In this article, an axiomatic lattice field theory is introduced. Its fundamental algebra is based on an irreducible integer representation of the rotation group. Its non-abelian properties and the existence of zero divisors allow the definition of a simple space-time structure and a creation process for fundamental particles of matter. An invariant expression for the definition of interaction is proposed and used to evaluate the mass-energy of potential particles. The concept of interaction at a distance is introduced using the fundamental properties of the algebra. A representation for electromagnetic, weak and colored gluons is submitted together with a potential explanation of the mechanism behind the electric charge and color concepts. A creation process for the fundamental particles, the electron, up and down quarks, based on the properties of the algebra, is proposed. These representations explain naturally the concept of spin and radiated electric charge for the same fundamental particles and provide a theoretical value for their mass. A possible explanation for the chiral properties of the weak vector boson is presented. A decay process for the quarks is defined. A representation for the three types of neutrinos is proposed together with a study of their properties and mass. Finally, some symmetry rules are introduced and a structure for elementary and complex particles such as leptons, mesons and baryons is proposed. A scheme for charge quantization is discussed. These concepts are then used to suggest a representation for the muon, tau leptons and hadrons. Their theoretical mass is evaluated and shown to agree extremely well with experimental values. Some theoretical energy distributions are compared to the experimental ones and shown to be well within the experimental precision. Finally a qualitative connection with conventional quantum field theory and gravitation is proposed. The fundamental lattice length R is then evaluated by comparing the mass of various fundamental and elementary particles. It is shown that the lattice length essentially constitutes the only parameter of the theory and could be related to the usual fundamental physical constants: the speed of light c, the Plank constant h and the gravitational constant G.

Lately, quite a few scientists have written about the end of physics. There is even a popular book entitled just that. The lessons of scientific history are indeed quickly forgotten. Other recent popularized books on fundamental physics are entitled "The God Particle" or "The Mind of God" partly to remind us of the futility to try to make sense of the Gödelian limit to rational thinking. I do not pledge allegiance to any of these extreme schools of thoughts. It is true however that the extreme complexity of the mathematical tools required to formalize the concept of super-string theories and their off-springs tends to dwarf the minds of most individuals, including mine. Similarly, the apparent infinite granularity of the fundamental structure of matter and space-time together with the proliferation in the number of fundamental particles, from the Higg=s particle to the super-partners of super-symmetric theories, make one wonders where all this will lead us. Will this mad scaling down toward finer and smaller matter structures ever stop?

To the optimist, present super-string theories are about to unravel the last secrets of the universe with the help of some duality concept that should eventually explain everything. To many, such theories are about to deserve the glorifying title of the Theory of Everything. But the task is far from being complete and most promises remain just that promises. Indeed, there are still important difficulties with the theories. The first one concerns the origin of space-time itself. The most commonly held belief is that the entire universe was born from a fluctuation in the fundamental energy level of the vacuum. The problem is that the vacuum exists only with or within the universe. The vacuum should actually "grow" from or be generated by the expansion of the universe. Before or exactly at the absolute time "ZERO", the vacuum did not exist. It was therefore impossible for the universe to be born from a vacuum that the not-yet-expanded universe had not yet generated.

Furthermore, all known present theories, fundamentally all based on the principles of axiomatic Quantum Mechanics, cannot explain the fundamental nature of mass and electric charges or permit precise theoretical calculations of these fundamental values for the multitude of elementary particles presently discovered. At best, they can only account for, like in Q.C.D., for the values of some quantum number such as spin, iso-spin, strangeness, color, etc. through the application of the symmetry rules of Group Theory. There is also a problem of proliferation of free parameters. In the standard model there are between 22 and 31 free parameters depending whether or not one includes neutrino's masses. There are also numerous other non-explained phenomena ranging from why there are three generations of quarks and leptons to the true nature of mass and electric charges.

There are numerous other difficulties or unresolved problems in all existing theoretical models be it with the commonly used Standard Model, Quantum Chromodynamics, the Super-symmetric models or with the most advanced, complex and sophisticated of them all, the Super String theories. One common problem in the first three is the apparent impossibility to include gravitation without generating infinities that cannot be renormalized away. A common problem is also that a new type of strange and yet unobserved particles, called the super partners, are required when one attempts to push the formalism to its limit to include gravity. They also require a substantially enlarged space-time manifold to do that, anywhere from 11 to 26 space-time dimensions are needed. This necessitates the introduction of new symmetry breaking mechanism and compaction techniques to roll up the extra dimensions and hide them from a potential observer. In the Super-string theories such mechanism have to be applied twice in order to get the job done.

Many theorists are beginning to believe that a new paradigm is required to explain fully and completely our present experimental knowledge of particle physics and its cosmological consequences. Steven Weinberg reminds us that: "quantum mechanics by itself is not a complete physical theory. It tells us nothing about the particles and forces that may exist...it is possible that there is only one logically isolated theory, with no undetermined constants, that is consistent with the existence of intelligent beings capable of wondering about the final theory." As David Lindley puts it: "The ultimate goal in physics seems to demand, paradoxically, a return to old ways. … The ideal of a theory of everything...is a mathematical system of uncommon tidiness and rigor, which may, if all works out correctly, have the ability to accommodate the physical facts we know to be true in our world"

In this article, I will first establish the mathematical foundation for a theory of space-time and matter by showing that it is mathematically possible to generate from almost nothing an algebra that will reveal itself extremely rich in terms of internal symmetry. It will be shown that such algebra can be used quite naturally to generate the basic fabric of the universe, i.e., the vacuum. It will be shown that out of the only two natural invariants offered by this algebra, a general expression for the evaluation of the fundamental energy of the vacuum and matter can be deduced. It will be shown that a natural representation of all elementary particles can be developed from the above fundamental algebra and that such representations can be generated directly from the vacuum. Similarly, a mechanism to construct representations for all known particles will be presented. The invariant expression used to evaluate the energy of the vacuum will be applied to the representations of particles to predict their theoretical mass. Such predictions will be shown to agree extremely well with experimental values.

We wish to construct a simple model of the universe based on as few axioms as possible. Let us propose the following two simple axioms:

AXIOM 1: At an absolute time ZERO, the universe contained one and only one mathematical point denoted ZERO, that we elect to represent by a null 6x6 square matrix.

This axiom arises from my incapacity to develop anything from the much simpler and desirable starting point that the universe must or could have been, prior to time zero, an empty set. The dimensionality of that zero arises from the necessity to be able to generate an algebra rich enough to allow the creation of a space-time with properties similar to what we observe and the existence of a fantastic diversity of elementary particles with all their properties and physical characteristics. The above dimensionality appears to be the minimal one offering all the required richness. It also appears to work.

AXIOM 2: The universe is a closed system.

This axiom arises from the necessity to have a physical description of the universe which allows as possible physical quantities only those values which are themselves contained in that same universe. Consequently, the algebra used to describe it should be closed. The term "closed" is used here in a more restrictive sense that what is commonly understood. Having define an operation Å over a set X, I will not only ask that " x, y Î X, (x Å y) Î X but also that, given z, w Ï X, (z Å w) Ï X, (z Å x) Ï X and (x Å z) Ï X " xÎ X are always true. Of course z and w must belong to a set of proper dimensionality such that the operation (x Å z) and (x Å w) are defined.

The easiest and possibly only way to generate an almost infinite number of particles from a single null element is to adopt an algebra which has multiplicative zero-divisors. Consider the following closed algebra:

where s _i are 2X2 matrix which satisfies the following conditions

s ₀ = I , s _i ² = s ₀ for i = 0,1,3, and s ₂ ² = -s ₀

such that -s _i = -1 s _i ,

and, for i¹ j¹ k and i, j, k = 1,2,3, we have s _i s _j = e _ijk s _k where e _ijk = 1 for i>j

e _ijk = -1 for i<j.

Because of the form of e _ijk, the s are not equivalent to the Pauli's spin matrices. The s 's and their negatives form a non-abelian group of order 8 under matrix multiplication and, consequently, so do the S 's. The s 's are linearly independent and are orthogonal matrices i.e. s _i s _i ^T = I. It is interesting to note that the set c of all (k₀ s ₀+k₂ s₂), with k_iÎ R , is isomorphic to the field of complex numbers C while the set z of all (k₁ s ₁+k₃ s ₃), with k_jÎ R , would also be isomorphic to a set C ' having similar properties to C but not closed under multiplication. z is called the set of hyper-complex numbers. It should also be noted that the above algebra requires all four elements in order to be closed under multiplication. In other words, it appears that the complex number system alone, represented by c , would not be large enough to accommodate the description of our universe.

We define a three-dimensional lattice of equal and arbitrary unit length h and for which all the nodes are represented by the matrix Z^v defined by Z^v = S ₀ +S₁. When the lattice is composed only of these Z^v at the nodes, it will be considered to be "empty" of all matter and as such will be used to represent the space-time vacuum. Therefore such a vacuum or space-time is endowed with a structure. In fact, Z^v will be considered to be the generator of space-time. In that context, -Z^v could be considered to be the annihilator of space-time. It is important to realize that the above does not constitute the definition of a coordinate system but rather the definition of the structure of space-time. The choice of Z^v to represent or generate the space-time vacuum was done based on multiple considerations. The most important being that S ₀ and S ₁ form a proper subgroup and that s ₀ could be associated with the real part of C while s ₁ could be associated with the "real" part of C '. This would then allow for a real representation of space and a "real" representation of time.

The above structure of space-time can be visualized in a 3 dimensional lattice, if one considers that the determinant of S ₀ is +1 and the determinant of S ₁ is -1. In three dimensional geometry, a positive determinant suggests a right handed oriented volume of space while a negative one suggests left handed oriented volume of space, as spanned by their respective column vectors. Because of the isomorphism discussed above, it becomes possible to associate S ₀ with a right handed three dimensional real unit coordinate system defining a space-volume element and S ₁ with a left handed three dimensional "hyper-real" unit coordinate system defining a time-volume element. Both of these unit coordinate systems, or space and time volume elements, can be superposed at the same three dimensional point to generate a three dimensional cubic lattice. A consequence is that S ₀ becomes associated with the usual three dimensional space structure, while S ₁ becomes associated with some kind of three dimensional "proper" time structure. A three dimensional time generated by S ₁, while unusual, would completely symmetrize the universe with respect to both space and time coordinates. It has been shown, but the proof is not presented here, that a 6 dimensional space-time can support the principles of causality and time ordering.

Figure 1: The space-time structure

NOINTERACTION BETWEEN ADJACENTDES

Definition: The interaction between two adjacent nodes on the lattice, respectively represented by two 6x6 matrices A and B, will be completely determined and characterized by the result of the matrix product AB.

From that definition, we infer the following postulate.

Postulate: A given node would interact directly only with the six adjacent nodes.

We define the following operator E between two adjacent nodes of the lattice, occupied respectively by the matrices A and B, by

Definition: E (A B) = k₁ Det (A B) + k₂ Tr (A B). Where the k_i are some dimensionality constants which we elect arbitrarily to set to 1.

We note that since the determinant and trace of matrices are the only invariant under a change of coordinates, this operator will also be invariant under a change of coordinates. Since the determinant in a three dimensional space is normally given the geometrical interpretation of a volume spanned by the three column vectors forming the 3X3 matrix considered, we wish to associate the determinant of the product of two S matrices to the concept of energy density. While the trace does not have a classical similar interpretation, I believe that it relates to some flux or displacement of energy between the two matrices considered. The above operator is therefore taken to be the energy operator which gives the total energy produced by the interaction of two nodes of the lattice occupied respectively by the matrices A and B. Furthermore, since Det (A.B) = Det (A). Det (B) and Trace (A.B) = Trace (B.A), it follows that E (AB)= E (BA) even if, generally, AB ¹ BA. Applying the above definition of energy to the so-called vacuum nodes, we note that the energy between two vacuum nodes would be given by

The vacuum would have a null energy density in agreement with our intuitive understanding but would enjoy a non-zero flux or exchange of energy between the nodes. This non-zero flux of energy could be considered a necessity to ensure that the vacuum or space-time structure enjoys some degree of stability.

The presence of matter in the vacuum would be represented by the replacement of the Z^v at some specific nodes of the lattice by singlets (any one of the S ) and/or by doublets ( any one of the other combinations of two S matrices) in some specific configuration which still need to be determined. The presence of multiple singlets or doublets at one node of the lattice can only occur when their matrix representations are different or are superposition of different matrices. This restriction is valid even if a specific S appears as components of two different doublets. This constitutes some kind of exclusion principle. It is justified on the basis that the S 's generate space-time or matter and that these physical entities cannot be generated twice at the same "place and time". For reasons that will become self-explanatory as we progress, the singlets S _i will be called respectively:

S ₀ the fundamental space charge S ₁ the fundamental time charge,

The fact that -S ₀ and -S ₁ are specifically not included in the above is not accidental. Both will appear later and will turn out to be associated with the representation of the neutrinos.

Let us simplify the notation. From now on, I will use the symbols at table 1 to represents the various doublets. Except for H , c and z the similarity with already widely accepted physical symbols is voluntary and will be better understood as we progress. It should also be noted that the superscript ~, " and ' are defined by the following similarity transformations:

It has been demonstrated that globally these 3 transformations can be associated with some modified form of the combined CPT operators, but that P and T operators cannot be separated in this theory.

Table 1: Symbology for the doublets.

THE CREATION PROCESS

The algebra provides the following zero divisors:

The above zero-divisors are exactly what we were looking for to generate an almost infinite number of elements from the original and unique number 0. Indeed, formally speaking the element 0 can always be represented by:

Equivalently, using any of the pairs of zero-divisors d_i.d_k, we also have:

These pairs of zero-divisors can occur as numerously as one wish. One could say that the universe was born in a flash of g 's, H 's, Z's and W's.

In such a random process, it is probable that pairs of zero-divisors such as...g '.Z^v.Z^v.g .... get produced "side-by-side". Since Z^v.Z^v produces an interaction which is not zero (actually 12, based on our earlier computations) and therefore stronger than a null interaction such as g '.Z^v , there must be a strong probability that the Z^v's will remain bounded and that g and g ' would become free to interact or not with other doublets. Since Z^v.Z^v = 2 Z^v, we cannot have a fusion or superposition of these fundamental states, in accordance with the exclusion principle discussed previously. It is speculated that they will remain "side-by-side" at a "distance" of 1 A.U., generating as proposed earlier the structure of space-time. A similar argumentation applies to other doublets and even singlets. At that stage (not at that time, since time does not exist, yet), we would have some strange space composed of a mixture of S _i's, g 's, H 's, Z's and W's. All states and superposition of states can be created by variations on the same technique. It can be shown that quickly, using only the properties of the algebra, that the creation process will quickly favor the generation of g 's, H 's, Z's and W's that we will soon associate to photons and weak gluons. Such a conclusion is somewhat qualitatively verified in that the present universe contains billions time more photons than any other type of particles such as electrons, protons and neutrons combined.

INTERACTION AT A DISTANCE

While the result of the matrix product between two nodes A and B, denoted A.B, allowed us to evaluate what we believe to be the energy of the interaction between two nodes, that same product provides us with additional information. Indeed, we consider for example a quantity S ₂ located alone at a node and that is in interaction with the six surrounding adjacent vacuum nodes Z^v. We have

for each of the vacuum nodes. The resulting doublets have been put within brackets to indicate that such a doublet resulting from the matricidal product should have a different physical meaning than a "physical" doublet generating one of the nodes of the lattice. Whenever the result of such an interaction has the mathematical form of one of the 8 singlets or 24 doublets, we will refer to it as a "virtual" state.

In the particular case above, one notes that E (S ₂.Z^v )=E (Z^v.S ₂)= 0, as if there was no real physical exchange of energy between the quantity S ₂ and the vacuum nodes. However, one could consider that the virtual state g =[S ₂ + S ₃ ], for example, can travel on the virtual string connecting the different vacuum nodes since it has a mathematical form compatible with that of Z^v, i.e. they have different matrix representations, satisfying the exclusion principle stated earlier. Therefore, such g would be free to travel along the lattice and interact with other distant material quantities.

Consider two such distant quantities S ₂ and -S ₂ . The interaction with g would only be felt when g reaches a vacuum node adjacent to the target quantity. The value of that interaction would be given by

This result suggest that S ₂'s can interact at a distance by the intermediary of some virtual quantity g and that the energy value of that interaction is negative for equal polarity S ₂'s and positive for opposite polarity S ₂'s. We also observe that the only singlets with that property of interacting at a distance are S ₂ and S ₃. The difference between the two is that equal polarity S ₃'s would have positive energy and opposite polarity S ₃'s would have negative energy. It is then tempting to associate the S ₂'s with the concept of electric charge and g 's with photons. Since Electro-magnetic fields normally described photons, it is tempting to associate S ₃'s with the magnetic monopole concept. Noting that such entities have never been observed experimentally, it is an open question whether or not such physical entities would effectively attract each other when of equal polarity.

In the example above, the source charge S ₂ was surrounded by six Z^v, therefore, we could say that there are six virtual g 's produced simultaneously. Each of these g 's are free to escape within the surrounding space-time lattice. You would have noticed that the distance between the source and the target, as evaluated above, did not affect the value of the energy associated with the interaction at a distance. However, as the distance between the source and the target increases, the number of possible paths followed by these virtual photon increases significantly. As a result, the frequency with which the virtual photons reach a target or interact with it is reduced substantially as the distance increases. It is possible to show that strength of such a transmission of information respect the usual inverse square law. Our association of these virtual photons to the representation of the classical Electro-magnetic field appears well founded.

