SU4-flavour-model ppm

In our approach a 3-form is not an object that exist in addition to the metric, it is the only object that exist and in particular the 4D metric, is defined by the 3-form.

Note

We would like to say that our present use of G2 structures (3-forms in 7D) is different from whatone can find in the literature on Kaluza–Klein compactifications of supergravity.

  • We show that the resulting 4D theory is (Riemannian) General Relativity (GR) in Plebanski formulation, modulo corrections that are negligible for curvatures smaller than Planckian.
  • Possibly the most interesting point of this construction is that the dimensionally reduced theory is GR with a non-zero cosmological constant, and the value of the cosmological constant is directly related to the size of . Realistic values of Λ correspond to of Planck size.

Also, in the supergravity context a 7D manifold with a G2 structure is used for compactifying the 11D supergravity down to 4D. In contrast, we compactify from 7D to 4D. (General relativity from three-forms in seven dimensions - pdf)

3-forms in 7D

The the main reason of assigning two (2) profiles instead of only one (1) is that we have to accommodate the major type of primes numbers called twin primes.

Note

This is a necessary but not sufficient condition for N to be a prime as noted, for example, by N= 6(4)+1= 25, which is clearly composite. We note that each turn of the spiral equals an increase of six units. (HexSpiral-Pdf)

twin primes

This means that we have a mod(6) situation allowing us to write: N mod(6)=6n+1 or N mod(6)=6n-1 (equivalent to 6n+5).

Elementary particles and their interactions are considered by a theoretical framework called the Standard Model (SM) of Particle Physics.

Note

The Standard Model presently recognizes seventeen distinct particles (twelve fermions and five bosons). As a consequence of flavor and color combinations and antimatter, the fermions and bosons are known to have 48 and 13 variations, respectively. Among the 61 elementary particles embraced by the Standard Model number electrons and other leptons, quarks, and the fundamental bosons. (Wikipedia)

17 distinct particles = 12 fermions + 5 bosons = 48 + 13 = 61 variations

Standard_Model_of_Elementary_Particles

It is hypothesized that gravitational interactions are mediated by an as yet undiscovered elementary particle, dubbed the graviton.

How many quarks?

Answer-1: 3 generation x 3 color x 2 types x 2 each = 36 quarks

How many types of quarks are there and what are their names?

Answer-2: 6 flavour x 3 colors x 2 types = 36 quarks

image

Answer-3: 6 flavour x 3 colour x 4 bispinor = 72 quarks

There are 72 quarks

Note

In order to be four-spinors like the electron and other lepton components, there must be one quark component for every combination of flavour and colour, bringing the total to 24 (3 for charged leptons, 3 for neutrinos, and 2·3·3 = 18 for quarks). Each of these is a four (4) component bispinor, for a total of 96 complex-valued components for the fermion field. (Wikipedia)

IMG_20240108_045902

It is stated that each of the 24 components is a four component bispinor. A bispinor is constructed out 2 simpler component spinor so there are eight (8) spinors in total.

Note

Bispinors are so called because they are constructed out of two (2) simpler component spinors, the Weyl spinors. Each of the two (2) component spinors transform differently under the two (2) distinct complex-conjugate spin-1/2 representations of the Lorentz group. This pairing is of fundamental importance, as it allows the represented particle to have a mass, carry a charge, and represent the flow of charge as a current, and perhaps most importantly, to carry angular momentum. (Wikipedia)

((3+3) + 2x(3x3)) x 4 = (3 + 3 + 18) x 4 = 24 x 4 = 96 components

  Fermion  | spinors | charged | neutrinos |   quark   | components
   Field   |   (s)   |   (c)   |    (n)    | (q=s.c.n) |  Σ(c+n+q)
===========+=========+=========+===========+===========+============
bispinor-1 |    2    |    3    |     3     |    18     |     24
-----------+---------+---------+-----------+-----------+------------ } 48
bispinor-2 |    2    |    3    |     3     |    18     |     24
===========+=========+=========+===========+===========+===========
bispinor-3 |    2    |    3    |     3     |    18     |     24
-----------+---------+---------+-----------+-----------+------------ } 48
bispinor-4 |    2    |    3    |     3     |    18     |     24
===========+=========+=========+===========+===========+============
     Total |    8    |   12    |    12     |    72     |     96

Thus fermion is constructed out of eight (8) spinors that brings the total of 96 components consist of 12 charged leptons, 12 neutrinos and 72 quarks.

