Introduction

All elementary particles and antiparticles appear to come from the six permutations of a set of three elements: the three colour charges of quantum chromodynamics.

This concept stems from a comparison between a table containing the 48 sub-sets of the 6 possible permutations of the G3 group, and its image: the table of the 48 elementary particles and antiparticles grouped into a single set (see figures).

The underlying logic of the tables

The concordance between the figures and the similar order prevailing in both tables makes it unlikely that the analogy is simply a coincidence. However, this similarity does introduce into the Standard Model for Sub-Nuclear Physics – abbreviated as the “Standard Model” – novel features whose importance renders it necessary to justify them.

If all elementary particles and antiparticles stem from permutations of a set of three colour charges, they will necessarily all have colour, or be closely linked to colour. In actual fact, there is no colour in sub-nuclear physics, but charges which combine in a manner similar to the three primary colours. For this reason, they are described as colour charges. Rather than use the colours of physics, which are less familiar, artists' colours are used in this context: red, yellow and blue, matched up in the mathematical table with the letters a, b, and c.

Currently, of the 48 elementary particles and antiparticles, only the 18 quarks and the 18 antiquarks are said to have colour, unlike the corresponding 6 matter leptons and the 6 antimatter leptons. And the electron, like the other leptons, would be an interloper in the coloured family of quarks if the SU5 group had not made it possible to establish that, despite appearances, quarks and leptons are not irreconcilably different, but in fact belong to one and the same family.

In the mathematical table (cf. Figures), the electron corresponds to the set (a,b,c); that is to say, to the complete set of colour charges which produces white. This explains why the electron, currently described as being white, in fact carries three colours internally.

The electron, whose formula is (a,b,c), and which carries three colour charges, must not be confused with the proton, whose formula is ((a),(b),(c)), and which comprises three quarks. The colour charge is not a quark, despite the fact that the latter is characterised by that charge. Similarly, the antiquark, whose formula is (a,b) and not ((a),(b)), is not formed by two quarks, but by two colour charges.

The anti-electron, or positron, differs from the electron only by the direction in which it rotates. Indeed, if an electron were to reverse the direction of its rotation, it would become a positron. If an electron interacts with a positron, their rotations, in opposite directions, cancel out. As a result, they are transformed into photons: two gamma rays.

The mathematical table justifies the opposite rotational directions of the electron and positron: the electron's formula is (a,b,c), while the positron's formula– located on the line straight below – is (a,c,b), in which the elements b and c permute.

If we use a geometrical interpretation of group G3 – an equilateral triangle with a centre o, whose three colour charges, designated {a,b,c}, correspond to the triangle's three points –we can schematically represent the permutation of elements b and c, and the rotations of the set of elements (a,b,c).

- The permutation of b and c presents an orthogonal symmetry on axis oa : in simpler terms, a ‘flipping over' of the triangle (a,b,c) which, because it leaves point a in the same position, produces the triangle (a,c,b).

- If the triangle makes a full revolution around its centre o in the direction a à b à c, a direction conventionally termed ‘negative', then the triangle (a,b,c) will be an electron, and will have a whole negative electrical charge. If the triangle performs the same rotation, but in the opposite direction a à c à b, the resulting triangle (a,c,b) will be a positron, and it will have a whole positive electrical charge. The fractional charges, both positive and negative, of quarks and antiquarks can be explained as one-third or two-thirds rotations of the triangle in either direction.

The three electrons in the table, that is, the electron of the current universe and the fossil electrons (the muon and the tau), behave identically, and differ only in their respective masses which, especially in the case of the tau, are substantial. The muon and the tau existed in the burning heat (10 28 K) of the primordial soup of particles produced by the Big Bang. They have since disappeared and can be recreated only in very high energy colliders. The same applies to their respective antiparticles.

The Standard Model cannot explain the generational structure of the elementary components of matter. This has raised one of the thorniest problems of modern physics. In our case however, the generational structure clearly orders the mathematical table and is a necessary part of its image: the colour table.

Indeed, the recourse to the three electrons and their opposites enables us, if we take G3 as a model, to explain the existence of generations by showing that the charged leptons and antileptons created the three families in the table. Each family has two branches, and the leptons and antileptons hand down their respective rotational directions to the elementary particles and antiparticles.

We can therefore express as E the complete set of three colour charges symbolized by {a,b,c}.

Set G3 of the permutations of E contains 3 ! = 6 elements.