OTHER POTENTIAL GLUONS

Of all possible doublets, only two have been identified so far. We have tentatively associated one with the generator of the space-time vacuum and one with the photon, the gluon responsible for electromagnetic interactions. There remains five families of doublets, which have not been given proper names: (-S ₀ ± S ₁), (± S ₀ ± S ₂), (± S ₀ ± S ₃), (± S ₁ ± S ₂) and (± S ₁ ± S ₃). Luckily for us, if we exclude for the moment the hypothetical graviton, there still remains a few gluons as predicted by existing theories. The so-called Weak gluons which are believed to be responsible for the radioactive decay process of hadrons and leptons and the eight, poetically called, colored gluons which are believed to be responsible for the strong force which binds quarks permanently inside hadrons.

THE WEAK GLUONS

The weak gluons are usually denoted Z⁰, W⁺, W^-, B, W ^m , W ⁰, etc. It is believed that the weak gluons change the flavor of quarks, allowing by the same process decay of some unstable hadrons into, ultimately, stable ones. For example, a strange quark in an hadron would be converted into an up or down quark by interacting with a weak gluon. Furthermore, according to the Standard Model of elementary particles enunciated by Weinberg, Salam and Glashow, the Electro-weak force, mediated by the weak gluons, Z and W's, distinguishes between left and right through weak charged currents and weak neutral currents. It is therefore chiral. The weak W charge is nonzero for left-handed electron and zero for a right-handed one. As a result of this asymmetry, the nuclear beta decay, which is governed by the W force, produces mostly left-handed electrons.

To assist us in the proper assignment of our doublets to some known weak gluons, it might be useful to make a simple and brief summary of some of the findings of the Weinberg-Salam model for the weak electromagnetic interaction. In order to obtain a renormalizable and unitary electro-weak theory, it had been proposed by Glashow to introduce in addition to the electromagnetic field A _m , to the massless neutral vector particle Z _m and to the charged doublet of massless vector particles W ^± _m , two auxiliary massless neutral fields B and W⁰. This is required for a self-consistent gauge theory. While W⁰ couples to the matter fields with a coupling constant g, B couples with an intensity g'. After symmetry breaking, which amounts to a redefinition of the vacuum expectation values, the fields W ^± _m and B _m recombine and reemerge as the physical photon field A _m , a now massive vector particle Z _m and a charged doublet of massive vector particles W ^± _m . This is called the Higgs-Kibble mechanism. The photon field can be identified as a linear combination of W⁰ and B. The field Z can also be represented by a linear combination of these two. We can write

A = B cos q + W⁰ sin q , and (1)

Z = -B sin q + W⁰ cos q .

The weak mixing angle q is related to the ratio of g and g' by the relation

Furthermore, g and g' are related to the electric charge by

These last two relations lead to

The point of this quick revision is that the splitting relations for A and Z, could be rewritten in matrix form:

Noting that the matrix s ₂ is equivalent to a rotation operator with q = p /2 and given the following relations in our theory:

There is an obvious relationship between the results of the two theories. It appears reasonable to associate the W's, H 's and Z's of my theory to the weak vector boson fields of the electro-weak theory. However, the relationship is not one-to-one. Considering the following de-coupling relations:

it would appear reasonable to associate ± S ₂ ± S ₃ to the electromagnetic field, the photon, and -S ₀ and -S ₁ to some neutral fields realized by the intermediary of some other massless neutral particles.

The Weinberg-Salam model was specifically designed to describe the neutrinos and electrons fields in interaction. In that theory, the left-handed fermions form an iso-doublet consisting of the Weyl neutrino and electron: L = ( n _e , e )_L , while the right-handed sector consists of an iso-singlet, the right handed electron and no right-handed neutrino since it has never been observed: R = (e) _R. Such a curious feature would be a consequence of the fact that the weak interactions violate parity and are mediated by V-A interactions.

In my theory, -S ₀ could be associated with a left-handed coordinate system and -S ₁ to a right-handed one. Therefore, it is tempting to associate these two singlets to the anti-electron-neutrino and the electron neutrino respectively. But I will come back to the neutrino later.

The doublet Z = (-S ₀ - S ₁), an obvious neutral gluon, is possibly related to some mixture of Z⁰, B and W⁰, the neutral currents of the electro-weak theory. W^± = (-S ₁ ± S ₂) could be related to the charged massive vector bosons W^± and the electric field part of A^m of the same theory. The remaining H ^± = (-S ₀ ± S ₃) doublets are possibly related to a mixture of Z⁰ and the magnetic portion of A ^m . While this relationship appears complex, such complexity arises only when we try to find some analogy between the two theories. By itself, the theory being introduced in this paper is not complicated and appears to cover most of the domain of interest and applicability of existing theories, have similar predictive power and could eventually offer much more.

THE COLORED OR STRONG GLUONS

The colored gluons were introduced in Quantum Chromodynamics. The main idea is that quarks (the Up, Down, Strange, Charmed, Top and Bottom) also come in three color-charges. The colored quark-binding gluon couples to the color-charges in the same way as the photon couples to electric charges. According to the theory, the forces mediated by the colored gluons are so strong that all quanta that possess a colored charge are permanently bound together. Consequently, the quarks, which all have color-charges, are permanently bound together. The problem is that nobody knows what the "colors" are. The concept was originally introduced in the theory in an ad hoc manner to take care of the Fermi-statistic problems related to the spin properties of the quarks. Indeed, without this additional degree of freedom provided by the concept of color charges, the existence of some simple hadrons would have violated the Pauli exclusion principle.

Since the value of the determinant and trace of (± S ₀ ± S ₂) and (± S ₁ ± S ₃) are ± 8 and -6 or 0 respectively, we have the possibility of stronger interaction and heavier gluons. They are more likely to be associated with the colored-gluons. The colored-gluons could then be represented by:

It is suspected that the so-called "color-charges" of the colored gluons are related to the appearance of the electric and magnetic charges in the mathematical representation of these gluons. A more convincing argument will have to wait until some basic particles such as the proton and neutron are defined.

COMPATIBLE OBSERVABLE

In Q.M. one would say that if A and B are two observables and their commutator is [A , B] = 0, then A and B are said to be compatible observables. On the contrary, if [A, B] ¹ 0, these observables are said to be incompatible. We can also introduce the concept of complete set of commuting observable by stating that if a set of operators {A, B, C,…} all commute with each other, we can define a unique orthonormal basis formed with the eigenvectors common to all the operators. As a consequence of the above Q.M. axioms and definitions, it is tempting to apply these concepts to the sets of singlets and doublets defined earlier. The existence of complete sets of commuting observables might indeed help us identify fundamental symmetries in the theory and may reinforce some of the physical interpretations given to these entities, individually or by families.

From the multiplication (or interaction) and commutation tables of the doublets or gluons, we deduce that there are three classes of gluons:: g =s with Z=s, W=s with H 's and c =s with z =s. We could take this as a hint that there are three fundamental classes of symmetry that could be associated with the three alleged fundamental forces of nature: the electromagnetic, the weak and the strong forces. Indeed, the electromagnetic forces are mediated by virtual photons, represented in our theory by the g 's propagating freely in space-time represented by Z^v. The weak forces are allegedly mediated by the weak bosons, represented in this theory by W^- and W⁺, H 's and Z's. Of course, the neutrinos would also play an important role in weak decay process. The strong forces are mediated by colored gluons, represented in this theory by the c 's and the z 's. The analogy is too strong to be only coincidental. Furthermore, since (H ⁺. H ^-) = 0, the column vectors of the two matrices are eigenvectors of their respective matrices and are, one to one, mutually orthogonal. The same thing could be said of a few other pairs of weak gluons. This relates admirably to the known fact that the weak vector bosons are mutually orthogonal, since they represents two states of polarization, as any massless spin one particle.

We also note from the same table that there are eight different quanta arising from the interactions of the eight different alleged colored gluons, namely: ± [2S₀], ± [2S ₂], ± [2S ₁] and ± [2S ₃]. All are traceless and have energetic values of [64|0] or [-64|0], except for the first pair, whose energetic values are [64|12] or [64|-12] depending on the sign of S ₀. The magnitude alone of these values suggest that they would play a somewhat different role than the previously studied "light" or weak gluons (g 's, H 's, W's and Z's). Because of the perfect and complete symmetry of the set {c }, it is expected that it would be more likely to find particles formed by them. It should be noted that {c } is not a minimal set of commuting operators. Indeed, the subsets {c ⁺} and {c^-} can be considered independently as minimal sets of commuting operators. While this choice is not unique, it appears to be the most practical one.

As stated before, Q.C.D. establishes the spectrum of observable or physically realizable particles by introducing an additional quantum number called color. At present, it is not known what exactly is a color or what gives color to the gluons and to the quarks. Quarks come in three colors and three anti-colors and each of them can transform into one another via the emission of 8 possible colored gluons. Color-charge of gluon leads to the consequences that not only quarks emit gluons, but gluon emission by gluons and gluon-gluon scattering are also taking place. In our theory, it appears that we could introduce a similar concept to that of "color". However, it would be based essentially on the type and signs of the charges composing the particle, rather than on some ad hoc super-selection rule.

In this theory, I could demand that the superposition (addition) of all singlets and doublets appearing in particle anti-particle pair creation should give identically 0, when taken two by two. This appears essential based on the relations deduced previously: g '.Z^v = Z^v . g = 0. Indeed, we would then be assured that the annihilation of a particle and its antiparticle produces a vacuum node, to fill the hole left by the annihilation, and a g , to dissipate the mass-energy. Similarly, when a "g " coalesces with a vacuum node, we get 0, from which we can get the production of a pair of singlets or doublets by additive superposition. Of course, from this 0, we can also get a pair of zero-divisors by the fission process discussed earlier. In this context, if I define c ⁺ to be a "strong interaction" gluon having a specific color, as determined by the type and signs of the fundamental charges composing it, its anti-colored gluon would be given by c ^- since c⁺ + c ^- = 0. The conjugate of electric and magnetic charges alone is not sufficient to provide for the complete annihilation of most doublets. Indeed, c ⁺ + c⁺~ = 2S ₀ ¹ 0. Furthermore, as stated earlier, if I restrict myself to the use of positive interaction quanta, i.e. to the set {c }, I would have three possible values for the binding energies: [64|12] = 76, [64|0] = 64 and [64|-12] = 52. Are these an indication that we have essentially three "colors" of energy quanta? Note that according to the interpretation of this theory only the value of that energy is observable and translate into the rest mass of a given particle.

As far as the colored gluons are concerned, it should be clear that all c 's commute with each other while the z 's commute with some z 's and anti-commute with other z 's. Note also that to each class of compatible (commuting) gluons we could add one or more of the fundamental charges, whenever the charges commute with the gluons. In particular, ± S ₀, ± S ₂ and the c 's form a particularly interesting group of commuting quantities which we should eventually exploit.

The determinant and trace of the product of an interaction was postulated to be a measure of the rest energy of a given distribution of singlets and doublets and therefore a direct measure of its rest mass. How should we interpret the value of the determinant and trace of the commutator? Note that forces generally appear in nature whenever a system is not in equilibrium or whenever the symmetry or stability of the system has been broken by the application of an external stimuli (an electro-magnetic field, a sudden acceleration, a potential difference, etc.). For example, two electric charges of opposite sign will tend to attract each other until they are side by side or annihilate one another restoring in the same process the global equilibrium of the system or its lowest possible energy level. Two identically charged particles will repel each other until both reach the "opposite limit" of space-time, again restoring in doing so a certain equilibrium or lowest possible energy level in the whole of space-time. Forces appear to be generated whenever the fundamental symmetry of a given distribution of energy is broken. Lack of symmetry is normally associated with non-commutative properties of two interacting entities.

The fact that two physical quantities, represented by some linear combinations of the S 's, do not commute and therefore are not compatible observable, in accordance with the Q.M. terminology, might be an indication that forces are at play while nature is attempting to restore symmetry. Remember also that in the Q.M. formalism, incompatible observables are normally associated with classical dynamical variables. Therefore, I feel justify making the following prescriptions for a physical interpretation of the theory:

Commutation rule: Whenever two physical quantities represented by linear combinations of the S 's commute, there are no attractive or repulsive forces originated from the physical quantities themselves and the value of the determinant and trace of the interaction contribute exclusively to the rest mass of the combined physical quantities.

The existence of three distinct families of commuting operators for the heavy gluons, as well as the existence of the previous sets of commuting operators for the light gluons, suggest a possible super-selection rule, which we could state as follows:

Selection rule 1: A stable fundamental particle or gluon cloud can only be composed of commuting linear combinations of the S 's.

From now on, for the purpose of this paper, particles will be classified as follows:

Fundamental charges: these are the fundamental eight singlets, the S _j's already studied.

Fundamental gluons (or gluons): these are all the doublets formed by a linear combination of two fundamental charges.

Fundamental particles: these will be all particles (physically realizable or not) formed by the distribution in space-time of seven doublets or singlets in the shape of a simple octahedron.

Elementary particles: all particles formed in space-time by the permanent or temporary binding of a certain number of fundamental particles and a certain number of gluons. The allowed or likely geometry of such particles will be determined later. An elementary particle of equivalent charge Q cannot be subdivided into smaller physically realizable particles of equivalent charge Q_{i ¹} 0 and such that S Q_i = Q.

Complex particles: all particles formed in space-time by the permanent or temporary binding of two or more elementary particles and a certain number of gluons.

Whenever a unique or group of fundamental particles are surrounded by gluons, it will form an elementary particle. In such a circumstance it appears reasonable to demand that the gluon cloud be composed exclusively of commuting gluons. Commutativity of the gluons themselves with the nodes of the internal fundamental particles might however not be required provided there is enough energy in the cloud to keep the particle together for a reasonable amount of time. This selection rule is dictated by the fact that if a potential particle contains non-commuting physical quantities, the particle would eventually be torn apart by the resulting internal forces. They also ensure that the energetic values of bound remains the same within a particle irrespective of the order of the multiplication or the presumed direction of the interaction of the constituting gluons.

A QUESTION OF SYMMETRY

Before proceeding with the crucial task of finding the proper representation for the electron and the strangely behaved quarks, we need to find some sort of rules of symmetry which will guide us in the construction of what we expect to be complex particles or particles with internal structures. Looking back at the representation of the vacuum, i.e. to the unit cube of vacuum formed by the Z^v's at each node, we observe that each node is connected to six neighbours. Each node can only interact directly with the six adjacent nodes. Therefore, the simplest and most natural first level of complexity, other than a unique singlet or doublet, would be the presence of seven such singlets or doublets collocated in the shape of a simple octahedrons: one singlet or doublet surrounded by six singlets or doublets at the six adjacent nodes.

Let us speculate on the different possibilities that nature would have to assemble fundamental particles in the form of an octahedron using, for example, two commuting fundamental charges. We choose the charges ± S ₀ and ± S ₂ since we are particularly interested in electrically charged particles. The most obvious and simple possibilities to create particles in the shape of a simple octahedron are illustrated at figure 2. It contains many of the possible symmetric distribution of up to 3 electric-charges ± S ₂, while the other nodes are occupied by mass-charges S ₀ or positive and negative S ₂'s. The particles are surrounded by vacuum nodes ( Z^v ). Some of the asymmetric representations, not shown in the figures, could possibly be formed but are likely to be very unstable since the "internal energy", given by the values of the quanta of energy resulting from the interactions between the nodes of the particles, would not be symmetrically distributed. The above examples offer very interesting prospect in terms of the fundamental particles that are of interest in this article. We can certainly recognise some potential candidates for the electron and the Up and Down quarks. Unfortunately there seems to be more than one candidate (not shown) for every one of the fundamental particles with different mass values. There are also some strange neutral but massive particles that could claim fundamental particle status. There is an obvious need for a more formal treatment of the fundamental particles.

Figure 2: Examples of simple octahedral structures to represent fundamental particles. They are constructed only from commuting charges.

CREATION OF PARTICLE PAIRS

Consider a localised area of the vacuum lattice free of all mater and/or charges. Since

the creation of a pair of opposite electrically charged particles is mathematically possible. Such a virtual creation process is indeed considered to be a normal and frequent occurrence in all standard and classical quantum field theories. Provided these particles are allowed to move in exactly opposite direction at the same distance from the original pair's birthplace, the total impulsion remains zero and the total energy of the vacuum is still conserved. There is a possibility that each individual charge would move to the adjacent vacuum nodes on both sides of the original node. Generally, unless this area of the vacuum is polarised by some external means or nearby charges, the two charges, attracting each other by the process of interaction at a distance, will eventually come back to the original node and cancel each other by simple superposition.

Such a creative process could also occur at two adjacent vacuum nodes. In such a case, as illustrated at figure 3, we have two interesting possibilities that occur when the produced pairs moved away form their respective original vacuum node along parallel paths. In the first case, the two fundamental charges of opposite polarity could end up co-located at two adjacent nodes and remain bounded for some unspecified period of time, forming two neutral dipoles. Based on the mechanism of interaction at a distance, it is likely that the two neutral dipoles would ignore each other and remain as independent composite "particles". Such dipoles, if superposed to the vacuum lattice should remain essentially unobservable at large distances since there are no resultant electric charge and no resultant space-time disturbances.

In the second case, two fundamental charges with the same sign are co-located at adjacent nodes, forming electrically charged dipoles. Classically, one would say that such dipoles are unstable and should not last very long since two opposite electric charges should repel each other.

Important result: It should be noted that according to our method of calculating interactions, two electric charges of the same sign located at adjacent vacuum nodes would not repel each other since the two fundamental charges commute. In accordance with our interpretation (commutation rule) only mass would result from the available energy.

Note that this would not be true if the two charges were separated even by only one vacuum node. In such a case there would be repulsion as a result of the mechanism of interaction at a distance. The entity formed of two positive S ₂'s will be called a positive dipole and the one formed by two negative S ₂'s, a negative dipole. It is exactly the same mechanism of interaction at a distance which will make the positive and negative dipole attract each other and eventually annihilate each other, unless they are prevented from doing so by some other external forces, gluon cloud or vacuum polarisation. Indeed it is easy to see that the positive dipole has a total virtual photonic radiation field equal to +10[g ] while the negative dipole has one of 10[-g ] = -10[g ]. They would interact on each other at a distance by the intermediary of the virtual photons.