Note

All perfect squares within our domain (numbers not divisible by 2, 3 or 5) possess a digital root of 1, 4 or 7 and are congruent to either {1} or {19} modulo 30. When the digital root of perfect squares is sequenced within a modulo 30 x 3 = modulo 90 horizon, beautiful symmetries in the form of period-24 palindromes are revealed. Here’s one modulo 90 spin on perfect squares.

  • parsing the squares by their mod 90 congruence reveals that there are 96 perfect squares generated with each 4 * 90 = 360 degree cycle,
  • which distribute 16 squares to each of 6 mod 90 congruence sub-sets defined as n congruent to {1, 19, 31, 49, 61, 79} forming 4 bilateral 80 sums.
  • each of the 6 columns has 8 bilateral 360 sums, tor a total of 48 * 360 = 40 * 432 (much more on the significance of number 432, elsewhere on this site).

There’s another hidden dimension of our domain worth noting involving multiples of 360, i.e., when framed as n ≌ {1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53 59, 61, 67, 71, 73, 77, 79, 83, 89} modulo 90, and taking ‘bipolar’ differentials of perfect squares (PrimesDemystified)

16 × 6 = 96

96 perfect squares

Beyond the 96

Note

Fermion particles are described by Fermi–Dirac statistics and have quantum numbers described by the Pauli exclusion principle. They include the quarks and leptons, as well as any composite particles consisting of an odd number of these, such as all baryons and many atoms and nuclei. Fermions have half-integer spin; for all known elementary fermions this is 1⁄2. In the Standard Model, there are 12 types of elementary fermions: six quarks and six leptons.

  • Leptons do not interact via the strong interaction. Their respective antiparticles are the antileptons, which are identical, except that they carry the opposite electric charge and lepton number. The antiparticle of an electron is an antielectron, which is almost always called a “positron” for historical reasons.IMG_20240108_032736
    • There are six leptons in total; the three charged leptons are called “electron-like leptons”, while the neutral leptons are called “neutrinos”.
    • Neutrinos are known to oscillate, so that neutrinos of definite flavor do not have definite mass, rather they exist in a superposition of mass eigenstates.
    • The hypothetical heavy right-handed neutrino, called a sterile neutrino, has been omitted.
  • Quarks are the fundamental constituents of hadrons and interact via the strong force. Quarks are the only known carriers of fractional charge, but because they combine in groups of three quarks (baryons) or in pairs of one quark and one antiquark (mesons), only integer charge is observed in nature.IMG_20240108_033012
    • Their respective antiparticles are the antiquarks, which are identical except that they carry the opposite electric charge (for example the up quark carries charge +2⁄3, while the up antiquark carries charge −2⁄3), color charge, and baryon number.
    • There are six flavors of quarks; the three positively charged quarks are called up-type quarks while the three negatively charged quarks are called down-type quarks.

All known fermions except neutrinos, are also Dirac fermions; that is, each known fermion has its own distinct antiparticle. It is not known whether the neutrino is a Dirac fermion or a Majorana fermion.[4] Fermions are the basic building blocks of all matter. They are classified according to whether they interact via the strong interaction or not.

Electrodynamics

Free Parameters

The physical evolution of neutrino parameters with respect to energy scale may help elucidate the mechanism for their mass generation.

Note

The most general Lagrangian with massless neutrinos, one finds that the dynamics depend on 19 parameters, whose numerical values are established by experiment.

  • The 19 certain parameters are summarized below:IMG_20231230_232603
  • The neutrino parameter values are still uncertain.
  • The value of the vacuum energy (or more precisely, the renormalization scale used to calculate this energy) may also be treated as an additional free parameter.

The renormalization scale may be identified with the Planck scale or fine-tuned to match the observed cosmological constant. However, both options are problematic. (Wikipedia)

