For a logic of elementary universe

The theory of groups plays a major role in elementary particle physics. Thus, group SU5 has made it possible to demonstrate that, contrary to appearances, quarks and leptons are not irreconcilably different, and that they belong to the same family.

It is group G3 that is used here.

It can be seen that the group of 48 sub-sets of the six permutations of group G3 (See page Figures) is built on the same model as the table grouping the 48 elementary particles and antiparticles of the Standard Theory, which proposes a theoretical representation of the sub-nuclear universe (See page Figures).

After analysis of correspondence between the two tables, the implications of this similarity are examined.

 

The mathematical table

Let E = {a,b,c} , a set with three abstract elements. G3 is used to designate the six permutations of E. If each of these six permutations is used to give the image of E, six successive arrangements are obtained that break down logically into three families, each with two branche:

1st family
2nd family
3rd family
(a,b,c) (a,c,b)
(b,a,c) (b,c,a)
(c,a,b) (c,b,a)

Each family is defined by the invariable element common to its two branches, whereas each branch differs from the other by the inversion of its two permutations. Thus, in the first family, (a,b,c) (a,c,b), element a is invariable, whereas elements b and c permute. The other two families are built on the same mode: the second has b as its invariable element, the third, c.

Each of the six images of E is a set with three elements and therefore, like any three-element set, has eight sub-sets. The sub-sets of the image (a,b,c) ar: (a), (b), (c), (bc), (a,c), (ab), (a,b,c), Ø .

The image (a,b,c) appears in the series because any set is a sub-set of itself ; the empty set Ø, is its complement.

As each of the six images of E has eight sub-sets, there are, in all, 48 sub-sets, which make up a table of six lines and eight columns.

The sub-sets of each image of E are placed on a line in the order of the image elements. Thus, the order of the eight sub-sets referred to above is determined by the order of the image elements (a,b,c). This is immediately followed by the sub-sets with one elemen: (a), (b), (c) ; then, in corresponding order, sub-sets with two element: (bc), (ac), (ab), inscribed in the table using the mathematical notation (), (),() to show that they are complements of (a), (b) (c). The sub-set with three elements, (a,b,c), accompanied by its complement, the empty set Ø, closes the line.

The image (a,b,c) is placed at the start of the line, because it is the first in alphabetical order. It is not to be confused with the set {a,b,c} , where the curly brackets indicate that it is not ordered.

There is a geometrical interpretation of group G3.

Let {a,b,c} be the set of three apexes of an equilateral triangle with center o. Six isometries of plane leave the triangle invariabl: coincidence, three symmetries, two rotations with center o and angle +2/3 and - 2/3.

Each permutation corresponds to one of the six isometries of plane.

 

The Standard Table

The Standard Theory gives 24 elementary particles in three families, each with two branches. A table of 24 elementary antiparticles in three families, each with two branches, corresponds to this table. It is possible to merge these tables into a single one, using the mathematical-model table with 48 sub-sets of six permutations of group G3.

To build up the table grouping the elementary particles and antiparticles, it is only necessary to determine the physical correspondents of elements a, b and c from which the mathematical table is constructed. Once the three physical correspondents are obtained in accordance with the model of mathematical elements a, b and c, a table can be built grouping the 24 elementary particles and 24 elementary antiparticles.

Before going any further, it must be underlined that a sub-set is itself a set and, like all sets, it is not the same as its element or elements. Thus, the mathematical table is not built up from single-element sub-sets (a), (b), (c), but from their elements a,b,c. For example, the complement of (a), the two-element sub-set, is not written ((b), (c)), but (bc).

Similarly, the Standard Table is not built up from three physical, homologous quarks of (a), (b), (c) but, as we will see below, from their three homologous physical color charges of a, b, c.

In fact, each of the three sub-sets (a), (b), (c) carries an element, a letter that characterizes it. In the same way, each of the three corresponding quarks carries an element, its color charge, which differentiates it from the other two quarks.

The color charge therefore corresponds to elements a, b, c and, in accordance with their model, can build the Standard Table.Quantum chromodynamics can be used to describe the role of color in filling in each of the boxes in the "total" Standard Table grouping all of the constituent elements of matter and antimatter.

The quarks -elementary particles- each carry one of the charges known as color because they associate in the same way as colors. Their antiparticles, the antiquarks, each carry a charge that is the complementary color to that of the corresponding quark.

The three colors usually used to designate the quark charges are red, violet and green. The colors attributed to the antiquarks -also known as anticolors- are cyan (antired), yellow (antiviolet) and magenta (antigreen).

Quantum chromodynamics describes the internal structure of hadrons, i.e. baryons and mesons. Baryons are formed from three quarks of different colors which associate in the same way as normal colors, thus leaving the baryon colorless, also termed white. This is also the case for mesons, formed from a quark and the corresponding antiquark. Thus the color of the pi meson, carrying a violet quark and a yellow antiquark, is white.