Figure 3: Dipole creation out of vacuum.

MINIMAL CHARGE

These three minimal charge distributions, the neutral dipole, the positive and negative dipoles essentially establish the measuring standard for all electrically charged distribution of matter. Other than being neutral, the smallest charge distribution should be equivalent to ± 10[g ]. It will be shown later that the radiation field of the electron and proton are indeed equal to -10g and +10g respectively. Why such small dipoles have apparently never been reported in numerous experiments? A lot of effort has been spent trying to find fractional electric charges or other similar light charged particles (c.f. Heavy lepton searches and fractional charge searches in Particle Data Group review). Therefore, I believe that such a particle would have been found if it could exist for a reasonable length of time. Note that such dipoles have essentially no space-time extension in two directions. They would be essentially "string-like" objects of very short length, possibly of the order of 10^-15 cm. or smaller. Since the mass or energy of such objects is all contained between the two fundamental charges and has a value of 7 A.U.M., when the dipole is superimposed to the vacuum nodes, it is possible that its presence can only be detected as a small extremely localised "linear" fluctuation in the fundamental energy level of the vacuum which is equal to 12 A.U.M.. Since, such fluctuations are indeed known to exist, the short-lived existence of these dipoles could help explain the phenomena. It is not obvious at this point what role these basic three dipoles will have to play in the charge quantization process. But before attempting to give some answer to this fundamental issue, I need some more potential particles.

A fundamental particle was defined as being formed of the minimal number of interacting points in the space-time lattice: seven points arranged in the shape of an octahedron i.e. one node of the lattice surrounded by the six adjacent nodes. What are all the possibilities of forming such a stable fundamental particles i.e. using only fundamental charges that commute? Since we are particularly interested in the electrically charged particles, it appears that we will have to limit ourselves to the use of ± S ₀ and ± S ₂. These two fundamental charges are particularly interesting because they are part of the larger class of commuting singlets and gluons:{ ± S ₀ , ± S ₂ , c ⁺ , c ⁺~, c ^-, c ^-~}. This last characteristic will prove essential in the building of more complex particles such as the proton and neutron and will even explain the "absolute" stability of the former. For the time being, just note that we have two other such "extended" classes of commuting quantities: {± S ₀, ± S ₁, ± S _{0± S 1}} and {± S ₀, ± S ₃, ± S ₀ ± S ₃}. None of these two classes contains heavy gluons, alleged to be required to form hadrons.

I have constructed all non-equivalent possibilities of forming fundamental particles with seven charges, using only the ± S ₂ and S ₀ charges. By non-equivalent, I mean all configurations that cannot be obtained from another one by a series of proper or improper rotations. I have found 63 such non-equivalent fundamental particles. Of course the electric charge conjugate of all the representations could be added to the list but would not change the results, except in terms of the sign of the electric charge of the particles and the sign of the resulting photonic radiation.

External radiation or radiated charge. This is the photonic radiation produced by the given representations. It is calculated in relation to the interaction of each electric charge with the surrounding non-electrically-charged nodes.

The equivalent radiated field (or internal radiation). This is the photonic radiation that would be produced if the net charge (number of positive - number of negative charges) was put in the most compact configuration. This is obviously a dipole if the net charge is 2. It is three charge in line if the net charge is 3, and so forth up to the maximum net charge of 7 which results when all seven nodes are occupied by electric charges of the same sign. The value of the equivalent radiation (Q_eq) with respect to the external radiation (Q_ext) will play a central role in the issue of selecting physically realisable particles. Three specific cases can actually occur. By virtue of the generally accepted principle of conservation of energy, it appears impossible that a physically realisable particle would have a greater external radiation field than what the net charge could produce. If it was allowed, one could assemble in a specific way a certain quantity of charges and draw more energy from that assembly than what was required to produce it. Such particles or assembly of particles could possibly exist for a very short period of time or longer if they are insulated from the vacuum by some charged gluon cloud. We will say that they are "allowed" but that they are not physically realisable. If Q_eq is larger than Q_ext we will say that such a particle is potentially physically realisable. However, it is believed that some additional conditions would have to be met to make them stable free particles. If Q_eq is equal to Q_ext, we have the borderline case.

Formed by dipoles. Physically realizable particle must obey the rules of our fundamental algebra arising from the fundamental axioms, the exclusion principle and the standard concept of conservation of energy, electric charge and total spin. Particle can be formed naturally out of the vacuum if the principle of pair creation introduced earlier is respected. This process is best illustrated at figure 4. Two pairs of identical charges have to appear simultaneously based on the algebra of the S 's:

And pairs of opposite sign should form simply on the basis of conservation of electric charge, which is a direct consequence of the principle of conservation of energy, and of our peculiar fundamental algebra. It should be noted that the relation at equation (1) above is physical, since the superposition at the same space-time node of two identical -S ₀ is possible by virtue of our exclusion principle (Z^v + (-S ₀) + (-S ₀) = -S ₀ + S ₁ = Z'). Mathematically, it is also a logical step leading to equation (2), which is physically relevant since it contains two positive S ₂ and two negative S ₂ that can be superposed at the same node in space-time. By contrast, equation (3) and (4) are not physically realisable based on the same exclusion principle.

Figure 4: Superposition and double pair creation rules.

Mathematically, pairs of opposite electric charges could also appear out of the vacuum by a similar process

However, equation (5) is not physical since two positive S ₀ cannot be superposed at a vacuum node, which already contains a S ₀. Therefore, equation (6), resulting from it, is not physically realisable.

Figure 5: The correlation rule allows only certain particle pair creation.

Indeed, one should realise that there is a fundamental difference between the process described by equation (6) and the process described by equation (2). First of all, there are only four nodes to fill with charges and only two pairs are required. You will note that relation (2) essentially describe a symmetrical creation process (charges of the same electric sign move to the opposite sides of the node where they are created). While equation (6) describe an asymmetrical creation process (charges of opposite electric sign move to the opposite side of the node where they are created). An immediate consequence is that, in the case of equation (2), the relative space-time orientations of the two pairs are strictly correlated, while the relative space-time orientations of the two pairs provided by equation (6) are not correlated. Indeed, in this last case, the relative orientation of the two charges of the first pair does not guarantee a specific orientation for the two charges of the second pair (See fig.5). As illustrated, in the first case, the positioning of the positively charged pair in the first particle does not leave any alternative for the positioning of the negatively charged pair of the second particle. Since the two pairs in the second particle are correlated, there is only one possible orientation. As a result the spin of both particles are directly correlated and the principle of conservation of energy is always satisfied. In the second case, positioning of the first neutral pair in the first particle forces the orientation of the first neutral pair in the second particle. However, since the two pairs in that second particle are not correlated, there are two choices of orientation for the second neutral pair, as is also the case for the second pair in the first particle. There is no guarantee of spin conservation. As a result, the use of the creation process provided by equation (6) would have an impact on the total spin conservation of the system formed by the two conjugate particles being created and, therefore, on the energy conservation of the whole process. Put differently, and this will become obvious while studying all the possibilities of creating fundamental particles, a creation process described by equation (6) would not in general respect parity conservation. If that process was allowed, to ensure that the principle of conservation of energy is preserved at all times, one would have to accept the fact that some form of instantaneous information is exchanged between various parts of the system of fundamental particle pair to ensure conservation of energy. It follows that the process described by equation (6) must be rejected as unphysical. The above discussion appears to suggest the following general principle:

Correlation principle: If two physical events originate from the same space-time node they must be correlated by some symmetry law.

And, as alluded to at the end of the last paragraph, this principle is closely related to the familiar concepts of causality and locality. The restriction of the creating process to that described by equation (2) ensures that the theory remains causal and strictly local. There is a theorem in Q.M. (Bell's theorem) which states that any theory obeying the axioms of Q.M. must contain non-local observable phenomena. In Q.M. the now famous Einstein-Podolsky-Rosen paradox and a series of modern experiments that tend to support Bell’s theorem exemplify this. This theory does not appear to obey Bell's theorem because it is not based on the principles of Q.M.. Our theory will remain local provided the proper particle creation mechanism is used.

Clearly, in addition to the above production of electric charges, we could have the production of other singlets. For example, two opposite nodes adjacent to the same pair-production node could become respectively S ₀'s or S ₁'s depending on which of the two following identities are considered:

However, the following pair-production identities are non-realisable on the basis of the exclusion principle which forbid the appearance of two of the same fundamental charges at the same node:

SIX SPECIAL FUNDAMENTAL PARTICLES

Based on the selection rules enunciated above, only a few configurations were found to be realisable. These were illustrated at figure 2. Out of the original 63 possible configurations (125, if I include the electric charge conjugates) there are only six configurations which are allowed in accordance with the creation process based on equation (2) above. If, for the time being, I exclude the two rightmost particles with neutral net charge, I am left with four very special candidates who might have a very special role to play in nature. It is rather fantastic that each and every one of the four remaining particles correspond exactly, in terms of electric charge content, to the three most basic fundamental particles that appear to be the building blocks of all matter in the universe: the Up quark, the Down quark (two possible candidates) and the electron. Of course, the electric charge conjugate of these three representations would represent respectively the anti-up quark, the anti-down quark and the positron. You will also note that, except for a minus sign in the case of the electron, all have the same mass value of 6 A.U.M.. Comparing the similarities of these three fundamental particles, I can only conclude that the electrons and the quarks must be in the same family, contrary to what is normally stated in most modern physics textbooks.

THE ELECTRON

At figure 2, I presented a list of possible representations of the fundamental particles, list which would include the electron formed by five -S ₂ and two S ₂ . Evaluating the electron's mass according to the definition given earlier, one finds -6 A.U.M.. The representation appears to have the right symmetry to justify a spin of ± 1/2, that is, they have two specific orientations given by the symmetric distribution of charges around an axis of symmetry. Each representation would have two distinguishable orientations with respect to a constant and oriented magnetic field. If we evaluate the perturbation in the fundamental energy level of the vacuum that an electron would generate, we get:

for a total energy of [0|0]. The electron has a null interaction with the vacuum. It is almost like if the electron was confined in a "bag" of zero energy. It generates a definite discontinuity in the homogeneity of the vacuum.

The fact that the energy of the electron is -6, i.e. 18 units below the energy level of the vacuum, could be an indication that the electron is "buried" deeply inside the vacuum. This might be also a possible answer to the question concerning the point charge behaviour of the electron. In the conduct of high-energy collision experiments, the electron behaves as if it was a integral point charge. The more energy one puts in the projectile particles, the more virtual pairs of particles are created, shielding the electron even better against attempt to break it apart. If the electron is really buried deep inside the vacuum, this is exactly what would be observed by such high-energy experiment. Indeed, since the higher energy level of the vacuum provides the first potential energy barrier to the projectile, the vacuum would tend to absorb all the energy provided by the experiment, creating more and more pairs of particles. Despite, its relatively weak structure and low binding energy, the electron would appear as an unbreakable point like particle.

Another possible or complementary interpretation for this negative value of the rest energy of the electron may be related to the concept of "self-energy". Indeed, when we compute the mass of the electron, we find the following results:

Obviously the negative value for the rest energy comes from the value of the trace. If my earlier interpretation of the determinant and trace is correct, the electron rest energy is composed of six units of matter density and resultant -12 units of flux of energy or internal self-impulsion between the various electric charges composing the electron. If we were to neglect this "internal" flux, the mass would be 6. However, the value of the trace cannot be neglected and we have a negative value for the rest mass. Other than some possible consequences in term of gravitational forces, which I have not defined yet, is there an observable difference between a value of +6 and a value of -6 for the rest mass of the electron? I suggest that for the portion of the theory covered so far, an eventual observer would not see a difference between a negative or positive value. Furthermore, it is precisely because of this negative value that decay processes, such as those that will be shown shortly, are possible.

Let us assume, for now, that this negative value is acceptable and that, for all practical purposes, an observer measuring it would read +6, on his instruments. Gravitationally, a negative mass is not necessarily dramatic. One generally considers that, according to General Relativity, a gravitational field around a given body is generated by or is equivalent to a distortion in the space-time metric proportional to the mass of that body. I have already shown that the electron would effectively provide us with a local distortion in the structure of space-time. Its presence could therefore affect the surrounding space-time and any other particles in its immediate surroundings in a way indistinguishable from what would do a particle with positive rest mass. Experimentally, the rest mass of an electron is possibly the quantity which has been more often measured and which is the most accurately known. As such, I will use the following correspondence to evaluate the rest mass of all other particles in units of Electron volts:

EFFECTIVE ELECTRIC CHARGE VS ELECTRIC CHARGE

In this theory, the electron is represented by a distribution in space-time of a total of seven charges. In accordance with our discussion of the interaction at a distance of a point charge in the vacuum, the two external positive charges forming the electron will interact with the surrounding vacuum and emit a total of 10 virtual photons [g ]. The four negative charges will emit a total of 20 virtual photon [-g ] = [g ~]. The interaction of all six external charges with the centre negative charge will produce virtual negative or positive space-charges of which the resultant total energy of -6 constitute the rest-mass of the electron. The resultant external virtual photonic "field" is equivalent to 10[-g ] and according to our explanation of interaction at a distance it is that resultant photonic field which is responsible for the observed value of the electric charge at a distance. It appears that the scientist is not observing the algebraic sum of the S ₂'s but the photonic field of 10[g ~] and defined to be equivalent to a charge of -1 in the usual units. At the scale of the observer, this field is more likely to be perceived as one originating from a punctual source. The high-energy probe would have to penetrate this high-density field in addition to the potential energy barrier produced by the difference of energy between the electron and the vacuum. This might well result in the creation of even more particle pairs and an increase shielding of the true nature of the electron.

INTERACTION WITH W AND Z

One of the most curious properties of the weakly charged gluons W± concerns their interactions with electrons and neutrinos and the resulting parity violations. Indeed, while left-handed electron experiences an attraction from the W-force, a right-handed one does not. As a result, nuclear beta decay, which would be governed by the W force, according to the Standard Model proposed by Weinberg, Salam and Glashow, produces mostly left-handed electrons. At present, there does not appear to be a good understanding of these properties of weak forces.

Figure 6: The interaction of electrons of different spin with weak vector boson.

Consider two electrons of opposite spin, as illustrated at figure 6, subjected to a close range interaction with a W⁺. Note that at long range, W would interact with the electrons via the usual radiative photonic field and the interaction would not be distinguishable from ordinary electro-magnetic interaction. It is only when located at the adjacent vacuum node, i.e. at short range, that there would be a different type of interaction distinguishable from normal electro-magnetic interaction and possibly given a new name, weak-forces, by an observer. In the vacuum, since W⁺ + Z^v = c ⁺, the nearest electric charge of the electron will not feel a force from that node since ± S ₂ commute with c . However, there would be a temporary increment in the mass-energy of the system "electron-weak gluon" as long as the two particles remain in close contact. The distant charges within the electron would still be under the influence of the radiative photonic field of W. As such, it would sustain a net repulsive or attractive force, depending on the sign of W in one of the orientation and a net zero force in the other relative orientation. Based on the value of the interaction of W⁺ with positive and negative electric monopole and on the principle of action at a distance, the non-vanishing nature of the resultant of all forces acting on the electron will depend on its relative orientation with respect to W⁺. And, as alluded to above, the structure of the electron would be directly responsible for this strange behaviour.

THE BASIC QUARKS

If I choose the representations at figure 2 as the valid representations of the Up and Down quarks, I am confronted immediately with the problem of observability of fractional charges. Indeed, nothing in the given representations suggests that such entities cannot be realised as free particles. Why can’t we observe them as free particle? On the contrary, if they are permanently confined and cannot be realised as free particles what is the confinement mechanism? Can a neutral colour concept of confinement, such as the one presented in Q.C.D., be explained on the basis of the fundamental axioms of my theory and without demanding some act of blind faith?

In an attempt to answer some of these questions, let us assume, for now, that free Up and Down quarks can exist. What would we observe? The first thing to note is that the representation of the Up quark has a radiation field of [+10 g ] and as such would appear as an entity of charge +1e to an observer at a large distance compared to the fundamental unit of length. Furthermore, the Up representation has a mass of +6. You will notice that except for the sign, the mass of the Up quark corresponds exactly to the mass of the probable representation of the electron. The question is then how would an observer distinguish such an Up quark from a positron, or an anti-Up quark from an electron? I believe that they would be indistinguishable. In other words, an Up quark (Anti-Up quark) would behave exactly like a positron (an electron) in the presence of an external electromagnetic field. You will note that, if our definition of spin in terms of symmetrical distribution of electric charges is valid, our representation of an Up quark has also a spin ± 1/2, like an electron. In addition, the light representation of the Up quark is also shielded from the vacuum with an envelope of null interaction exactly like an electron.

Another possibility has to do with the fact that each of the pairs of charge in the up-quark are separated by a space charge resulting in the generation of a repulsive radiation field [± S ₂ ]. Consequently, each pair of identical charges in the up-quark tend to repel each other leading to the breaking apart of that particle in the absence of containment by other entities such as gluons. The lifetime of free up quark could be so short that they are virtually unobservable. However, while the down-quark mass is also +6, it has an observable photonic radiation field of [-2g ]. It should be distinguishable from the representation of the electron in that it should register as a particle with an electric charge equal to 1/5 that of the electron. Why is it never observed as a free particle? To answer this question, we need the introduction of one more concept, that of decay.