$True Prime Pairs:
(5,7), (11,13), (17,19)
 
layer | node | sub |  i  |  f.                                       MEC 30 / 2
------+------+-----+-----+------      ‹------------------------------ 0 {-1/2}
      |      |     |  1  | --------------------------
      |      |  1  +-----+                           |    
      |  1   |     |  2  | (5)                       |
      |      |-----+-----+                           |
      |      |     |  3  |                           |
  1   +------+  2  +-----+----                       |
      |      |     |  4  |                           |
      |      +-----+-----+                           |
      |  2   |     |  5  | (7)                       |
      |      |  3  +-----+                           |
      |      |     |  6  |                          11s ‹-- ∆28 = (71-43)
------+------+-----+-----+------      } (36)         |
      |      |     |  7  |                           |
      |      |  4  +-----+                           |
      |  3   |     |  8  | (11)                      |
      |      +-----+-----+                           |
      |      |     |  9  |‹-- ∆9 + ∆18 = ∆27         |
  2   +------|  5* +-----+-----                      |
      |      |     |  10 |                           |
      |      |-----+-----+                           |
      |  4   |     |  11 | (13) --------------------- ∆32
      |      |  6  +-----+            ‹------------------------------ 15 {0}
      |      |     |  12 |---------------------------
------+------+-----+-----+------------               |
      |      |     |  13 |                           |
      |      |  7  +-----+                           |
      |  5   |     |  14 | (17)                      |
      |      |-----+-----+                           |
      |      |     |  15 |                           7s ‹-- ∆24 = (43-19)
  3*  +------+  8  +-----+-----       } (36)         |
      |      |     |  16 |                           |
      |      |-----+-----+                           |
      |  6   |     |  17 | (19) ‹-- parameters ✔️    |
      |      |  9  +-----+                           |
      |      |     |  18 | -------------------------- ∆68 - ∆18 = ∆50
------|------|-----+-----+-----  ‹----------------------------------- 30 {+1/2}

The Standard Model with massive neutrinos need 7 more parameters (3 masses and 4 PMNS matrix parameters) for a total of 26 parameters.

Note

In principle, there is one further parameter in the Standard Model; the Lagrangianof QCD can contain a phase that would lead to CP violation in the strong interac-tion.

  • Experimentally, this strong CP phase is known to be extremely small, θCP ≃ 0, and is usually taken to be zero.
  • If θCP is counted, then the Standard Model has 26 free parameters.
  • The relatively large number of free parameters is symptomatic of the StandardModel being just that; a model where the parameters are chosen to match the observations, rather than coming from a higher theoretical principle.
  • Putting aside θCP, of the 25 SM parameters, 14 are associated with the Higgs field, eight with theflavour sector and only three with the gauge interactions.

Likewise, the coupling constants of the three gauge interactions are of a similar order of magnitude, hinting that they might be different low-energy manifestations of a Grand Unified Theory (GUT) of the forces. These patterns provide hints for, as yet unknown, physics beyond the Standard Model. (Modern Particle Physics - pdf)

(24-5) + (24-17) = 19 + 7 = 26

  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     |     ❓
-----------+---------+---------+-----------+-----------+------------+-----------
bispinor-4 |    2    |    3    |     3     |    18     |     24     |     ❓
===========+=========+=========+===========+===========+============+===========
     Total |    8    |   12    |    12     |    72     |     96     |     ❓
Note

Because the value 30 is the first (common) product of the first 3 primes. And this 30th order repeats itself to infinity. Even in the first 30s system, therefore, the positions are fixed in which the number information positions itself to infinity. We call it the first member of the MEC 30.

  • The numbers not divisible by 2, 3 or 5 are highlighted. We call them prime positions, hence 1, 7, 11, 13, 17, 19, 23, 29. Important for our work is that in the following the term prime refers only to prime numbers that are in the prime positions. So primes 2, 3 and 5 are always excluded.
  • These positions: 1 7 11 13 17 19 23 29. We refer to this basic system as MEC 30 - “Mathematical Elementary Cell 30”. By repeating the positions we show the function of the basic system in the next step. If we extend the 30th order of the MEC, for example, to the number 120, the result is 4 times a 30th order and thus 4 × 8 = 32 prime positions.
  • Hypothetical assumption: If the product of the primes (except 2, 3, 5,) would not fall into the prime positions, thus be divided by 2, 3 or 5, the information would have 120 = 32 primes in 32 prime positions: 1, 7, 11, 13, 17, 19, 23, 29, / 31, 37, 41, 43, 47, 49, 53, 59, / 61, 67, 71, 73, 77, 79, 83, 89, / 91, 97, 101, 103, 107, 109, 113, 119
  • These forms gives prime positions: 1, 7, 11, 13, 17, 19, 23, 29, / 1, 7, 11, 13, 17, 19, 23, 29, / 1, 7, 11, 13, 17 , 19, 23, 29, / 1, 7, 11, 13, 17, 19, 23, 29. The 30th order is repeated in the number space 120 = 4 times, 4 × 8 = 32 prime positions, thus 4 terms.