If we represent the image of E with each of these arrangements, we obtain a series of 6 images which, as in the Standard Model, divide up into 3 families each with 2 branches, and defined by an invariable element (their first letter), as follows:

Family a : ( a,b,c) ( a, c,b)   Family b : ( b,a,c) ( b,c,a)    Family c : ( c,a,b) ( c, b,a)

of which, given that a = red, b = yellow and c = blue, the images (see attached tables) are as follows:

Red family: e – e +      Yellow family: µ - µ +     Blue family: t - t +

Coloured sub-sets

Each of the six permutations, like any three-element arrangement, has eight sub-sets, which we can see in the first line of the mathematical table:
(a),(b),(c) (b,c),(c,a),(a,b) (a,b,c) Ø

The two-element sub-sets are complementary to the corresponding single-element sub-sets. For example, the two-element sub-set (b,c) is the complement of sub-set (a), with which it forms a full set of three colour charges: ((a)(bc)). Following standard mathematical practice, the complement of a can be symbolized by a : an ‘a' with a bar. Similarly, in the colour table, the complement of the quark (u) is the antiquark ().

The first line of the sub-sets in the colour table can therefore be transcribed as:
(u), (u), (u) (), (), () (e-) v- with, reading from left to right, the three quarks, the three antiquarks, the electron and the electronic neutrino, since the eighth sub-set of the mathematical table – the empty sub-set Ø –corresponds to the complement of the electron (the electronic neutrino), whose absence or emptiness of colour can be symbolized by an ‘anti-colour': black.

The electron is included among the sub-sets it generates because any set is a sub-set of itself. It therefore appears twice in the construction of the table: once outside, as the origin of the family it creates, and once inside, as a sub-set. The same is true of the five other charged leptons and antileptons aligned vertically below it (see Figures).

In the table of elementary particles and antiparticles, the antiquarks have a colour complementary to that of the quarks to which they correspond. Therefore, the green, violet and orange antiquarks have colours complementary to the red, yellow and blue of the quarks.

In quantum chromodynamics, the rule is that particles and antiparticles combine according to their colours to form all three primary hues. For example, if a green antiquark combines with a red quark, it is because green is made up of blue and yellow, and the addition ‘red + green' gives ‘red + yellow + blue', or the three base colours, which cancel each other out to form white.

The positron, whose formula is (a,c,b), orders the second branch of the first family, just as the electron governs the first. The same arrangement is found in the second and third families.

Nevertheless, a difficult issue now arises: how can the electrons and their antiparticles create their sub-sets, which form the 48 elements in the colour table?

Astrophysicists offer up an answer to this question. According to them, in the primordial soup which resulted from the Big Bang, temperature conditions were so extreme that any particle could change into another: an electron could thus become a quark, and vice versa.

It is possible to visualize the phenomenon using the triangle representative of the electron, page "Elementary Structure of the Universe".
One supposes that the paramount heat of the Big-bang tries to strip the triangle of its charges of color, 4 cases are then possible:
- the triangle resists heat: it remains electron
- it loses 1 charge, it its remains 2: it is transformed into antiquark
- it loses 2 charges, it its remains 1: it is transformed into quark
- it loses 3 charges, it its remains 0: it is transformed into neutrino, the particle without color

If the triangle turned in opposite direction, it would be antimatter, positon, and would have the 4 same options as in its electron form...

The creative photon

According to the hypothesis examined in this paper, the forty-eight elementary particles and antiparticles come from the rotational characteristics of a single type of element – the electron, the muon or the tau – and their opposite counterparts, whose single origin, the photon, can be seen in the colour table.

The nature of the photon is revealed if we refer to its counterpart in the mathematical table: the non-ordered set {a,b,c}, which appears here between curly rather than normal brackets to indicate its non-ordered nature, and to avoid the risk of confusion with (a,b,c), which symbolizes the arrangement corresponding to the electron.

Indeed, it is impossible to list the elements of a set without conferring an order on them. The purpose of the curly brackets is to circumvent this difficulty.

Since the element set of the photon at the origin of the table is not ordered, its three charges necessarily coincide. Each is therefore within each of the two others. But how can the non-ordered photon transform itself into six permutations: three pairs of charged leptons and antileptons?