THE DECAY PROCESS

Let us define a new operation on the fundamental particles. Assume that, given two octahedral structures, it is possible to multiply two by two each of the nodes situated at the same relative position, as illustrated at figure 7. Mathematically, such a process is fully justifiable. Physically, it would make sense only if the two structures resulting from the fission or decay process at each stage can be de-coupled while respecting the basic principle which states that two identical fundamental charges cannot co-exist at the same location in the space being considered. Note that such decay is made possible because of the mathematical equivalency of the two sides of the following equation, provided one accept that doing a node by node multiplication is a valid operation:

Figure 7: One of the decay modes for the down quark.

Where "D" stands for Down quark, e^- for the electron, "U" for Up-quark and "NC" for the neutral cloud. Considering the possible decay process at figure 7, any down-quark representation should eventually decay to a combination of electron, Up quark and some neutral cloud. Note the sign of the charge S ₀ at the centre of the Up quark and neutral cloud. Therefore, it is possible that free Down quarks would not be observed because they decay very rapidly releasing electrons and other particles: the Up-quark which could be mistaken for positrons and some neutral particle. Note that such a neutral particle was identified at table1 as a potential neutral quark. And the mystery of the confinement of quarks would then simply be a case of mistaken identity. The resulting shower of electrons, positrons, electron-like and neutral particles which would be observed when one is attempting to break hadrons or mesons is exactly what is often observed in high energy collision experiments. Note that the mass-energy of the initial particle is +6 and the mass-energy of the resulting particles are respectively -6 for the electron, -6 for the slightly modified Up-quark and -6 for the neutral cloud. The decay would give up a certain amount of energy likely to be used as kinetic energy by the resulting particles.

The neutral cloud that would be created in the decay process of a Down quark is perplexing. How should it be interpreted? Is it a new fundamental particle not yet detected? Note that our theory is essentially static since the phenomena that are being represented are those seen from the photon's frame of reference. As such, there are only one way to express kinetic energy or momentum, it is by representing it as mass-energy. Looking at the neutral cloud produced in the reaction above, we notice that it has a -S ₀ charge at its centre. This will turn out to be exactly the representation that will be proposed for the anti-neutrino. I would like to propose that the "massive" neutral cloud is only the energetic representation of a massless anti-neutrino that would be expelled from the reaction with the corresponding kinetic energy. Such an interpretation will turn out to be quite useful when attempting to explain the decay process of the neutron (Up, Down, Down) into a proton (Up, Down, Up), an electron plus an anti-neutrino.

EXCITED STATES OF QUARKS

It is possible to construct some heavier representations for the Up and Down quarks. They are presented at figure 8. Heavy Up and Down quark formed with the c 's are simply considered in this theory as exited states of the lighter representations. The various heavy representations of the Up-quark would have a mass of 58 and 84 A.U.M., or converting to the more familiar units, we obtain a mass equivalent to 4.939658 MeV and 7.153987 MeV.. This is quite in good agreement with the reported mass of the constituent Up quarks of a proton as evaluated in Q.C.D. theory. They estimate the mass to be approximately 4 MeV. The heavy down (or Anti-down) quarks have a mass of 34 and 48 A.U.M. that is equivalent to 2.89566134 MeV and 4.087993 MeV. It is likely that these are the representations that will be used to form the hadrons. As for the light quarks, the exited states can also decay in accordance with a similar process as we studied earlier.

Figure 8: The excited states for the quarks.

During the study of the weak gluons, while making a comparison with the Electro-weak theory of Weinberg and Salam, I stumble over the possible representations of some neutral particles which would be formed simply by two of the fundamental charges -S ₀ and -S ₁. There was also a strange neutral particle, represented also by -S ₀, appearing in the decay process of the down quark. I speculated at the time that these particles could be the elusive neutrinos. The neutrinos are believed to appear in three different sizes according to which of the three known leptons they are associated with, i.e. an electron-neutrino, denoted n _e, a muon-neutrino, denoted n _m , and a tau-neutrino, denoted n t . They all have spin 1/2 but they are believed to have a very small mass, if any. Present theories and experiments reveal only an upper limit to the mass of the neutrinos:

a. for the electron-neutrino, m _n < 2 x 10^-8 GeV

b. for the muon-neutrino, m _n < 2 x 10^-4 GeV

c. for the tau-neutrino, m _n < 0.035 GeV.

This problem of the mass of the neutrinos is crucial in modern cosmology. Indeed, neutrinos are so numerous that, if they even possess a tiny mass, it could account for as much as 90 % of the total mass of the universe. It would give enough mass in fact to ensure that the universe is closed and that it would eventually collapse upon itself in a gigantic black hole. This is what the "Big Bang" theory would predict.

Let us investigate the possibility of selecting n =-S ₁ and n ~ = -S ₀ as contenders for the representations of the neutrino and anti-neutrino. An important known characteristic of the neutrino is that it almost never interact with observable matter and is, as a consequence, very hard to detect. Note that the neutrinos are considered to be leptons. They are differentiated from the gluons by the fact that they are of spin 1/2 and are therefore fermions, while gluons have spin 1 and are therefore bosons. Only the graviton, if it exist, is alleged to have spin 2. To simplify, the values of the determinant X and the trace Y will be indicated as [X|Y].

Superposing a neutrino or anti-neutrino to the vacuum, I get

Therefore n (n ~) is, unlike the photon, not in equilibrium with the vacuum. When occupying a node by itself, it is not in equilibrium either. Their trajectory in the vacuum would be "visible" by the local cancellation of the space or time-charge in the fundamental structure of the vacuum. And it is also because of this negative sign that n can travel as freely as a photon within the vacuum.

Figure 9: The three types of neutrinos.

ON THE CHIRALITY OF THE NEUTRINO

One of the most interesting and troubling properties of the neutrinos is that it is a chiral particle. Early investigations of the beta decay led to the discovery of the neutrino and antineutrino, electrically neutral particles travelling at the speed of light. Like the electron, the antineutrino, emitted by radioactive matter, has a spin; but, unlike the electron, it appears that antineutrinos can exist only in the right-handed form. On the opposite, radioactive antimatter emits only left-handed neutrinos. So far, right-handed neutrinos and left-handed antineutrinos have never been observed in nature and, to this day, none of the existing theories have been able to explain why it is so.

I would like to represent < and < ~ by two singlets. From the definition of the CME operator, I get:

n ~ = -S ₁ n = -S ₀ n ~ = S ₁ n ~ S ₁ n = S ₁ n S ₁

n = S ₃ n S ₃ = S ₂ n (-S ₂) -n ~ = S ₃ n ~ S ₃ n ~ = S ₂ n ~ (-S ₂).

These equations indicate that under the C, E and M operator, < transforms like another n . However, n ~ changes sign under E or M. This last consideration is of no significance. E being the combination of time reversal and conjugate of electric charge, it should be no surprise to find out that if time is reverse everywhere in the universe, then the anti-neutrinos would have to change sign to remain free moving entities. Note that earlier we associated the representation of S ₀ to that of a right-handed co-ordinate system and S ₁ to a left-handed one. Our interpretation and representation of both particles preserves the experimentally verified chirality of the neutrinos. The existence of chiral neutrinos should be considered as proof that the arrow of time in our universe points definitively in the positive direction and that space is a positive definite quantity. We clearly established earlier that space-time, in the universe represented in this theory, was composed exclusively of S ₀ + S ₁ and was therefore exclusively non-negative in time as well as in space. It appears that the possibility of having negative n (i.e. S ₀) and n ~ (i.e. S ₁), as free particles, is formally excluded on the basis of the two fundamental axioms of the theory. Furthermore, if -n and -n ~ are allowed to exist, since mathematically they are acceptable entities, they would not enjoy the same freedom of movement as do n , n ~ and the g 's since positive S ₀ or S ₁ would prohibit free displacement at the speed of light in the vacuum. They would remain trapped in the vacuum. And, as we know by now, they combine in a doublet to form the vacuum.

At this point, it is tempting to conclude that the selected representation for neutrinos is the right one. In particular, if most matter in the universe was formed of +S ₀, +S ₁ and ± S ₂ (and it will turn out to be just that), the present choice of representation for n and n ~ would explain their unusual ability to go through matter without interacting with it most of the time. Note that the proposed representation would allow for a simplification of the debate concerning the possibility that the neutrino is a contender for the dark matter concept. Since its mass would be zero, at least when taken in isolation, it cannot participate directly to the total energy density of the universe and, therefore, to its closure.

THE TWO OTHER TYPES OF NEUTRINOS

There are supposed to be two other neutrinos in this elusive family, the muon-neutrino and the tau-neutrino. While the first one has been observed repeatedly in experiments, a direct detection of the tau-neutrino is still lacking. We do not know much about these two neutrinos. However, it appears possible to propose representations for both. The muon and tau neutrinos have possibly very similar characteristic as the electron-neutrino. Therefore, it is likely that they are formed with the same two fundamental charges, but bounded together in a more complex structure than what is provided by the point-like fundamental charges.

The first level of complexity that one encounter passed the single-point charge is the doublet. Forming the muon-neutrino as a doublet -S ₀ - S ₁ is tempting but it would make it equivalent to the gluon Z studied earlier. Furthermore, such an assignment would make the muon-neutrino a boson which is unacceptable. The next level of complexity happens to be a dipole: the two different fundamental charges are placed 1 A.U. apart in the lattice, as shown at figure 9. Such a particle would have two distinct orientations in space-time, which would qualify it as a fermion of spin ± 1/2. Furthermore, these two charges would interact with one another giving the muon-neutrino a mass equal to -1 A.U.M.. This is equivalent to 1/6 the mass of an electron or 0.0851665 MeV. Well under the experimental upper limit of 0.2 MeV. If such a configuration is valid, there would be no distinction between a muon-neutrino and an anti-muon-neutrino except its space-time orientation. I believe this would make it a Majorana neutrino.

Following the same recipe, the next level of complexity for a particle is to place charges at seven nodes in the shape of octahedrons, as shown at figure 9. The choice for the distribution of -S ₀'s and -S ₁'s is dictated by a requirement of simplicity and symmetry. Such a particle would have a mass of -6 A.U.M. or 0.51099906 MeV, again well below the experimental upper limit of 35 MeV.. There are no absolute reasons to accept the two configurations as the likely representations of the muon and tau neutrinos. However, it will become obvious, as we progress that they make a lot of sense. Furthermore, you should notice that ± S ₀ and ± S ₁ form a minimal set of commuting observables, if we exclude the trivial set form by the identity alone. All the neutrino representations obtained would therefore be stable in accordance with our previously established interpretation.

I believe that the essential geometric properties of the fundamental particles will be copied by nature while building more complex particles. It is after all the most compact and economical form over the lattice. It is therefore likely that complex particles would essentially be build around the general shape of an octahedrons, as shown at figure 10, and that any gluon cloud, required to make these heavier structures stick together for a reasonable time, will be neutral in terms of total electric charge content. This last property could possibly be associated with the requirement to have neutral colour particles as in the SM. That cloud would also be symmetrically distributed around a main axis of symmetry containing, as for the fundamental particles, the total equivalent charge of the elementary particles.

Figure 10: The octahedral structure of more complex particles.

Figure 11: The two configurations for the gluon cloud giving equal mass but non equal color.

Figure 12: The building process toward complex particles.

The main difference between the fundamental and the elementary particles would be that the "main axis" of symmetry would be more like a cylinder containing all the required quarks necessary to account for the equivalent or net electric charge of that particle. Similarly the neutral cloud would be formed by weak or coloured gluons depending on the mass and other properties of the particle being considered instead of the single charges or gluons used for the light and heavy fundamental particles. The three fundamental particles were formed, on their main axis, by respectively one, two and three charges. Therefore it would appear normal, in a first attempt, to use respectively one, two or three quarks on a main axis to form the elementary particles. A similar argument would probably be valid for the complex particles formed by the combination of the elementary one's. The concept is illustrated at figure 10, 11 and 12.

The structure at figure 10 does not appear to be strong enough to be stable. Indeed, it is more than likely that the mass of a simple octahedral glueball is larger than the mass-energy provided by the bound between any pair of internal nodes. Therefore, for stability, more "glue" is required to hold a particle together. Two simple ways of achieving that come immediately to mind. They are illustrated in successive layers at figure 11 A and B. The first one is preferred because all nodes are either at the centre or at the summit of simple octahedrons. Therefore, it is not only the most natural structure but it is also the most economical in terms of gluons. It will also turn out to be the only possible one whenever an elementary particle will be formed of gluons which are all carrying an electric charge. The second one (figure 11B) requires exactly four more gluons than the previous one but cannot be ruled out simply on that basis. The only essential and inescapable requirement is that the total gluonic cloud be of neutral electric charge. Note that both configurations would have the same mass. Based on the above considerations, I put together the following symbolic representation of how the different elementary and complex particles could be constructed using the quarks (Q = Up, Down and Electron), the weak and/or strong glueballs (G). This is illustrated at figure 12. The order and symmetries of the process are obvious.

SOME SUPERSELECTION RULES

Earlier, I established some rules to decide whether or not a given distribution of charges over simple octahedrons could correspond to a fundamental particle. The idea is to extend these rules to the higher level of particles. I want to see what combination of two or three quarks on a main axis of symmetry surrounded by a neutral cloud of gluon could be interpreted as potential physical particles, directly observable or not. It should be noted that the type of cloud would play a crucial role in this exercise. Clearly, a cloud formed by electrically charged gluon will have to contain an equal number of positive and negative S ₂. Furthermore, such a cloud would necessarily affect the photonic radiation field of the internal constituent quarks, since the S ₂ charges inside the gluons would prevent g 's from escaping the cloud and an eventual detection by some observer. In other words such a cloud would be masking the true charge content and the true radiation field of the internal quarks. While for the fundamental particles, Q_ext was necessarily equal to the radiated photonic field (Q_rad) of the whole particle, in general, it will not be so for the elementary and complex particles. Q_rad will be the resultant field from the charged gluons and the charged nodes of the quarks that are in direct contact with the space-time lattice or the weak gluon cloud.

CHARGE QUANTIZATION

Earlier, I also briefly discussed the concept of charge quantization. This mysterious rule which seems to force nature into building observable particles with a radiation field equal to an integer multiple of 10g . It was also stated that the up-quark could incidentally be mistaken for a positron since it has the same radiation field. It was also shown to be unstable in a free state. It was also argued that the best argument in favour of the non-observability of the down-quark, having a radiation charge of -2g , was their probable rapid decay into an electron, an up-quark and some neutral particle that could possibly be associated with the neutrino. Unfortunately, it does not appear that similar arguments could be used to explain the fact that more complex particles have never been observed with a measurable radiative charge different from an integer multiple of 10g . In particular, only those particles which have an internal equivalent charge (Q_eq) of ± 3k (k Î ø) fundamental unit of charge, producing a radiation field (Q_rad) equal to an integer multiple of the charge of the electron, have ever been observed experimentally. The experimental measurement of the photonic radiation fields would be the only possible verification of the charge content of an observable particle.

The charge quantization requirement should probably be preserved. However, one should note that most particles listed by the Particle Data Group are fairly unstable and their charge has been inferred in almost all cases from the charge of the stable particles remaining after their decay. It is not an absolutely verified fact that all observed and short-lived particles have effectively a charge which is an integer multiple of the charge of an electron. We only have accurate direct measurements for the charge of the electron, proton, the neutron and the electron-neutrino, as indicated by the Particle Data Group latest listing. This is a relatively weak sample to attest of the absolute validity of the integral-electric-charge quantization principle. Notwithstanding this lack of experimental direct measurement, there are theoretical arguments that can be invoked to support the charge quantization requirements. Classically, electric charge quantization is somewhat introduced axiomatically. Indirectly, Quantum Chromodynamics theory, while providing some justification for it by stipulating that all observable particles must be singlets realisation of the three "colours" group SU(3), is based on the axiomatic premises that quarks appear with an electric charge of ± 1/3e and ± 2/3e without providing justification for this choice other than the undeniable fact that it works quite well. On the contrary, electric charge conservation is intimately connected to the fact that the messenger responsible for the transmission of electromagnetic interactions at a distance, the photon, is a boson of mass zero.

So far, my theory is fully consistent with these classical notions of fundamental electric charge and their conservation and interaction properties. There are however major differences. An important one is that classically the vacuum, within which electromagnetic propagation occurs is a continuum. My theory is built on a vacuum lattice concept. Such a distinction should have some impact on the properties of the propagating photon responsible for electromagnetic propagation at a distance and therefore on the resulting concept of charge quantization and associated observable consequences. Actually, it can be shown that the photonic radiation field produced by more than one electric charge is not conservative contrary to the classical case.

Some will not accept the statement that the familiar superposition principle for electrostatic forces is not valid at the scale of 1 A.U.. It will be shown that this scale of length is much smaller than the order of 10^-15cm. You will note that the electrostatic theory is based on a few approximations and has only been tested to scale of about 10^-13 cm. or the dimension of the classical radius of an electron. In essence, the principles and laws of electrostatic are based in large part on macroscopic considerations. Charges are idealised as point charges and it is assumed that the limiting process of shrinking infinitesimal lengths, areas and volumes to zero is valid at nearly all scale. To that effect, you will note that strictly speaking, Gauss's divergence theorem is not necessarily applicable to the case of the 1/r potential since it has a singularity at r = 0. By contrast, such a singularity will never occur in my theory since the electric field and potential are not defined at the six nodes adjacent to a single charge. In fact, if one brings another charge at any of these six nodes, mass will be generated as a result of the interaction of the two charges instead of another increase in the potential energy of the system. However, for linear distribution of two electric charges spaced 1 A.U. apart, there are strips of width 1 A.U. in the xz-plane and also in the yz-plane where the field is locally conservative. By comparison, the field produced by three point charges on a lattice is not conservative, except on the one-dimensional line corresponding to the three axis of the co-ordinate system, centred on the central charge. The field produced by four charges would not be conservative except for the same strips and the z axis found in the case of two charges. The situation repeats itself for all odd and even configurations. I believe that these strips of almost conservative field area would essentially affect the formation of complex charge distributions by favouring the creation of linear fundamental particle distributions.