From our consideration we can conclude that the distribution of prime numbers must have a static base structure, which is also confirmed logically in the further course. This static structure is altered by the products of the primes themselves, since these products must fall into the prime positions since they are not divisible by 2, 3 and 5. (Google Patent DE102011101032A9)

Note

Now we show the interplay of the finite system of prime positions with the 15 finite even positions in the cyclic convolution. Consequently, we only need to fold a 30’s cycle as so that we can identify the opposite prime positions that form their specific pairs in a specific convolution.

13+17 = 11+19 = 30

  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     |     ❓
===========+=========+=========+===========+===========+============+===========
     Total |    8    |   12    |    12     |    72     |     96     |     ❓

The pairwise disjoint

Note

The Lie group structure of the Lorentz group is explored. Its generators and its Lie algebra are exhibited, via the study of infinitesimal Lorentz transformations.

  • The exponential map is introduced and it is shown that the study of the Lorentz group can be reduced to that of its Lie algebra.
  • Finally, the link between the restricted Lorentz group and the special linear group is established via the spinor map.

The Lie algebras of these two groups are shown to be identical (up to some isomorphism).

270355_1_En_7_Fig1_HTML

Note

The four pairwise disjoint and non-compact connected components of the Lorentzgroup L = O(1, 3) and corresponding subgroups:

  • the proper Lorentz group L+ = SO(1, 3),
  • the orthochronous Lorentz group L↑,
  • the orthochronous Lorentz group Lo = L↑ + ∪ TL↑+ (see below) and
  • the proper orthochronous Lorentz group L↑+ = SO+(1, 3), which contains the identity element.

Of course, the sets L↓−, L↑− and L↓+ do not represent groups due to the missing identity element. ([The-four-pairwise-disjoint)

19 + i(13+5) = 19 + i18

  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 ✔️

A bispinor is more or less "the same thing" as a Dirac spinor. The convention used here is that the article on the Dirac spinor presents plane-wave solutions to the Dirac equation.

19 + 7 = 26

The-four-pairwise-disjoint-and-non-compact-connected-components-of-the-Lorentz-group-L

 Bispinors | 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
-----------+---------+---------+-----------+-----------+------------+-- 17
bispinor-2 |    2    |    3    |     3     |    18     |     24     |   i5+i7 ✔️
===========+=========+=========+===========+===========+============+===========
bispinor-3 |    2    |    3    |     3     |    18     |     24     |   11
-----------+---------+---------+-----------+-----------+------------+-- 19
bispinor-4 |    2    |    3    |     3     |    18     |     24     |   i13+i5 ✔️
===========+=========+=========+===========+===========+============+===========
     Total |    8    |   12    |    12     |    72     |     96     |   66+i30

ReciprocalTransform

Taking 19 as a certain parameter we can see that the left handed cycles are happen on 5th-spin (forms 4th hexagon, purple) and 6th-spin (forms 5th hexagon, cyan). Both have different rotation with other spin below 9th spin (forms 6th hexagon, yellow).

Note

Proceeding, the number line begins to coil upon itself; 20 lands on 2’s cell, 21 on 3’s cell. Prime number 23 sends the number line left to form the fourth (4th) hexagon, purple. As it is not a twin, the clockwise progression (rotation) reverses itself. Twin primes 29 and 31 define the fifth (5th) hexagon, cyan. Finally, 37, again not a twin, reverses the rotation of the system, so 47 can define the yellow hexagon (HexSpin).

7th spin - 4th spin = (168 - 102)s = 66s = 6 x 11s = 30s + 36s

IMG_20231221_074421

Thus it appears that the cosmological models] derived from compactification of 11d supergravity on a manifold with G2 holonomy have some hidden E7 symmetry.

Counterpart of Bispinor

A Dirac fermionis equivalent to two (2) Weyl fermions so it is not the same as bispinor. The counterpart is a Majorana fermion, a particle that must be its own antiparticle.

Note

Because particles and antiparticles have opposite conserved charges, Majorana fermions have zero charge, hence among the fundamental particles, the only fermions that could be Majorana are sterile neutrinos, if they exist.