Both astrophysics and common experience offer the beginnings of an answer to this. According to the Standard Model, the creation of the universe originated with the Big Bang. However, the laws of physics governing space and time did not apply to the primordial entity, which can be considered a point, or cosmic singularity, since time itself was created by the Big Bang. Nevertheless, we can be certain of the following: for the purpose of symmetry, the Big Bang created a matter universe and an antimatter universe.

The minimum process for creating matter and antimatter is known: it involves the disintegration of a photon possessing a sufficiently high energy level to form an electron and a positron. A photon with a much higher level of energy might be capable of creating the three pairs of charged leptons and antileptons which produce the table of forty-eight elementary particles and antiparticles, each of which corresponds to a permutation of the three colour charges (see tables).

By the same token, the primordial photon carrier of all energy would certainly have been capable of creating the three primordial matter-antimatter pairs. This hypothesis would be in line with the one which posits the existence of a single primordial cosmic singularity, since the photon is always a single particle, regardless of its energy level.

The three matter-antimatter pairs, electrons-antielectrons, created by the primordial photon, pre-existed within it in the form of non-ordered colour charges which permuted when the Big Bang occurred, thus creating matter and antimatter according to a process that we can visualize.

We begin by considering the opposite situation: when an electron interacts with a positron, their opposed rotational directions cancel out and they are transformed into photons.

The G3 group triangle, of which we take the points to symbolize the three colour charges, provides us with a partial representation of the reaction.

When an electron interacts with a positron, the rotations of the triangles (a,b,c) and (a,c,b), which symbolize the particle and the antiparticle, cancel out.

Since the three colour charges, which correspond here to the letters a, b and c, cease to rotate, the centrifugal force which has been keeping the triangles in the same state vanishes. Dragged in by the centripetal force, the colour charges fall to the centre of each of the two triangles, where they continue to coexist as a photon. This is a mini-Big Crunch (Big Crunch is the antiBig Bang).

The ordered colour charges of the electron and the photon have been transformed into non-ordered charges in the two photons.

The contrary phenomenon transforms a photon with sufficient energy into an electron and a positron, and creates, for the purpose of symmetry, two triangles which are initially reduced to their photonic central point, but then explode in a second phase to form, via a symmetrical process, the two three-colour triangles. This is a mini-Big Bang, which creates a particle-antiparticle pair, since the non-ordered charges of the photon have been transformed into the ordered charges - endowed with mass - of matter and antimatter. The Higgs boson could be the vector of this transformation, as of any identical transformation. It would be then be symbolised by each of the six arrows originating from the creative photon (cf. figures).

The same applies to the initial Big Bang, but, given the extreme energy in that context, it resulted in three matter-antimatter pairs.

In order to trigger the Big Bang, either the photonic universe must have reached the critical level for its disintegration at the very moment of its origin, or, more probably, a localised, fortuitous disintegration must have spread to the whole universe, thus serving as a detonator.

Conclusion

By comparing the table containing 48 sub-sets of the 6 possible permutations of group G3 with its image, the table of colour charges, we can:

- sort the 48 elementary particles and antiparticles logically into three families each with two branches and provide an explanation of the nature of each;

- justify the existence of the fractional electrical charges of the quarks and antiquarks;

- suggest an explanation of the equivalence of matter and energy, since both are, in the present hypothesis, formed by the three colour charges. However, the charges of the photon, each included in each of the others, would not, according to this view, be ordered, whereas those of matter, like antimatter, are apparently ordered by permutation;

- symbolise graphically the Higgs boson;

- surmise that the Big Bang did not create one, but rather, three matter-antimatter pairs, as confirmed by the attached star diagram, with the photon at the centre symbolizing the original photon;

- see that the equality of the colour charges among themselves requires that no one permutation be given priority over any of the other possible arrangements. Immediately after the Big Bang, the three pairings would therefore be rotating around the centre of the star in a manner such that each, in turn, occupied the place of the others. Mixed by this rotation, the three pairs would then, according to this hypothesis, have become the ingredients of the astrophysical ‘primordial soup';

- provide a schematic representation of h, Planck's quantum of action, a constant described by de Broglie as the “essential physical discontinuity” and which appears in the formula e = hv, in which e is the energy of the photon and v its frequency,

and in fact, according to the hypothesis set out here: h = {a,b,c},

since Planck's constant, the minimum quantity of energy, is in this context the minimum quantity of non-ordered colour charge. The photon's frequency is determined by the number of copies it possesses of the set {a,b,c}.

nota: the original text is in french