The crucial argument is the following. A charged distribution cannot consistently be realised and cannot appear as free particle (i.e. unshielded by charged gluons), unless an arbitrary linear charge distribution has a radiative field equivalent to ± 10g or 0g and unless this field has a similar radiative pattern to that of a charged or neutral dipole. Note that the above conditions arise from physical considerations on the radiative field originating from a given charge distribution. As such, these conditions could be considered as "physical" or macroscopic conditions.

Earlier, we discussed the creation of fundamental particles based on the properties of the fundamental algebra and on the two fundamental axioms that imposed the principles of conservation of energy, conservation of charge and conservation of spin. Only six different fundamental particles were allowed. The possible radiation fields of these fundamental particles were determined to be 0g , ± 2g or ± 10g . These results were established strictly and exclusively from mathematical considerations, by contrast with the previously deduced "physical" conditions. The essential argument is that we have two sets of conditions that must be satisfied by fundamental distributions of charges: the first set requires that the radiation field be equal to 0g , ± 6g or ± 10g and the second that it be equal to 0g , ± 2g or ± 10g . These two sets were deduced from completely different premises and are therefore independent statements. The only way that both restrictions could be mutually consistent is that all physically realisable charge distribution must satisfy the conditions given by the intersection of both sets of conditions. This translates in the following result:

Charge quantization lemma: a necessary condition for a fundamental particle to be physically realisable as a free particle is that it generates a radiation field equal to 0g or ± 10g .

Note that the requirement that all physically realisable charge distribution must obey the above charge quantization rule does not imply that a fundamental particle like the down or anti-down quark having a radiation field equal to ± 2g cannot be realised. It just states that such a particle cannot be realised as a free particle. The allowed fundamental particles can all be realised within a more complex particle as long as that particle has a radiation field equal to 0g or ± 10g . Upon disintegration of a particle containing such a down quark, the down quark would almost instantly decay into other fundamental particles whose existences are allowed. Based on the earlier discussion on the decay of down quark into up quark, electron and neutrino, we know that such a decay is possible and that the resulting particles respect the above quantization requirements. An obvious limitation to the above stated quantization law is that the conditions on the radiation field are "necessary" but do not appear to be "sufficient" to limit the choice of possible complex particles to those which are normally considered to be acceptable by the current quark standard model.

Remember that, from the fundamental axioms and the properties of the algebra of the S 's, we had deduced that only six fundamental particles could be formed. These fundamental particles had the following equivalent linear charge distribution if we make abstraction for the moment of the neutral cloud:

It is possible that this restriction should be extended to elementary particles, which do correspond after all to a linear distribution of quark surrounded by a neutral gluon cloud. As we pointed out earlier, the non-conservative property of the radiation field over a lattice, except along specific axis, would favour certain charge distributions. It is therefore possible that all physically realisable particles must be build from those fundamental linear charge distributions. Consequently, the charge quantization rule stated above would have to be amended. It would need to reflect the requirement that, in addition to having a radiation field of 0 and ± 10g , the total equivalent charge determined by the quark content of the particle should be reducible to one of these five equivalent linear charge distributions. For example, for the charge distributions presented earlier, only the representation (1) and (3) would then correspond to physically realisable particles. The addition of this reducibility criterion to the quantization rule would permit the addition of the word "sufficient", giving:

Charge quantization theorem: The necessary and sufficient conditions for a given charge distribution to be physically realisable as a free particle is that it generates a radiation field equal to 0g or ± 10g and its equivalent charge is reducible to that of one of the fundamental particles.

ELECTRIC AND MAGNETIC DIPOLE MOMENT

It should be noted that the geometry of a charge distribution is closely related to the concept of electric and magnetic dipole moment of a particle. In particular, one could define the electric dipole moment as a function of the distance between the geometric centres of the respective negative and electric charge distributions. Contrary to the classical case, where the principle of electric charge superposition is always valid, the contribution of a given charge in a distribution could be completely or partially shielded. Consequently, it appears necessary to distinguish between a geometric dipole moment and an effective dipole moment. The first one does not take into consideration possible shielding effect and should correspond to the classical case, while the second only take into consideration those charges whose electromagnetic effects are measurable. Since some of the individual electric charges in a given charge distribution can interact electro-magnetically by the intermediary of "virtual" photons or "virtual" electric and magnetic charges, there is a resultant "virtual" electric current circulating within or around the charge distribution. As a result there should be resulting induced magnetic fields that are possibly equivalent to some virtual magnetic charge dipole moment.

It is easy to see that both the geometric and effective electric dipole moment for all fundamental particles, except the down quark with a neutral radiation field, would be identically zero. In particular, this zero value for the electric dipole moment for the electron is compatible with current experimental results and the present theoretical understanding that it must be so to satisfy both T and P invariance requirements. It is beyond the scope of this paper to go much deeper than the above in the discussion or evaluation of other electric or magnetic dipole moments. We will limit ourselves to mentioning those particles that have an obvious zero value for their electric dipole moment.

THE MUON

According to the literature, the muon, denoted m , is a "fat" electron. It is the second lepton and was observed for the first time in 1937 by Anderson and Neddermeyer. The muon is a major component of cosmic radiation and has the same electric properties as the electron. Its spin is ± 1/2 like the electron. Its mass is estimated at 105.65946 MeV.

The muon, itself the decay product of p ± and k ± mesons, has a short mean life of approximately 2.197134 x 10^-6 sec and decays into electrons, neutrinos and gamma rays. Its interactions have been specified to great accuracy by quantum electrodynamics. Since, a priori, there are no reasons why the symmetric properties of the electron should not be carried over to the muon, and since the muon is considered to be a fat electron, I have attempted to build the muon by simply enlarging the electron. As for all heavier particles, whose construction will follow, this enlargement will be done following as closely as possible the symmetry of our elementary structure, the octahedron along the general principles established earlier. We will also make use of the gluons, weak and strong, to glue these heavier and larger structures.

A muon decays most of the time (branching ratio • 100%) in accordance with the following relation: m ^- ® e^- + n _e~ + n _m . Therefore, it appears quite natural to demand that a muon be composed of some of the above particles held together by some gluons. Since the muon is relatively light, it is likely that light gluons will be required. In accordance with the construction rule stated above and the selection rule on the commutativity of the gluons composing the gluon cloud, it appears that W⁺, W^-, Z' or Z^v' are the preferred weak gluons to be used in such a construction. Indeed, if we wanted to use the H , only an equal combination of H ⁺ and H ^- would ensure that the gluon cloud is magnetically neutral. However, a symmetrical distribution of such a combination of gluons would give too many null energy bounds to assure us of a reasonably stable particle. Of course, I could use only H ⁺ or H ^-. This would result in having a particle heavily charged magnetically, contrary to the construction rule stated above and experimental evidences. Similarly, the use of g 's would necessitate an equal number of g and g ~ and again too many resulting null energy bounds.

Figure 13: A possible muon configuration

On the contrary, the pairing of the W⁺ and W^- or the exclusive use of Z' or Z^v' should allow the construction of neutral gluonic clouds, provide reasonably strong bindings in accordance with the following relations:

W⁺ * W^- Þ [0|12],

Z' * Z' Þ [0|12],

Z^v' * Z^v' Þ [0|12],

and ensure a clear energetic boundary between the outer layer of the particle and the vacuum lattice.

A close examination of the interaction between the outer layer of such representations and the vacuum lattice militate in favour of using the Z^v' option. Indeed, it offers, contrary to Z', a positive space component which is more in agreement with our intuitive concept of particles which should have spatial extension in real and therefore positive space. Also, it offers a null interaction with the vacuum lattice in accordance with the relation

As a result, particles formed with such a gluonic cloud would be, as for the electron, enclosed in a "bag" of null energy bounds. Furthermore, contrary to a W⁺ - W^- gluonic cloud which possess a non-zero gamma radiation field, a Z^v' gluonic cloud has such a neutral radiation field. In fact a Z^v' based gluonic cloud would interact very little with the radiation field of whatever elementary particles, electrons or quarks, that one wishes to place in the middle of it. This will be illustrated shortly. Furthermore, unlike a cloud formed of Z^v or Z^v', W⁺ does not commute with W^- and both would be required to form a neutral cloud. Because of this non-commutativity, a neutral cloud of W's would be unstable based on arguments presented in an earlier article.

Another argument in favour of the choice of Z^v' as the proper light gluon to form the muons is that they would appear naturally in nature as cosmic rays. Remembering the speculations about the original creation of the fundamental states, gluons and space-time, we chose the doublets Z^v as the generator of space-time, leaving its associated zero-divisor Z^v' free to be used for something else which was not specified at the time. It is conceivable that such unused Z^v' gather around free electrons or other fundamental particles and are one of the possible sources of cosmic rays. We are continuously receiving a copious rain of cosmic rays. Their continuous formation could then be an indication that new regions of space-time are still being created between galaxies. It is also possible that the fluctuations in the background radiation of the universe, as observed recently, is a further evidence of such a continuous creation process. In accordance with the above discussion and the construction rules stated earlier, the simplest neutral light gluon cloud whose space-time distribution and structure closely match that of an octahedron is given at Figure 13. It will be noted that, as suggested by the dominant decay mode of the muon, I have placed an electron in the middle of the cloud. Evaluating the rest mass of the resulting particle, I get 1242 A.U.M.= 105.776805 MeV. The presently accepted experimental value for the muon is 105.658387 MeV. The difference is less than 1/4 of the mass of an electron or 0.1% of the experimental value.

To explain the observed decay process, one would have to assume that some of the gluons Z^v' transform into n _e~ and n _m when the particle breaks apart. Such a mutation is mathematically possible, since:

By first ejecting n _e~ = -S ₁, the relation becomes:

It is then possible to cancel out (-S ₀ - S ₃) with (S ₀ + S ₃) provided both (-S ₀) and (-S ₁) are emitted simultaneously. I get:

where n _m is assumed to be given by the two charges -S ₀ and -S ₁ bounded together in a neutral dipole. One advantage of this representation is that the decay process involve the gluons W's, which are, in accordance with our earlier interpretation, the mathematical representation of the so called weak-gluons which are alleged to mediate weak decay processes in the formalism of Q.E.D. theory. Alternatives to the above choice for the representation of the muon-neutrino, might be limited to making it absolutely identical to the electron-neutrino. Such a proposal appears to go against experimental observations that suggest that the two neutrino flavours are indeed different as demonstrated by Lederman, Steinberger, Swartz and collaborators at the AGS accelerator of Brookhaven national Laboratory. By observing that the neutrinos from the (p m ) decays produced only muons but no electrons they concluded that muon-neutrino and electron-neutrino are different.

THE TAU

The next and possibly only remaining leptons are the Tau, denoted t , and its associated neutrino, n _t . If the muon was a fat electron, the Tau is considered to be a fat muon, or an obese electron. Discovered in 1976 by Perl and collaborators at SLAC in an e⁺ - e^- collider experiment, it has all the same electrical properties as an electron and the muon: same spin and same electric charge. Its rest mass is estimated at 1784.1 (± 3.6 MeV). As for the muon-neutrino, not much is known about the tau-neutrino. It is believed to have a mass inferior to 35 MeV. A direct experimental proof of the Tau-neutrino existence is still lacking. Because of its large mass, the Tau has many more decay modes than the muon. Its principal leptonic decay modes are

Note that the two decay modes could be considered equivalent since

it follows from (2) that

Assuming that the two muon-neutrinos cancel one another, we get relation (1). There are also many hadronic decay modes that could be studied only after a proper representation of the mesons is completed.

Figure 14: A possible Tau configuration.

The representation of the Tau, corresponding to the first decay mode, should be constructed around the same general principles as those used for the muon. However, considering its large mass, it is unlikely that light gluons will be sufficient to build it. Indeed, if we were to use them, more than 15 times the amount of doublets that was used for the muon would be required. The complexity and dimension of the resulting particle would be such that the probability for its formation would be very small. Therefore, I would like to propose something like the representation found at figure 14. It is suggested by the construction scheme proposed earlier. This representation of the Tau would give us a predicted theoretical rest mass of 20954 A.U.M. or 1784.57905 MeV compared to the experimental value of 1784.1(± 3.6) MeV, a difference of 0.47905 MeV. or 0.027%, which is well within the precision afforded by experiments.

The experimentally observed decay modes given earlier could be considered to make sense if one realises that the heavy gluons c 's are respectively equivalent to:

c ⁺ = Z^v + W⁺ , and c ⁺~ = Z^v + W^- .

Or equivalently, based on our algebra:

c ⁺ = Z^v + (-S ₁ )(S ₀ + S ₃ ) = Z^v + n _e~ H

c ⁺~ = Z^v + (-S ₀ )(-S ₀ - S ₃ ) (-S ₁ ) = Z^v + n _e (-H ) n _e~ .

Provided that the n _e and n _e~ are first expelled in a configuration suggested at figure 9, the remaining H and -H are free to cancel out by simple superposition. Considering one of the glueball formed of six c ⁺ and one c ⁺~ in the middle, as seen in the representation of the tau, one could visualise the decay process illustrated at figure 15. This would suggest that a tau-neutrino could be represented by an octahedron formed by 6 n _e~ and one n _e in the middle, as proposed in an earlier article. This would give a mass of -6 A.U.M. for n _t or 0.51099906 MeV. that is well below the experimentally allowed maximum. It is also identical to the electron mass. Of course, the remaining n _e~ would correspond to the observed anti-neutrino. The representation for the tau is also consistent with our earlier representation of the muon. You will note that the electric-charge content of the Tau-neutrino respect our construction rule about the electric neutrality of the gluonic cloud of a physically realisable particle.

The photonic radiation field of the proposed representation of the Tau is also equal to -10g , as one would expect since it is alleged to have the same measurable electric charge as the electron and the muon. However, you will note that the experimentally accepted electric charge of the Tau has probably never been measured directly, but is inferred from the resulting decay process, based on the principle of conservation of electric charge. Indeed, a direct measurement of the charge would be rather difficult considering the short mean life of the Tau, approximately 10^-12 sec..

Figure 15: Possible decay of heavy glueball

SUMMARY

An important feature of this theory is that electrons and quarks would be in the same family of elementary particles. This is somewhat reminiscent of the results obtained in the grand unified theory proposed by Georgi and Glashow (1974) using the SU(5) group symmetry, where the quarks and leptons appear together in the same multiplets. The super-heavy gauge bosons of the SU(5) unified theory can cause transitions between leptons and quarks. In my theory, similar transitions between quarks and leptons are also evidenced in the decay processes presented earlier. Another important characteristic of the fundamental particles (electrons and quarks) studied so far and of all more complex particle representations that we will construct in the next chapters is that they have DIFFERENT DIMENSIONALITY span than what we are used to. We are accustomed to view material bodies or entities as occupying space alone and lasting for so much time. By virtue of our definition of space-time over a lattice, we have to accept the consequence that complex particles could have also temporal, electric or magnetic dimensionality as well as a spatial dimensionality. Any two adjacent points in space-time for example occupy not only different portion of space but also different portion of time. While surprising, such a result should probably have been expected. In relativistic field theory, space-time is a continuum and an essential property of relativistic field equation is that space and time must be treated on an equal footing, both appear with the same order for the derivatives. Why should particles have only spatial extension in such a continuum and be necessarily punctual with respect to the other possible dimensionality?

Such a remarkable property should have observable consequences. Consider a high energy photonic probe, energetic enough to allow deep penetration within a proton, which is alleged to be constituted of three permanently bounded quarks ( Up, Up and Down). Isn't it possible to consider that, since our probe, being a photon and as such not having spatio-temporal extension according to our representation of a photon as a doublet could only "see" one doublet or singlet at a time within our target proton and not feel the others? If this is so, the probe would view that single part of the whole particle as being in relatively weak interaction with the six neighbouring points. Isn't that essentially the basic idea behind the concept of asymptotic freedom that is alleged to characterise the behaviour of quarks inside hadrons?

I am quite satisfied so far that electrons, basic quarks, the neutrinos and heavy leptons are adequately represented by the elementary structures given above. A verification of the adequacy of the above representations of the Up and Down quarks in describing true physical quantities will have to await the attempts to build mesons and hadrons, the object of a subsequent article. The other flavour of quarks, i.e. the Strange, Charm, Bottom and Top quarks, will be dealt with at the same time than heavy hadrons and mesons that are alleged to be made from them.

Physicists and cosmologists are presently speculating about the existence of only three families of neutrinos. In fact, they are convinced that there is no more than three families of leptons and three families of quarks. Our present understanding of the Big Bang theory and other cosmological considerations seem to support this thesis. In my theory, there appear to be no valid reasons to exclude larger entities formed by the same two fundamental charges. Except possibly that much larger structures built exclusively with S ₀ and S ₁ would not have very strong energy bounds and as such would become quite unstable, collapsing essentially under their own mass. This has been seen before and provides the essential reasons why there are no "natural" elements beyond the 106 presently confirmed one's. In a subsequent article, I will use the same basic principles to illustrate that it appears possible to build representations of all known hadrons. In all cases it will be shown that it is possible to obtain very good reproductions of the experimental mass spectra and credible explanations of the various decay modes of those particles.