In this text, we have preferred the attribution of primary colors red, yellow and blue to quarks, and the complementary colors, green, violet and orange, to the antiquarks. This is because, if the Standard Table is the image of the mathematical table, the quark corresponds to the single-element sub-set, for example (a), and the antiquark to its complement, the two-element sub-set (bc). This is why the quark is here assumed to carry "one"color and the antiquark "two," which when added together form the complementary anticolor.

In this system the quark belonging to the pi meson mentioned above is yellow (one color) and the antiquark is violet (two color: blue + red), not the reverse. However, in both cases, the rule for adding colors is respected and the particle remains white.

This brief overview of the structure of hadrons constitutes a cursory summary of the ways of associating colors, but it does not assume the existence of hadrons in the Standard Table ; this latter only groups the elementary building blocks of matter and antimatter.There is a third way of associating colors, which will emerge during the subsequent examination of the Standard Table.

The "total" Standard Table comprises 24 particles and 24 antiparticles in three families, each with two branches, arranged in six lines and eight columns.

Each of the six branches of three families carries, in its first three boxes, three quarks of different colors, arranged each time in a different order. There are therefore 18 quarks in the first three columns of the table. These are followed, in the three adjacent columns, by the 18 complementary antiquarks.

It is more difficult to justify the presence, in the seventh column, of charged leptons and antileptons, from top dow: the electron and positron in the first family ; the muon and antimuon in the second ; and the tau and antitau in the third. Of course, as images of the three-element sub-sets, they logically fit in after the quarks and antiquarks, themselves images of single- and two-element sets respectively. But how can it be accepted that charged leptons and antileptons, carriers of three color charges, are elementary particles and, above all, are not subject to the color force, also known as the strong interaction ?

Firstly, it is possible to reply that the antiquark, a colored sum of two color charges, is an elementary particle, and that this is also true for the charged lepton or antilepton, the white sum of three color charges.It is more difficult to justify the lack of sensitivity of charged leptons and antileptons to the color force. This has its source in the hadrons, for example a baryon, like a proton in which particles (three quarks) mutually sensitive to their presence, exchange gluons, the vectors of the color force.

On the other hand, given their elementary nature, charged leptons and antileptons obviously cannot be the source of exchanges of gluons between included particles nor, as a consequence, be surrounded by messenger pions external to gluons.

Beside the charged leptons and antileptons, in the eighth column of the Standard Table, the leptons and antileptons without electrical charge -accordingly called neutrinos and antineutrinos- have empty sets as their mathematical equivalent. Empty is an evocative adjective which means that these particles could also be seen as mathematical fiction for which there could not, reasonably, be a physical equivalent.

However, as the table system works for the other 42 components, it would be surprising if it failed to do so for the uncharged leptons and antileptons, and they can, in fact, be shown.

The fact that the neutrinos and antineutrinos are "empty" implies that they do not carry the color charges of the corresponding leptons and antileptons.

The triangle representing group G3 makes it possible to understand this emptiness. To simplify the expression, in what follows we designate the primary color charges by the letters of the elements of the mathematical table to which they correspond, i.e. red = a, yellow = b, blue = c. The triangle (a,b,c) therefore represents the charged lepton of the first famil: the electron. If a, b and c are removed from the triangle, this, now empty, becomes an electronic neutrino. In the proposed Standard Table, the charged lepton, the sum of three primary colors, is therefore white. Conversely, the neutrino is black, to symbolize the absence of color.

 

Color and electricity

The geometrical interpretation of group G3, the equilateral triangle {a,b,c} with center o, can also demonstrate the existence of electrical charges.

We know that two of the permutations of the six elements of group G3 are rotations through an angle of 120° around the center o of the triangle, one in the positive direction, the other in the opposite, negative, direction. Composed or not, these positive or negative rotations, either full, one third or two thirds of a circle, correspond to all of the electrical charges, positive or negative, ful: -1 and +1, or fractiona: - 1/3 and + 2/3 or + 1/3 and - 2/3 for all of the charged leptons or antileptons and all quarks and antiquarks.

This would explains why the electrical charges of the quarks and antiquarks are set fractions of those of the charged leptons and antileptons. It would therefore be the rotations of color charges at the apexes of the triangle which create the electrical charges. This explains why the neutrinos and antineutrinos without color charge are also without electrical charge.

Why describe electrical charge by a rotation ?

In quantum physics, it is sometimes convenient to represent an electrical charge by an arrow rotating around an axis. It is thus possible to symbolize the electrical charge +1 by a rotation from 12 o'clock to 12 o'clock of the second hand of a chronometer. This movement seen in a mirror would be in the opposite direction and would symbolize the -1 charge.

Here, we use this comparison, replacing the arrow with the triangle representing the electron (a,b,c). Its rotation around the center o is termed negative if it is in the a-->b-->c-->a direction. It is positive if it is in the a-->c-->b-->a direction. But the electron has become a positron, the charged lepton of the second branch of the first family.

In the first case, the electron creates an electrical charge of -1 ; in the second, the opposite charge of +1 is created by the positron. Here it is the permutation of b and c that, by reversing the direction of rotation, transforms the electron into a positron. The same process transforms the charged leptons of the other two families into charged antileptons.