If they do, then at low energy (after electroweak symmetry breaking), by the seesaw mechanism, the neutrino fields would naturally behave as six Majorana fields, with three of them expected to have very high masses (comparable to the GUT scale) and the other three expected to have very low masses (below 1 eV).

matrices-interpreted-2

 Majorana  | spinors | charged | neutrinos |   quark   | components | parameter
  Fields   |   (s)   |   (c)   |    (n)    | (q=s.c.n) |  Σ(c+n+q   | (complex)
===========+=========+=========+===========+===========+============+===========
majorana-1 |   2x2   |    -    |    18     |     -     |     18     |   13+5
-----------+---------+---------+-----------+-----------+------------+-----------
majorana-2 |   2x2   |    -    |    12     |     -     |     12     |   5+7
-----------+---------+---------+-----------+-----------+------------+-----------
majorana-3 |   2x2   |    -    |    ❓     |     -     |     ❓     |   i66❓
===========+=========+=========+===========+===========+============+===========
     Total |   12    |    -    |    ❓     |     -     |     ❓     |   30+i66❓
Note

In physics, a subatomic particle is a particle smaller than an atom.[1]

subatomic particles

Experiments show that light could behave like a stream of particles (called photons) as well as exhibiting wave-like properties. This led to the concept of wave–particle duality to reflect that quantum-scale particles behave both like particles and like waves; they are sometimes called wavicles to reflect this. (Wikipedia)

the 12 fermions and 5 bosons are known to have 48 and 13 variations, respectively

 Majorana  | spinors | charged | neutrinos |   quark   | components | parameter
  Fields   |   (s)   |   (c)   |    (n)    | (q=s.c.n) |  Σ(c+n+q   | (complex)
===========+=========+=========+===========+===========+============+===========
majorana-1 |   2x2   |    -    |    18     |     -     |     18     |   13+5
-----------+---------+---------+-----------+-----------+------------+-----------
majorana-2 |   2x2   |    -    |    12     |     -     |     12     |   5+7
-----------+---------+---------+-----------+-----------+------------+-----------
majorana-3 |   2x2   |    -    |    13     |     -     |     13     |   i13 ✔️
===========+=========+=========+===========+===========+============+===========
     Total |   12    |    -    |    43     |     -     |     43     |  30+i13 ✔️
Note

Leptons do not interact via the strong interaction.

  • Their respective antiparticles are the antileptons, which are identical, except that they carry the opposite electric charge and lepton number.
  • The antiparticle of an electron is an antielectron, which is almost always called a “positron” for historical reasons.
  • There are six leptons in total; the three charged leptons are called “electron-like leptons”, while the neutral leptons are called “neutrinos”.
  • Neutrinos are known to oscillate, so that neutrinos of definite flavor do not have definite mass, rather they exist in a superposition of mass eigenstates.

The hypothetical heavy right-handed neutrino, called a “sterile neutrino”, has been omitted. (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
-----------+---------+---------+-----------+-----------+------------+-- 17
bispinor-2 |    2    |    3    |     3     |    18     |     24     |   i12 ✔️
===========+=========+=========+===========+===========+============+===========
bispinor-3 |    2    |    3    |     3     |    18     |     24     |   11
-----------+---------+---------+-----------+-----------+------------+-- 19
bispinor-4 |    2    |    3    |     3     |    18     |     24     |   i18 ✔️
===========+=========+=========+===========+===========+============+===========
  SubTotal |    8    |   12    |    12     |    72     |     96     |   66+i30
===========+=========+=========+===========+===========+============+===========
majorana-1 |   2x2   |    -    |    18     |     -     |     18     |   18 ✔️
-----------+---------+---------+-----------+-----------+------------+-----------
majorana-2 |   2x2   |    -    |    12     |     -     |     12     |   12 ✔️
-----------+---------+---------+-----------+-----------+------------+-----------
majorana-3 |   2x2   |    -    |    13     |     -     |     13     |   i13
===========+=========+=========+===========+===========+============+===========
  SubTotal |    12   |    -    |    43     |     -     |     43     |  30+i13
===========+=========+=========+===========+===========+============+===========
     Total |    20   |   12    |    55     |    72     |    139     |  96+i43 ✔️

Since the total of parameters is 66+i30 then according to renormalization theory the 12 boson fields should have the total complex value of 30+i66.

Note

When describing spacetime as a continuum, certain statistical and quantum mechanical constructions are not well-defined.