The Muon was formed essentially from a cloud of Z^v' surrounding an electron. Since the p ± mesons are known to decay into muons and muon-neutrinos, it appears reasonable, as a first attempt, to build the potential representations of the charged and neutral p mesons, using essentially the same technique. However, inside the gluonic cloud, we would probably find u, u~, d and d~ quarks in various paired combinations depending on the electric charge of the mesons. This last statement is based on the premises that the quark content of such particles has been essentially verified by experiments and that the essentials of the standard quark theory should be preserved. The build-up process described earlier also supports it. The difference with standard Q.C.D. theories is that we will not need a superposition of colored states, which explicitly requires six quark states to form a simple meson. Figure 16 represent some of the many possibilities respectively for p ⁰ and p ± . These have been built on the basis of the possible linear two quark distributions that are compatible with the charge quantization rule discussed previously.

Figure 16: Some configurations for the neutral and charged pion.

The first thing to note is that I am getting much more possibilities than what is expected from experimental evidences. While the mass values of the resulting particles are of the same order of magnitude as the accepted experimental values i.e. from approximately 124 MeV. to 145 MeV., such diverse values do not appear to be supported by current experimental results which are of 134.9739 (± 0.0006) MeV. for the neutral pion and of 139.5675 (± 0.0004) MeV. for the charged pion. Experimentally, there does not appear to be other still undiscovered particles of similar masses in the same energetic area. The exception to the above considerations appears to be provided by the allowed uu~ representations of p ⁰ (figure 16 shows a typical one) which have a mass of 1584 A.U.M. or 134.9038 MeV. that compares very well with the experimental value of 134.9739. The difference is only of 0.05%. This would indicate that we are possibly on the right track. If you remember, we argued in previous articles that non-commutativity of charges forming a particle was an indication that such a particle would be unstable. It appears reasonable to consider that the determinant of the commutator provides some measure of the energy produced by some force which tends to tear apart a given particle whose internal constituents do not commute with their immediate neighbors. Of course such destructive energy could be counterbalanced by the energy made available to generate the mass of the particle as provided by the determinant and trace of the interaction. You will note that all gluons and naked electric charges composing the central quarks do not commute with the Z^v' doublets forming the neutral cloud of the given representations of p ⁰. The mass energy of each of the c -Z^v' bounds is [0|6] while the "energy" of the commutator

is [-1|0]. The argument is that [0|6] is greater than [-1|0] and should provide reasonably stable bound. However the mass

energy of each of the ± S ₂ - Z^v' bounds is [0|0] while the energy of the commutator

is [-1|0], which would provide for an unstable condition within the particle, contributing to its eventual decay. If this assessment is correct, there should be a correspondence between the experimentally verified mean-life of a particle and the number of its relatively unstable bounds. According to this hypothesis, the above representations of p ⁰ would be quite unstable. Experimentally, the half-life of a p ⁰ is of the order of 10^-17 sec. Considering now the representations of the p ⁰ based on a down and anti-down quarks, as illustrated at figure 16, it appears that we have quite a few more representations than we actually need. Furthermore, none of the masses, while being in the appropriate energy range, coincide with the experimental value. Considering that this region of the mass spectrum has been extensively studied, it is unlikely that such neutral particles with mass of 1610 A.U.M. (137.118 MeV.), 1608 A.U.M. (136.9477 MeV.) and 1636 A.U.M. (139.3324 MeV) respectively would have been missed by experimentalists.

Also you should note that some of the representations are built using the asymmetrical down quark. It is possible that nature does not allow such construction especially when the gluon cloud is composed exclusively of weak gluons. However, I could not find strong theoretical or algebraic arguments in favor of such a rule. What should we make of it? Earlier, we introduced the concept of electric charge quantization that allowed us to classify as unphysical numerous combinations of quarks. I also made extensive use of the properties of the algebra and the two fundamental axioms to restrict the number of possible fundamental particles. The principles of conservation of energy, charge and spin were also used to discriminate between different possibilities. It is unlikely that the algebra could again be invoked to provide us with additional exclusion rules. The only remaining characteristic of a particle that has not been invoked yet to discriminate between allowed and forbidden representations is the mass itself. Are all mass values permissible for stable particles? I have tried to find a consistent way to eliminate some unwanted pion configurations but without much success.

Is it possible then that we must accept all possible configurations, i.e. those that respect the radiated charge quantization requirement, the global neutral color requirement and the requirement that Q _rad be smaller than or equal to Q _eq? This would be a major departure from the generally accepted paradigm that assumes that all possible representations of a given physically realizable particle are fundamentally and absolutely non-discernable. It is also generally assumed that the experimentally observed spread of the mass spectrum distribution, as shown in the PDG article, is a consequence of measurement imprecision and of quantum mechanical effects. If multiple representations for a given physically realizable particle were acceptable than we would have to systematically build all acceptable possibilities, evaluate each of the masses and average them using some weighting factor to account for their relative probability to be realized. Such a probability would likely be based on the relative symmetry of each representations, on the number of different ways that a given representation or charge/gluon distribution can be assembled and/or on the number of non-commuting bounds within the particle. That theoretical mass average would then need to be compared to the experimental one. Similarly the theoretical and experimental mass spectra would have to be comparable i.e. with peaks and troughs appearing at relatively the same mass values.

THE NEUTRAL PION

In an attempt to validate or invalidate this last possibility, I have constructed all possible neutral radiated charge and neutral color representations of hypothetical neutral pions along the general construction principles enunciated earlier. I have even allowed those representations for which the colored gluon cloud of the internal quarks were not individually neutral (see particle F at figure 16). The only requirements were that the total gluon cloud had to be of neutral color, the radiated electric charge (Q rad) had to be smaller or equal to the equivalent charge (Q eq) and the global radiated electric charge had to be neutral. The frequency of occurrence of each geometrically distinguishable representation is obtained by various rotations of the internal quarks that respect the previously stated rules of charge quantization. Rotations that result in undistinguishable representations are also accepted. I have identified, as shown at figure 17, 568 such possibilities whose mass and frequency of occurrence are as listed. It is rather remarkable that the theoretical mass average obtained is 134.9763 MeV which is within 0.0000568% of the experimental value of 134.9764 (± 0.0006) MeV.

Figure 17: The neutral pion. All possibilities are included.

THE CHARGED PION

At figure 16, I was presenting some of the possibilities for the representations of the charged pion p ± . Proceeding in a similar manner as we did for the neutral pion, I have identified 1970 representations for the charged pion. We obtain a theoretical mass average of 136.3352 MeV. that is unfortunately only within 2.31% of the experimental value of 139.56995(± 0.00035)MeV.. This is definitively not as good a result as the one obtained for the neutral pion. Why is there suddenly such a difference in predictive power?

The first thing that should be noted is that by allowing all configurations for the charged pions we have allowed quite a few configurations which have naked electric charge inside the particle such as the configurations m at figure 17. In many cases, especially when asymmetrical down quarks with naked electric charges are involved in the construction, the asymmetrical stress imposed by the non-commuting bounds could be such that the mean life of such particles would be much shorter than for the others. Note that the extremely short lifetime of the neutral pion (8.4 x 10 ^-17 sec) was explained by the numerous non-commuting bounds generated by the naked electric charges inside the particle. The charged pions have a typical observed life time of approximately 2.6 x 10 ^-8 sec which is nine order of magnitude above that of the neutral pion. If we allow in the evaluation of the mass of the charged pion the configurations with asymmetrical naked electric charge inside the particle such a difference could not be explained. Note that this exclusion does not imply that the rejected configurations are not physically realizable. Rather, it means that those configurations are so short lived with respect to the other ones that they essentially do not contribute to the experimental mass spectra.

In the original computation, I had even included some representations where the central positive charge of the anti-down quarks had been replaced by a negative charge, making it effectively a down quark, whenever the total resulting electric radiation of the particle was not affected. While this is contrary to the principle enunciated earlier that Q _rad must be smaller or equal to Q _eq , I felt that the validity of that principle should be verified further. These representations have also been removed from the final series in addition to the representations with naked asymmetrical electric charges inside the particle and those with vertically oriented naked electric charges. The results are shown at figure 18. Note that the resulting averaged theoretical mass for the charged pion is 139.57014 MeV. that is within 0.0001% of the experimental value. This exceptional result must be more than coincidental.

An important consequence of the above is that a given particle would not have a unique or non-discernible configurations. Again, this is a major departure from the accepted paradigm. If we accept this conclusion and the resulting mass spectra there are still a problem to reconcile them with the reported experimental distributions. Experimentally, it is observed that most particles occur respectively within s = 0.0006 MeV for the neutral pion or s =0.00035 MeV for the charged pion of the stated experimental mass values. It would be strange if high peaks, such as those appearing at 1534(40), 1610(56) and 1636(52) AUM, would not show during scattering or production experiments of neutral pion and if only the one at 1584(80), which is within 0.05% of the experimental value, would effectively be produced. The same thing could be said about the peaks 1610(72), 1622(66), 1648(76), 1672(60) and 1696(60) for the charged pion, noting that the average of 1634(24) and 1646(54) would give us a mass of 139.8696 MeV. that is within 0.2% of the experimental value. Is it possible that experimentally all these peaks would have been missed or considered something else like background noise?

Figure 18: The charged pion. Restricted data.

An alternative would be to consider that by some symmetry breaking mechanism and some super-selection rule not yet found, the only acceptable charged pion configurations would be those appearing at figure 19. In both cases one of the colored gluon has been replaced by a weak gluon W. The spin and the electric radiation field are preserved. The mass of these charged pions becomes 1638 A.U.M. (139.50274 MeV.) which is within 0.046% of the experimental value. The difficulty resides in accepting that a specific W should be used for a given polarity of the meson charge. For a positive pion, c ⁺ becomes W⁺ or W⁺" simply by the absorption of a Z or Z', since

The decay process would be initiated by the reversed relations

where the minus sign could be an indication that a Z or Z' is ejected. The anti-down quark would then decay through the usual process

leading to the experimentally verified decay mode

Note that if our earlier representations of the neutrinos were correct, a Z would be nothing more than the sum of an electron neutrino and an anti-electron neutrino. The superposition of the two entities could then be considered as a neutral current represented classically by Z⁰ that is either ejected or absorbed. Z' = n _e - n _e~ would be seen as the "simultaneous" ejection of an anti-neutrino and absorption of a neutrino. Since Z can be ejected freely into space-time Z^V, it appears normal to consider that it is the preferred occurrence. This seems to indicate that the interaction results provided by the presence of W⁺ are more compatible with the structure of space time than that of W⁺".

Note that in addition to allowing the particle represented at figure 19 to have a mass which is closer to the experimental value, the replacement of c ⁺ by W⁺ slows down the decay of the particle: the anti-down quark cannot decay through the above process unless W⁺@ is first converted to c ⁺. The decay is also slowed down by the presence of the heavier up quark which must also decay to the normal excited representation found in p ⁰. This decay could be justified simply by the creation of a (-S ⁰ )(-S ⁰ ) pair out of the center of the quark similarly to what was done in the creation process of the fundamental particles. These delays would indicate that the charged pion should decay much more slowly than the neutral pion. Experimentally, we do observe such a large difference in the decay rate of both particles: 10^-8 sec. compared to 10^-17 sec.

Figure 19: Alternate configurations for the charged pion.

Similarly, the proper representations for the neutral pion could be restricted to that shown as configurations D at figure 16. Note that this does not exclude the possibility that the dd~ configuration (B at figure 16) might appear within composite particles. Its non-observability in the experimental spectra could be explained by a rapid decay to the lower mass configuration of 1584 AUM. Indeed, we have shown that the down and anti-down quarks could decay almost "instantaneously" to up and anti-up quarks by virtue of the process defined earlier:

From which it follows that

This would suggest that the original heavier p ⁰ is some sort of excited intermediate state which, upon release of a certain amount of energy in the form of g or electron-positron pair, becomes a neutral uu~ pion having the proper experimentally observed mass. The theoretical values for the masses obtained would be very close to the experimental values. Note that it is possible that all other configurations would remain acceptable within complex particles. There might even be advantages in using all these representations. It could help explain some of the multiple peaks appearing in the mass spectra of many complex particles whose decay modes include charged pions, such as the r 's, the S 's and many others.

Note that for all representations the resulting gamma radiation fields are equivalent to ± 10 g for p ± and 0 g for the uu~ and dd~ representations of p ⁰. This is what one would expect to measure at a distance from the particles. However, it should again be noted that in most cases the only experimental certainty that we have about the "measurable electric charge" of such short lived particles is what is inferred from the charge of the particles resulting from the decay. It is then assumed, by virtue of the principle of conservation of electric charge, that the original particle had the same measurable electric charge, even if it is essentially impossible or very difficult to measure that charge. You will also note that the total electric charge of the gluons forming the internal quarks is neutral.

AN ALTERNATE CONFIGURATION FOR THE GLUON CLOUD

As an alternative to the previously chosen light gluon cloud distribution, as depicted at figure 16, it is possible to take the one having four additional weak gluon in a slightly different distribution, as shown at figure 20, without affecting the mass or electrical properties of the particles. However, this choice offers a representation which is less compact than the previous one and which demands four additional gluon doublets. While the previous one is preferred, for reasons of economy, I cannot exclude the possibility that this different representation could actually occur, especially, as will be seen shortly, in the representation of more complex mesons whenever glueballs will be used in the construction.

Figure 20: An alternate configuration based on type B gluon cloud (see fig. 11).

EXCITED STATES FOR THE PIONS

Figure 21 shows some of the possibilities for the formation of excited states of the basic neutral pions. Similar constructions would apply for all pions. I believe that such excited states generally occur only within more complex mesons and are, as a consequence, not usually observed. For that reason, so long as the overall gluonic cloud of the complex meson is electrically neutral, the previously essential super-selection rule might not have to be imposed on the individual excited states. Until I have some evidence that the super-selection rules must also apply to the individual excited states, I will keep all their representations. However, it must be recognized that it is probable that excited states with high divergence from the charge, mass and cloud neutrality super-selection rules would have much lower probability of occurrence than those whose radiation fields are closed to the values 0 or ± 10g and their gluonic clouds are neutral. Another characteristic of these excited states is that the central up (anti-up) quarks can also be in an excited state. This is achieved by the replacement of the central S ₀ by c ⁺ or c ⁺~ gluons. Note that by virtue of the much stronger energetic bindings provided by the liaisons between the c 's, it is possible to have much more massive particles within the same space-time volume.

Figure 21: Some of the excited states for the neutral pion.

THE MESON FAMILY

I wish to demonstrate that a simple superposition of two elementary or excited pions, in a manner explicitly suggested by their geometrical shape (an apparently missing quark on top), could generate particles having the mass of most of the known mesons. In some cases, the reinforcement of the binding energy of the resulting particle by the symmetrical distribution of an appropriate number of gluonic glueballs between the two pions would be required. The process was suggested by the scheme described earlier. Using the suggested method, we will demonstrate with a few example that it is possible to construct representations of all the mesons having a dual charged mesons decay mode. Actually, it has been possible to construct representations for all experimentally observed mesons irrespective of their decay mode.

THE NEUTRAL K⁰_S MESON

Except for the charged K± mesons, at 493.677 MeV., the K⁰ meson is the lightest of all complex mesons at 497.7131 MeV.. Experimentally, it appears in two apparently equal mass configurations denoted K⁰_S and K⁰_L . The only differences between these two particles are their lifetime (5.17 x 10^-8 sec for K⁰_L and 0.8926 x 10^-10 sec for the K⁰_S) and their decay modes. While K⁰_S decays almost exclusively into dual pionic modes: p ⁺ p ^- , branching ratio 68.61% and p ⁰ p ⁰ , branching ratio 31.39%; K⁰_L decays primarily into three particles modes: 3 p ⁰, p ⁺ p ^- p ⁰, p ⁺ m ^- n _m ~ and p ⁺ e^- n _e~. The neutral K-mesons are alleged to be formed of d S~ or d~S quarks pair. Let us assume that triplets of quarks, as suggested earlier, can indeed represent a strange quark:

where the order of the quarks might not be important. From four fundamental quarks, say u, u~, d and d~, it is possible to construct quite a few linear quark combinations. For a neutral particle, only the quark combinations having a resultant equivalent radiated electric charge of zero would be allowed. Furthermore, let say for now, without further justification, that a realizable particle must have the possibility of decaying into pions configurations. A representation such as udd~u~ does not appear to meet these criteria. However, it should be noted that the combination [u-d] can indeed produce a particle which has the same mass as the positive pion [u-d~] and a photonic radiation field equal to +10g which would make it non-discernible from a conventional p ⁺. Such a configuration was considered unphysical in the creation process of a single pion, since the radiated charge of the particle would be larger than the charge content as provided by the quarks. However, a priori, their temporary existence might be allowed when such a pion appears within a larger complex particle. Furthermore, in a given representation such as [d-u-d~-u~], one of the pseudo constituent pion (the bottom [d~-u~] part) does not have quite the right mass and radiation field when considered independently. However, it is possible that upon decaying one of the two quarks rotates by 90 degrees releasing energy in the form of neutrinos in a decay process similar to what was described earlier. It follows that some of those representations are acceptable candidates.

Figure 22: A possible configuration for the neutral Ks Meson.

The building concept is illustrated at figure 22 with a representation of what is believed to be a K⁰ meson that is known to decay into the indicated combinations of p ⁰, in one of its primary decay modes. The K⁰ meson would be composed, in accordance with current quark theories, of a down quark and a strange (S) quark. Note that, in the representation provided, the K⁰ is effectively composed of a down quark and a secondary group of quarks (d~u~u) of charge -1/3 which could easily be associated with the concept of a strange quark. One will note that the predicted theoretical mass (498.7351 MeV.) is within 0.2% of the experimental value (497.671 MeV.). Such a particle would obviously decay into two different representations of neutral pions. Again the total color is neutral. The total radiated charge is also 0g , which is what one would expect from a neutral particle. It should be remembered that this electric charge would correspond to that of a neutral particle only at a relatively large distance from the particle. The issue whether or not all representations should be accepted as valid remains. The same could have been said for the charged pion. This construction process would suggest that there is more than one possible representation for a given particle. And these representations would be oscillating around a central mass value that would correspond to the experimentally observed mass. The experimental mass spectra would be nothing more than a statistical average over all the possible representations of a given particle. To verify this possibility, I constructed all possible representations of the K⁰_S meson., On the basis of the experimental decay products, It was assumed that the main difference between the K⁰_S and K⁰_L is that the first one is composed of four quarks, while the second is composed of 6 quarks. We will see later that such an assumption appears to be well founded.