 

Origin of the Standard Table

The Standard Table can, as we have seen, be constructed from the image of the table of 48 sub-sets with six permutations of group G3, replacing in the respective boxes the letters a, b and c by the three color charges of quantum chromodynamics.

In this case, just as all of the components of the mathematical table have the six arrangements of the set {a,b,c} as their origin, the origin of all of the components of the Standard Table is the six arrangements of the corresponding set of three color charges.

However, a mathematical process cannot always be easily transposed into physics, and questions arise.

How can the six arrangements of color charges -the three charged leptons and antileptons -be the origin of the 48 components of the Standard Table ? How can they have as their origin a single set of three color charges ? And to what does this set correspond ?

To answer these questions, we must consider the universe at its beginning. At present, there are no isolated quarks or antiquarks, but this was not always the case. Astrophysicists, exploring the past by extrapolation of the present-day universe, have described cosmic events, simplified to the extreme, as a "soup" bubbling with elementary particles and antiparticles within which colossal temperatures prevented quarks and electrons from associating to form matter in its present form.

Therefore, any particle or antiparticle could be transformed into any other, e.g. from an electron into a quark. Thus, all of the components of the Standard Table could be created from the charged leptons and antileptons. But what was the origin of these charged leptons and antileptons ?

According to the mathematical model, the charged leptons and antileptons, grouped in three families, were created in pairs and stemmed from a single set of three color charges. In fact, in three families, the electron-positron, muon-antimuon and tau-antitau pairs stem from a single particl: the photon.

Disintegration of the photon into an electron-positron pair has long been known, but it was necessary to make use of the high energy released in an electron-positron annihilation in a synchrotron to obtain disintegration into a muon-antimuon pair and, with even more difficulty, into a tau-antitau pair.

In all cases, the lepton-antilepton pair is created by disintegration of the photon, which is therefore its particle of origin, even if, at present, we cannot achieve a sufficiently high level of energy to create all three pairs at once. However, this level was certainly attained just after the Big Bang.

Like the origin set of the mathematical table, the origin set of the Standard Table, the photon, via charged leptons and antileptons, has the formula {a,b,c} , replacing the letters by color charges. But what then is the difference in relation to the electron with formula (a,b,c) ?

As for the letters of the set corresponding to the mathematical table, the electron color charges are ordered and are placed in brackets. On the other hand, like the letters of the origin set of the mathematical table, the photon color charges are not ordered and are therefore placed between curly brackets.

It is impossible when mentioning three elements not to give them any order, e.g. a,b,c for the photon. However, the curly brackets override this concept of order ; they signify three discrete, but not ordered, elements.

As the photon color charges are not ordered, they must all be at the same place. Their union within the photon in this way is probably linked to the fact that this is a boson.

As the photon and the electron have the same composition -the color charges- it can be seen how they can transform into one another. For example, in their collisions, electrons and positrons probably annihilate one another, because their opposite rotations neutralize one another and their color charges, now freed from rotation, attract and then fuse to form photons.

 

Conclusion

Given the perfect analogy between the table of 48 particles and antiparticles and that of the 48 sub-sets of six permutations of group G3, it is assumed that the two tables stem from the same type of sequence.

In constructing the Standard Table from the model for the mathematical table, surprising results were obtained, although these do not contradict the present state of knowledge of particle physics.

Color charges would compose photons and elementary particles and antiparticles, with the exception of neutrinos and antineutrinos. This common nature would explain why matter and antimatter can convert into energy and vice versa.

The proposed table contains only isolated elementary particles and antiparticles. To find a previous instance of this, it is necessary to go back to the Big Bang. According to cosmologists, quarks were already beginning to assemble into protons one second after. This was the end of their isolation.

According to the logic of group G3 (see Figure 1b) a primordial photonic universe would have disintegrated into pairs of charge leptons and antileptons, probably beyond the Planck barrier, which astrophysicists have encountered in trying to go back in time, at 10-43 seconds after the Big Bang.

After this very brief leptonic period, the pairs, perhaps by fractional rotation, created the ingredients of the astrophysicists' "soup" the elementary particles and antiparticles, images of sub-sets of the mathematical table. We can also assume that many of them annihilated one another, thus recreating a large number of photons.

 

Discussion

Is this reasoning, which ranks the Standard Model elements mathematically, purely speculative or has the diagram it advances been proved by experiment ?

Here, experiment preceded speculation and provided the data on which analysis can be based. This analysis was restricted to using the G3 group of permutations to establish a relationship, direct or indirect, between the photon and all elementary particles and antiparticles of the Standard Theory.

The simplicity of a system in which all particles and antiparticles have the same origin should be sufficient proof of its authenticity now that unification of knowledge has become a priority of physics.

It should come as no surprise that the color charges attributed to the photon permute. This is a fundamental phenomenon found at the very core of the atom, in the proton for example, where the three color charges permute constantly, forming their six possible arrangements successively, through gluons.

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