  • To define them, or make them unambiguous, a continuum limit must carefully remove “construction scaffolding” of lattices at various scales.
  • Renormalization procedures are based on the requirement that certain physical quantities (such as the mass and charge of an electron) equal observed (experimental) values. That is, the experimental value of the physical quantity yields practical applications, but due to their empirical nature the observed measurement represents areas of quantum field theory that require deeper derivation from theoretical bases.
  • Renormalization was first developed in quantum electrodynamics (QED) to make sense of infinite integrals in perturbation theory.
  • Initially viewed as a suspect provisional procedure even by some of its originators, renormalization eventually was embraced as an important and self-consistent actual mechanism of scale physics in several fields of physics and mathematics.

Despite his later skepticism, it was Paul Dirac who pioneered renormalization. (Wikipedia)

image

Supersymmetry

images (14)

images (13)

images (15)

In the forty years since 11D on-shell supergravity theory was constructed in 1978, a lot of efforts have been made to understand supergravity in superspace.

Note

Inspired by the history of how Einstein constructed General Relativity, we study the linearized Nordstrom supergravity in 10- and 11-dimensional superspaces.

  • Valise adinkras, although an important subclass, do not encode all information present when a 4D supermultiplet is reduced to 1D. We extend this to non-valise adinkras providing a complete eigenvalue classification via Python code.
  • We found no obstacles to applying the lessons we learned in 4D to higher dimensions. We also derive infinitesimal 10D superspace Weyl transformation laws. The identification of all off-shell ten-dimensional supergeometrical Weyl field strength tensors, constructed from respective torsions.
  • We realize that Lie Algebra techniques, in particular branching rules, Plethysm, and tensor product, provide the key to deciphering the complete list of independent fields that describe a supersymmetric multiplet in arbitrary spacetime dimensions efficiently.
  • Thus, adinkra-based arguments suggest the surprising possibility that the 11D, N=1 scalar superfield alone might describe a Poincare supergravity prepotential or semi-prepotential in analogy to one of the off-shell versions of 4D, N=1.
  • All of these results strongly suggest adynkras are pointing in the direction of using series expansion in terms of Young Tableaux (YT’s) as a tool to gain the most fundamental mathematical understanding of this class of problems.

We show the explicit one-to-one correspondence between Lorentz irreps and field variables, leading to an adynkrafield formalism in which the traditional ζ (theta)-monomials are replaced by YT’s as shown below. (YangruiHu.com)

Higher-Dimensional Supergravity

Adinkrasupersymmetry

Similarity

Note

In supergravity theory, supersymmetry theory and superstring theory, Adinkra symbols are a graphical representation of supersymmetry algebras. The similarity between Adinkra in supersymmetry and Adinkra symbols is that they are both graphical representations with hidden meanings (Prof. Sylvester James Gates Jr.). (Adinkra Alphabet)

SM-SUSY-diagram

Note

Numerous connections have been observed between some, though not all, of these exceptional objects. Most common are objects related to 8 and 24 dimensions, noting that 24 = 8 · 3. By contrast, the pariah groups stand apart, as the name suggests. Exceptional objects related to the number 8 include the following.

  • The octonions are 8-dimensional. The E8 lattice can be realized as the integral octonions (up to a scale factor).The exceptional Lie groups can be seen as symmetries of the octonions and structures derived from the octonions;[19] further, the E8 algebra is related to the E8 lattice, as the notation implies (the lattice is generated by the root system of the algebra).
  • Triality occurs for Spin(8), which also connects to 8 · 3 = 24.Likewise, exceptional objects related to the number 24 include The Leech lattice is 24-dimensional.Most sporadic simple groups can be related to the Leech lattice, or more broadly the Monster.The exceptional Jordan algebra has a representation in terms of 24×24 real matrices together with the Jordan product rule.These objects are connected to various other phenomena in math which may be considered surprising but not themselves “exceptional”. For example, in algebraic topology, 8-fold real Bott periodicity can be seen as coming from the octonions. In the theory of modular forms, the 24-dimensional nature of the Leech lattice underlies the presence of 24 in the formulas for the Dedekind eta function and the modular discriminant, which connection is deepened by Monstrous moonshine, a development that related modular functions to the Monster group.

In string theory and superstring theory we often find that particular dimensions are singled out as a result of exceptional algebraic phenomena. For example, bosonic string theory requires a spacetime of dimension 26 which is directly related to the presence of 24 in the Dedekind eta function. Similarly, the possible dimensions of supergravity are related to the dimensions of the division algebras. (Wikipedia)

1200px-Exceptionalmindmap2