It would be impossible to provide a picture of all possible representations in this article. The reader will have to thrust me that a large amount of these were constructed and their masses evaluated. These were constructed on the basis of the allowed possibilities for 4 quark configurations. Amongst the possibilities one must also consider those given by the rotation of the heavy glueballs at the center of the particle and those given by various rotation of the constituent quarks which preserved the total neutral electric radiation field of the particle. Indeed, at this point, I would restrict the study to those representations with the exact proper radiation field. For the potential p ⁺ p ^- configurations, one must also consider all alleged proper representations of the charged pions as well as those without the W± . One must also consider the possibility that the complex particles can form with the heavy glueballs distributed asymmetrically with respect to the relative center of the particle. Remembering that the heavy glueballs can also be rotated around the particle main axis and could be also located at the bottom of the particle. It was also mentioned that an alternative acceptable configuration for the pions was that presented at figure 20. They must be considered with every one of the previous configurations. I did not consider the mixture of both pion configurations in a single particle since these would not change the averaged mass value and would not displace the mass spectra. The only impact would be a sharpening of the peaks in the theoretical distribution of mass-energy. I do not believe that all possibilities are physically realizable. The simplest restriction considers only the configurations which provide for Q_eq=0 (which amounts to applying the criteria of reducibility to one of the fundamental charge distribution) and for those which can decay to a two pions final state. A grand total of 7376 such possible different representations were found each having a neutral color and a neutral electric radiation field. The results are presented in the form of a mass (energy) distribution graph (figure 23) for all possibilities of four quarks with neutral radiation field and neutral color.

While the PDG does not provide any recent experimental mass spectra for the neutral Kaons, the shape of the distribution is generally speaking that of a typical experimental energy spectrum. You will note that the averaged mass of all theoretical possibilities is 497.5514 MeV which is within 0.024% of the experimental value of 497.672 MeV. What is even more surprising is that the theory would predict that 5056 of these would decay in the p ⁺ p ^- modes for a predicted branching ratio of 68.55% and the remaining 2320 would decay into p ⁰ p ⁰ for a branching ratio of 31.45%. The experimental results are 68.61% and 31.39% respectively, as provided by the latest PDG publication.

Experimentally, the above multiplicity of representations is suggested by the various decay modes of most complex particles. In a sense this model could be more satisfactory than the present standard model of Q.C.D. which claims that a unique particle, defined as a color singlet of a certain symmetry group (SU(3)) and described by a given solution or superposition of solutions of a relativistic wave equation, can have multiple and some time quite different decay modes. Such a representation of reality might be completely satisfactory on the basis of the mathematical formalism. However, it endows the absolute reality of the elementary particle world with an aura of uncertainty and mystery which has been and remains hard for many to reconcile with the hard evidences presented by the macroscopic reality.

Figure 23: Theoretical mass spectrum for the neutral Ks Meson.

THE NEUTRAL K⁰_L MESON

As stated earlier, we also have a long-lived Kaon, denoted K⁰_L, which appears to have almost an identical mass to that of K⁰_S but a longer lifetime and quite different decay modes. In particular, it appears to be decaying into three pions for about 34% of the time and combinations of charged pions, electrons and neutrinos (38.7%) or charged pions, muons and muon-neutrinos the rest of the time (27%). One possible representation is shown at figure 24 for the p ⁰ p ⁰ p ⁰ decay mode. Because of the extremely large amount of possibilities with which 6 quarks could be assembled while preserving the neutral color and neutral radiation field principles, I have not computed all possibilities.

Figure 24: One of the many possible configurations for the K(L) Meson.

One interesting aspect of the respective configurations of both types of neutral Kaon is that it tends to confirm earlier speculations about the primary reason for differences in decay rates of apparently similar particles. This concerns the apparent relationship between the relative stability of a particle and the abelian properties of the gluons and fundamental charges forming a given particle. You will note that all our representations of K⁰_S contains a larger amount of heavy gluon of the c ± type than the representations of the K⁰_L whose gluon cloud is composed exclusively of weak gluon of the Z^V' type. If our hypothesis that non-commutativity of the constituent gluons is partly responsible for the decay rate of the particles, we would have to conclude that K⁰_S must decay much more rapidly than K⁰_L. This happens to be exactly what is being observed in experiments. It should be noted however that other mechanism could also contribute to the decay rate of unstable particles. In particular, I suspect that the relative symmetry of the radiation field of a particle, which is directly related to the internal symmetry or distribution of the constituent fundamental charges and charged gluons, will also contribute substantially to the stability or instability of a particle. An immediate consequence would be that the representations of a given particle would not have necessarily all the same decay rate. This would translate in a certain indeterminacy or uncertainty in the observed decay rate of a given particle since in most cases one representation is hardly distinguishable from another using current technology.

OTHER CASE STUDIES

To illustrate further this multiplicity of possible representations for a given particle, I have chosen, within the Particle Data Group tables, a simple particles which offer a relatively simple spectrum of decay modes and for which the experimental mass spectra is provided. The goal is to compute as many representations as possible of a given particle in all its major decay modes and to compare graphically the results with the experimental mass spectra of the same particle as provided by the P.D.G.. For this purpose, I have identified the r ⁰(770) and the w (782) with the following decay modes and branching ratios:

r ⁰ ® p ⁺ p ^- (~ 100%)

w ® p ⁺ p ^- p ⁰ (88.8%)

® p ⁰ g (8.5%)

® p ⁺ p ^- (2.21%)

Figure 25: One of the many possible configurations for the Rho Meson.

I have completed the process for the two charged pions decay mode of the r ^0® p ⁺ p ^-, using the same pion representation than that used for the k ⁰ . One typical Rho representation is shown at figure 25. It is important to note that variations on these representations are obtained by rotation of the internal quarks or a rotation of the heavy colored glueballs around the vertical axis in a manner which preserves the overall charge symmetries, the color neutrality and the resultant neutral radiated electric charge. All possible configurations of the charged pions are used including those where the W weak gluon has been replaced by an heavy colored gluon. Plotting the number of discernable occurrences of a specific mass value versus the energy, I got results that did not fit well with experimental results. The spectra does cover the general area where the rho meson is located (around 9050) but it extends too far up covering even the area where one would expect to find the omega meson.

Figure 26: Mass spectrum for the neutral rho meson.

In an attempt to achieve a better match with the experimental mass spectra, I separated the lower mass values (three families) and the higher ones (the remaining 7 families) on two separate graphs. I got the spectra at figure 26 and 27. At figure 26, we see two distinct peaks located essentially where the experimental peaks of the neutral rho mesons are and as provided by the PDG data. The theoretical averaged mass is 9042.5333 AUM or 770.1210 MeV., which is within 0.08% of the experimental mass resulting from non photo produced mesons. I did not consider the photo-produced data since there are no experimental mass spectra curved provided. The experimental mass average of those is 768.1 MeV.. That is essentially within the same general area. Note that presently accepted theories cannot explain the presence of these two peaks. It is true that the first peak does not appear proportionally speaking as high as the first peak of the experimental spectra. We have assumed in our theoretical construction that all modes appear with the same frequency. This is not necessarily the case in nature or within the accelerators producing these particles.

As far as the rest of the configurations are concerned, as illustrated at figure 27, it appears to provide us with an averaged theoretical mass of 9182.368308 AUM or 782.0303 MeV. . That is within 0.011% of the experimental mass of the Omega meson. Our spectra covers a wider energy range than the experimental one, which has been reproduced to scale in the upper portion of the graph, but out of place for clarity (both vertical lines should essentially be superposed). However, its top part does match fairly well with the experimental data, noting that the three pions decay modes are only partially included in our graph under the generic title "other modes". While our theory would suggest that a p ⁰ p ⁰ mode should also exist for the rho meson, it has never been observed according to the PDG data.

Figure 27: The mass spectrum for the neutral Omega Meson

THE REST OF THE MESON FAMILY

To further test the amazing, even if only coincidental, power of the theory, I have succeeded in constructing the potential representations of all known mesons in their most important decay modes. In particular, I constructed a particular sequence of particles using only two charged pions in different excited states and using different glueballs to reinforce the energetic liaison between these two pions. The sequence follows exceptionally well the distribution of the mesons across the mass-energy spectra as provided in the P.D.G. listing, agreed very well with all experimental mass values and predicted all the large energy gaps were no such particles are observed. Considering that the number of different decay modes increases significantly with the mass of the meson being considered, I could not present in a single paper all the results obtained. Invariably, I was able to represent, using this technique, all decay modes of all the mesons for which it was attempted. And the resulting theoretical mass was always close to or well within the published experimental precision or one could see the emergence of mass spectra similar to what was done in details for the Rho, Kapa and Omega mesons. Without further discussions, I am presenting at figure 28, possible representations for the neutral Heta Meson in one of its primary decay mode. Note that the experimental mass is estimated at 547.45 (0.19) MeV.

A representation for the r (2210) appears to be the largest possible particle formed of p ⁺ and p ^- on the basis of the excited states defined earlier (see Fig 29). However, I have succeeded in constructing much larger particles with the same two pions and, experimentally, there appear to be such heavier particles with that decay mode. To get heavier pions than the heavy excited states of mass 10522 AUM, I surrounded them with alternatively

a. 12 heavy gluons (mass 1824) in a symmetrical way suggestive of the "squared" representation of the pions, or

b. 24 heavy gluons (mass 5472), three gluons for each of the eight inside corners of the normal representation, or

c. a combination of the two preceding configurations, for an additional mass of 7296.

Adding these 24 heavy gluons to the drawings would have added unnecessary complexity to them, since they would not normally be visible on the main vertical cut of the particles. Their presence is indicated solely by the addition of an extra corresponding mass value (1824, 5472 or 7296) to the energy computation column.

Figure 28: Two possible configurations for the neutral Heta meson.

Figure 29: The heaviest meson using standard excited states.

Figure 30: Adding 12 heavy gluons to each of the two pions allow a substantial mass increase.

The lightest of such particles is the f (2510) shown at fig.30, followed by the h _c(1s) with a mass of approximately 2980 MeV, shown at fig.31. The heaviest is the c _co(1P) with a mass of approximately 3415 MeV as shown at fig. 33. For this last one, note that I could identify only four different representations of respective mass 40012, 40060, 40140 and 40188 AUM for an averaged theoretical mass of 40100 AUM or 3415.177051 MeV. that compares perfectly with the experimental mass of 3415.1(1.0)MeV..

In support of this unorthodox process, you will note that, so far, there are no experimentally observed particle between the r (2210), the f(2510) and the h _c(1s) which would decay into the p ⁺ p ^- mode. It is rather amazing that our theory offers the possibility of jumping exactly at these three energy levels with no theoretical possibility for particles to be in between. A similar argument is also valid for the J/y (1s) at 3097 MeV.(Fig.32) and the c _co(1p) at 3415 MeV.(Fig.33). That last one is the heaviest possible particle using this process. Not surprisingly, there appear to be experimental evidence for yet heavier particles with a p ⁺ p ^- decay mode. They are the U (1s) at 9460 MeV.(fig. 34), the c _bo(2p) at 10.232 GeV (not shown)., the c _b1(2p) at 10.255 GeV (Fig 35) and the c _b2(2p) at 10.268 GeV.(Not shown). These are realized by surrounding a particle such as the c _co(1p) with large glueballs of appropriate color. Again we obtain particles with masses which agree very well with the experimental values while following the experimental data over these large energetic gaps where there does not appear to be any particle with the two charged pions decay mode. As a last remark on the subject, it should be noted that the number of possible configurations, offering both an acceptable radiation field and a neutral color tends to decrease substantially with heavier pion excited states. Therefore it is even more remarkable that the theory still offers very good match with the experimental mass for a particle such as the c _b2(2p): the theoretical mass value is within 0.0047% of the experimental one.

Figure 31: Adding 24 heavy gluons to each of the two pions.

Figure 32: Adding 12 and 24 heavy gluons to each of the two pions.

Figure 33: The heaviest meson using this technique.

Figure 34: Adding four heavy glueballs of total neutral color around the two pions.

Figure 35: Possibly the heaviest meson made in combining both techniques.

Figure 36: Could this be the representation of a negative weak vector boson W?

One would think that further increases in mass must be possible by continuously adding an equal number of positively and negatively colored gluons to some combination of mesons. For that matter, one could add such large glueballs to an electron and obtain some very massive particle. Figure 36 is just such an attempt to enlarge an electron. Curiously the particle that result from this process happens to be of the same mass than the experimentally measured W that decays into an electron (or positron) and neutrinos. This decay mode is not unlike that of the muon and tau that also decay into electrons and neutrinos.

One of the inescapable consequence of this section on mesons, is that contrary to the current belief, elementary particles would not appear in single mass configuration which could decay via the weak or strong interactions to multiple different decay products. On the contrary, a given meson resonance already contains the elementary particles into which it would eventually decay. The representations are not unique and the relative number of their occurrences determines essentially the branching ratio of the different decay modes. All these representations provide a theoretical mass spectrum that replicates very well the main characteristics of the experimental mass spectra. The so-called experimental mass appears to be provided by the average of all the representations contained in a given theoretical mass spectrum. And finally, it would appear that the present particle nomenclature would need to be changed drastically. A classification on the basis of the decay products would appear to be more appropriate.

THE PROTON and NEUTRON

Using essentially the same conceptual approach as for the mesons, considering the relatively large mass of the proton and considering that its photonic radiation field should be equal to +10g , one of the representations offering all these characteristics and satisfying our charge quantization rule is presented at figure 37. Note that the gluonic cloud is electrically neutral and that each quark would be of neutral colour, if our previous speculations on the representation of the concept of colour in my theory were correct. Considering the symmetric distribution of the gluonic cloud around the vertical axis and the other obvious symmetries of the proton configuration, it would appear to make sense to say that this particle only have two possible distinguishable orientations with respect to an external electromagnetic field. They should be spin ± 1/2 particles. Note that the central down quark does not play a role with respect to the spin since the heavy gluon cloud shields its interaction with such external fields.

Graph 1: The theoretical mass spectrum for the proton

Graph 2: The theoretical mass spectrum for the neutron.

Figure 37: One of the many possible configurations for the proton.

As for the pions, the unexpected result is that we obtain many possibilities. It is indeed possible to construct 40 different representations of particles with a radiation charge of 10g using all possible linear assemblies of three quarks and the heavy gluon cloud as shown at figure 37. This number is increased substantially if we allow for the various rotations of the quarks. The mass of the illustrated representation is 11016 A.U.M. or 938.19427 MeV., compared to the experimental value of 938.27231 (± 0.00028) MeV., a difference of 0.07804 MeV.. This is a difference of 0.008% with respect to the experimental value. If one considers all 40 possible representations together with those arising from the quark rotations and the respective probability of occurrences of each representation, we obtain an averaged theoretical mass of 938.25105 MeV. that is within 0.0022% of the experimental results (see graph 1). The high degree of symmetry, found in all possible representation of the proton, could help explain the stability of that particle and its remarkable lifetime. Note that all singlets and doublets forming the proton commute with one another reinforcing our confidence in the earlier interpretation of commutativity of constituent gluons and charges as a possible measure of particle stability. Another readily noticeable characteristic of all representations for the proton is that its electric dipole moment would be 0, as it should be, based on current understanding of P and T invariance requirements and on all available experimental evidences.

The neutron is often considered in modern theories just as one particular energy state of "nucleons", a generic term describing an elementary particle capable of two energy states, the other one being the proton. The practicality of such a description is based on the great similarity between both particles in their strong interactions. Indeed, it appears that as far as the strong interactions are concerned, there are no practical differences between a neutron-neutron, a neutron-proton or a proton-proton interaction. The repulsive electromagnetic forces between the two protons of the Helium isotope could indeed explain the difference of energy in nuclei such as 3H and 3He. There are, of course, other similarities such as their almost identical masses, their spin and their intrinsic parity. However, there are major differences which are related to their different quarks content, to their measured electric charge and to the fact that, while the proton can be considered as absolutely stable, the neutron decays into a proton, an electron and an antineutrino. More specifically, the neutron is considered to be a bound state of three quarks of different colour. It is alleged to be composed of one u_r , one d_b and one d_g, where r, b and g are colour quantum numbers of different values. Since the neutron decays into a proton, the two representations must be very similar except for the quark content. Even then, it is likely that one of the "down" quark of the neutron decays into an "up" quark releasing an electron and a neutrino, in accordance with the type of decay processes identified in earlier articles.

Figure 38: A possible configuration for the neutron and a modified configuration that match the experimental mass

Still following the earlier construction principles, we would have 16 possible representations satisfying all exclusion rules and quantization requirements. In this case the calculated theoretical value for the mass, allowing for various quark rotations, would be 11032 A.U.M. or 939.5569383 MeV. (see graph 2), a difference of 0.000925% with the experimental value. As we did for the alternative charged pion, we can replace the bottom heavy gluon c ⁺~ by W^- in one of the representation, as shown at figure 38. As a result we get a particle with the same neutral radiation field. Its mass would be also 11032 A.U.M. as for the average of all possibilities. Using this Aaveraged" representation, the neutron decay process can be explained in terms of the following sequence of reactions. The neutron first transforms its W^- into a c ⁺~ by virtue of the following algebraic identities: W^-= Z + c ⁺~. A free neutral weak vector boson is produced accompanied by a c ⁺~. In a second phase, the two "down" quarks are respectively rotated by +p /2 and -p /2. This transformation could be represented by two mutually cancelling operators, each being of neutral charge and which effectively cancel each other. Simultaneously, a proton is produced by the release of an electron, in accordance with the relation:

that was introduced in a previous article. The overall process is very similar to the experimentally observed reaction:

noting the appearance of an intermediate Z. Note that once the first stage of the reaction has occurred, there are virtually no limit to the speed at which the two next phases of the decay process could be realised. Note also that the intermediate particle produced in this process has the same quark content as the original neutron. Except for its mass of 11040 A.U.M., it is a state very close to that of the original configuration.

THE CONFINEMENT OF QUARKS

One might wonder why the same quark configuration but with weak Z type gluon forming the neutral cloud or without any gluon does not appear to be observed in nature. The reason is simply that the effective photonic radiation Q_rad of such particles would be much greater than the net charge content of the constituent quarks and as such is not permitted in accordance with the same construction rule that was deduced for the fundamental particles. In addition, such particles would not have the properly quantified electric charge. The shielding effect that is provided by the electrically charged coloured gluon is essential for the existence of both neutron and proton. As a result, the quarks appear confined within these particles.

THE OTHER BARYONS

The other baryons are produced in a manner similar to what was done to get the meson resonances. The difference is that mesons are added to either a proton or a neutron and, again, various glueballs are used to glue the constituents together. This was clearly suggested by the structure illustrated previously. As first examples, consider the particles at figure 39.They could very well be representations of L ⁰, S ⁰ and X ⁰. It is obvious that the three particles offer a gradual increase in mass while preserving a similar symmetry and the neutrality of the color of the heavy gluons. According to this model, all three particles would decay eventually to a proton and a negatively charged pion. This is indeed verified experimentally as provided by the P.D.G. data:

L ⁰ ® p p ^- , G = 63.9 %;

S ⁰ ® L ⁰ g ® [p p ^- ] g , G = 100 %; and

X ⁰ ® L ⁰ g ® [p p ^- ] g , G = 1.06 x 10^-3 % .

While the frequency of occurrence of the last one is small, it shows nevertheless that our theory has some potential. You will also note that the theoretical mass values are very close to the experimental values, actually within 0.03%. It is also interesting to note that according to Q.C.D., L ⁰ and S ⁰ are equally composed of three quarks (U d S). Based on our earlier speculations about a possible representation of the Strange quark, it appears that the grouping of the three bottom quarks U, U~ and d could play that role. Without further justifications, I am presenting at figures 40 to 43 the mass spectra of the L ⁰, S ⁰, S ^-, S ⁺ as predicted by this theory. As you will note, the predicted mass value is always very close to the experimental value. For the Sigma, you will note that the averaged theoretical mass is very close to the experimentally measured mass.

As for the mesons, I was able to reproduce all known baryons and their resonant states. It was also evident that our theory was predicting exactly the energy gaps were no particles are found experimentally. This has been verified for all N and D resonant states. Also, there are numerous resonances for the L and S baryons. They range in mass from a low 1405 MeV. to approximately 3200 MeV.. Most of them have as primary decay modes one or more of the following: N+k ~, S +p , L +p , L +h , L +w , N+k ^*(892). In other words, they decay into a nucleon and two or more pions. Our construction technique would still apply, except that the number of possible representations increases significantly. While, I do not intend to study these in detail, I am providing partial result for the S ⁺(1385) at figure 44 with a typical representation at figure 45. Figure 46 through 59 are different examples of configurations for various baryons. It is possible to make all the Delta and Nucleonic resonances by a method similar to what was done for the mesons.

Figure 39: Three baryons formed of one proton and a negatively charged pion.

Figure 40: The mass spectrum for the neutral Sigma baryon.

Figure 41: The mass spectrum for the positively charged Sigma baryon.

Figure 42: The mass spectrum for the negatively charged Sigma baryon.

Figure 43: The mass spectrum for the neutral Lambda baryon.

Figure 44: The partial mass spectrum for the positively charged Sigma (1385).

Figure 45: A possible configuration for the Sigma (1385).

One question that always comes back to mind, like for the complex mesons, is whether or not the electric charge quantization rule should apply to the complex hadrons. This is not a trivial question. Some of the results obtained above seem to militate against a blind application of such a rule for complex particles. While the charge quantization rule must apply to the fundamental particles and the elementary mesons and baryons, taken individually, it does not necessarily apply to the excited states of the mesons and could therefore not apply to the complex baryons and complex mesons. The measured electric charge of these extremely short-lived particles, if such a measure was possible, might not necessarily give zero or an integer multiple of 10g . An experimental verification would be extremely difficult considering their extremely short lifetime. The global measured electric charge of the resulting particles, after the decay process is completed, would however reflect an exact charge quantization and conservation.

OTHER POSSIBILITIES

Looking at the sequences of particles for the baryons as well as for the mesons, it appears that some other intermediate and heavier possibilities for particles are possible. It would also appear that some of these intermediate possibilities are not reported in the present table of experimentally accepted or even probable particles. While, I have produced much more representations than those submitted in this document, I cannot claim that I have exhausted all possibilities. Nor can I claim that the technique for constructing representations of particles gives exclusively physically realisable particles that should be eventually observed. Only a systematic construction of all possibilities accompanied by a systematic experimental search through all energy level in substantially small energy increments would clarify the situation.

A definite improvement in the experimental value obtained for the mass of many resonant states would also be most welcome. As explained earlier, the distribution of some experimental values suggest that there might be more than one discrete mass value in a given energy domain. In contrast, the Particle Data Group tries to average out the differences and force, in a certain way, the experiment to match the often purely qualitative theoretical predictions of the standard quark model. Such a possibility appears to be supported by our earlier studies of the energy spectrum of some basic mesons and baryons. It should also be remembered that the physical environment in a super collider is far from benign, especially in terms of the intensity of the ambient electromagnetic field and the momenta of the particles involved. The resulting production of particles could be irremediably tilted in favour of some preferential, or more likely, modes or excited states.

If the theory exposed in this article has some value, it is likely that the present particle nomenclature and classification scheme would have to be revised. A better one appears to be on the basis of the number of "elementary" particles (nucleons, pions and electrons) forming any given particle. You will note that for most of the representations of baryons and mesons, I have paid very little attention to the spin orientation of the constituent quarks. The reason for such neglect is that, in most cases, the relative spin orientation of the constituent quarks plays a negligible role in the total mass value of most particles.

Consider the almost exponential increase in the number of possible particles as the number of constituent pions increases. I have often wondered if the apparent predictive power and accuracy of this theory was only fortuitous and was arising mainly from the relative large number of available parameters or excited states of the pion representations. However, its relative simplicity and the relative beauty and symmetry of the representations of the basic elementary particles: electrons, quarks, leptons, gluons, pions and nucleons, lead me to believe that, while some aspects of the theory still need some additional work, there must be some validity to it. Especially when looking at a representation such as the one proposed for the meson c , which would be composed, in one of its decay modes, of essentially 7 different pions and glueballs. Indeed, it would be the result of an almost demoniac coincidence if the interaction of over 800 different mathematical points, leading to a difference between the theoretical and experimental mass of only a fraction of a percent, was only due to chance and had no physical significance.

In this article, it was seen that the theory offers much more possibilities of hadrons than what is normally believed to exist based on the present full listing of the Particle Data Group and on the unique representation paradigm. QCD appears to be more restrictive on the number and type of different particles even if the model does not clearly account for the mass spectra of all the hadronic resonances observed experimentally. Based on QCD, various attempts are being made to predict the hadron mass spectra; one of those is known as the Quark Potential approach. It is based on the assumption that "integrating out the gluon fields in the QCD action, one can hopefully come to an effective Hamiltonian with reasonable potentials which describes well the physics of hadrons.

As explained in this last reference, while such models obtain reasonably good predictions for some part of the hadronic mass spectra, it does so at some costs. First, the quark mass values are generally model-dependent since nobody knows how to estimate exactly the non-perturbative confinement effect. In general, these non-relativistic quark models predict that the spin of the proton is due mainly to the quarks. However, recent experiments tend to indicate that the mutually interacting coloured gluons and the alleged cloud of quark-anti-quark pairs inside the proton play an important role in the value of the spin. In most of the models of quark dynamics, confinement is introduced phenomenologically rather than as a consequence of the underlying field theory: the confinement potentials of various forms imply that an infinite amount of energy is needed to separate two quarks and gluons. In other words, the desired effect is forced into the theory. As a result, free parameters or arbitrary coupling constants have to be introduced and iteratively adjusted to gradually improve the model reliability in reproducing part of the mass spectra.

My theoretical model does not assume any value for the mass of the quarks but specifically provide a value for it. It does not have free parameters or arbitrary coupling constants to account for the interaction of some ad hoc potential and the matter field describing the particles. It offers a potentially credible explanation for the importance of the gluon cloud in the evaluation of the spin of the proton or other hadrons. This is done at a cost of "apparently" allowing much more hadronic resonances representations and exited states than what is currently reported experimentally or believed to exist on the ground of currently recognised theoretical models. Is this a real problem and/or can this undesirable multiplicity of representations of particles be reduced by the application of super-selection rules based solely on the basic principles of the theory enunciated so far or on some credible physical observations? Is the apparent success of the theory due only to chance in that by providing a lot of possible particles with a rich mass spectrum it covers necessarily all presently observed particles? The answer lies partly in the provision of better experimental data. This concludes our study of the baryons, and with it, the study of the mass and representation of the experimentally observed elementary and complex particles.

Figure 46: The positively charged Delta baryon.

Figure 47: The negatively charged Delta baryon.

Figure 48: The neutral Delta baryon.

Figure 49: The double positively charged Delta.

Figure 50: A nucleonic resonance.

Figure 51: The neutral Ksi baryon.

Figure 52: The negative Ksi baryon.

Figure 53: The Ksi baryon.

Figure 54: The Omega baryon.

Figure 55: The Lambda (Charmed)

Figure 56: The Sigma (Charmed)

Figure 57: The Ksi (Charmed)

Figure 58: The Omega(Charmed)

Figure 59: The Lambda (Bottom).

In this last section, I will attempt to offer a way of conceptually connecting this theory with conventional Q.F.T. and with Gravitation. In the previous article we first defined an algebra, containing zero-divisors, with which it was possible to construct a six-dimensional space-time lattice. Singlets and doublets were defined, using the fundamental S matrices of the algebra, and were used to represents the most fundamental physical elements such as fractional electric charges, photons, neutrinos and other gluons. These fundamental matter constituents were then used to build composite representations of the known elementary particles: quarks, leptons, mesons and baryons. A charge and mass quantization scheme was proposed and used to discriminate between physical and unphysical representations. It was seen that the mass of these alleged representations did not only provide an extremely good theoretical value for the mass of all elementary and other known particles but also provided a simple justification for their various decay modes.

It was also observed that some of these representations were not unique. While in some cases, such multiple possible representations were not a serious problem, as for the many possible representations of the neutral pion, often this multiplicity was either perplexing or simply annoying. The multiple possible representations of the proton and neutron fall in that category. For both these particles, it was stated in a rather ad hoc manner that only one representation was valid and that the others were excited states. Such states were then used profusely in explaining the energy spectra of some complex baryons.

It is always possible that this theory is completely inadequate. But it would be rather troubling that such complex and highly symmetrical structures would depict accurately some of the most important features of the acclaimed standard model of quark theory and quantum Chromodynamics, including the proper spin, mass and electric charge and be simply the result of some monstrous coincidence. This last statement is also supported by the fact that the simple proton, as an example, is composed of no less than 93 different mathematical entities. Whenever assembled together in a rather systematic and highly symmetrical way, their mutual interaction gives a theoretical prediction for the mass of the proton that is within 0.08% of the experimental value. And a comparable or better precision is obtained for the evaluation of the theoretical mass of most of the known particles, even when such representations contain much more than 93 mathematical points. It was also shown in some simple special cases that the multiplicity of some representations provided a satisfactory explanation for the unexplained characteristics of some particles energy spectra, such as the rho (770) meson.

The theory might be completely false, but it appears to give a rather credible explanation for the decay modes of most particles, at least for those representations that were attempted and were not necessarily presented in the previous articles. Yet, one would object that the theory would predict the existence of much more particles or at least representations of particles than those presently reported in the literature. I can see more than one possible explanation for this.

The first one is that the laboratory environment in which most of these "artificial" particles are produced is not what one would call either a benign or a natural environment. The electromagnetic fields used to contain, focus and accelerate elementary particles in all high energy accelerators are of extremely high intensity and highly directional. They could, as a consequence, favour the creation of certain particles rather than others. Secondly, I think that a systematic search for particles at all energies, that is by small increments of say 1/6 the mass of an electron, which is the theoretical precision available with this theory, has yet to be conducted. Such a systematic search could provide some surprises, as it was the case in the discovery of the J/Y meson. It is also possible that many production events have not been properly identified or even properly recognised since the energy peaks often appears in the middle of a lot of "background" noise.

But assume for the sake of an argument that the theory desd, you will rememcribed in this paper is really a good model to represent the physical reality, at least as far as the photon perception of that reality is. Indeeber that the theory appears to be described with respect to an observer who is travelling at the speed of light. And this presumption is predicated by the respective mathematical representations of the vacuum nodes and the photons. Then, what are the consequences of such a representation of the most common elementary particles, such as electrons, protons, neutrons and photons.

A FIELD THEORY INTERPRETATION

The first thing to keep in mind when a particle is represented by the structures suggested in the previous article is that the structure is always embedded in the vacuum lattice. As a result, all points in the universe are either described by arrays of numbers representing the six-dimensional "empty" space-time nodes or the "non-empty" matter nodes. Formally speaking, the universe and all that it contains would be described by a discrete or lattice tensor (or matrix) field, defined over the integers. The only difference between the nodes describing the vacuum and the nodes describing an area where matter is present resides in the mathematical representation of these nodes. All points are fundamentally massless. The familiar concept of mass, or energy, is brought about simply by a mathematical operation involving the determinant and trace, two invariant quantities, which characterise the strength of the interaction between two adjacent nodes. And it is stipulated that the energy density is provided by the value of the determinant while the trace is assumed to represent a flux of energy. This interpretation was justified by the fact that a determinant could be geometrically interpreted as a volume, as is often the case in three dimensional vector algebra.

This tensor field representation of space-time and all matter that it contains generate a six-dimensional lattice, which can be represented in a three dimensional space. The physical reality of such a lattice superstructure is made credible considering that it appears that, presently, the only possible theoretical mass predictions for elementary particles containing quarks must be calculated on super-computers using numerical methods based on lattice gauge theory. The lattice, within the framework of Q.C.D., refers to a three dimensional scaffold, on which point-like quarks are at the nodes and the bonds between the nodes represent gluons. Using such a technique, researchers at the IBM Thomas J. Watson Research Centre have been able to come up with predictions for the mass of eight hadrons. Their calculations agree with recognised experimental values within 1% to 6%, which is considered very good. But the calculations took a year of computer time on a GF-11, a massively parallel computer capable of 11 gigaflops. It should be added that the researchers claim that the uncertainties are for the most part imputable to the statistical algorithm used in the calculations. The algorithm rely on a simplification called the valence approximation method: it does not take into account the spontaneous creation of quark-anti-quark pairs, an important and non-negligible feature of Q.C.D.. Rather than incorporating the process, this approximation assumes that such virtual pairs act mainly to reduce the strength of the colour field and, therefore, compensates for the decrease. Notwithstanding the objection to this approximation by some physicists, it should be noted that my theory consistently provides excellent theoretical mass predictions: invariably in the 0.005% range. For example I get a comparative difference of 0.008% for the proton and 0.0009 % for the neutron. And such a precision is obtained in less than five minutes of manual calculation and without any approximation.

Of course, this so-called tensor field does not necessarily have all the properties that are normally assigned to fields in quantum field theories. For one, the usual superposition principle does not appear to be valid in most cases. Indeed, in general it is impossible to superpose at the same location two nodes which have common matrix representations, by virtue of the exclusion principle discussed in the first article. However such impossibility could be only a reflection of the common sense thinking that one cannot superpose at the same point in space-time two particles of matter. Its eventual representation in a continuum by some wave function that would be solution of some dynamical field equation is also not certain at this point. One could speculate that the conventional wave representation of matter is an approximation of the discrete physical reality described by the lattice and vice-versa.

The tensor field described above appears to provide an exact value for the mass of particles. In fact it is a mass-energy distribution since it has specific discrete values over a given space-time span corresponding to the "size" of the particle. Such an energy distribution in space-time could be approximated for each linear distribution of singlets and doublets across the particle by some continuous mass-energy-density function M(x,t). This continuous mass-energy-density function could clearly be formed using the technique of Fourier analysis. The Dirichlet theorem assures us that such an expansion can be done. Another possibility would be to say that the distribution of matter on the lattice between two nodes described respectively by the matrices A and B is given by some function F (x,t) such that:

where the integral would be performed over the distance "h" between the two nodes A and B.

Of course, one would likely impose the boundary conditions M(x,t)=0 or F(x,t)=0 at the nodes and some continuity requirements on the same functions in terms of the respective partial derivatives. One could then impose the requirements that these energy distribution functions are themselves the result of some operator acting on some wave-like functions obeying some standard but appropriately modified wave equations normally found in quantum mechanics and quantum field theories, such as the Dirac, Klein-Gordon, Laplace or Schrödinger equations.

For example, one could always consider that the total energy value between two nodes A and B, as provided by the determinant and trace of the matrix product A*B, is represented by the energy E = hn of some standing wave solution to a dynamical equation like the Schrödinger equation. In fact, because of the boundary conditions and the fact that a particle in our preferential frame of reference is essentially trapped by the surrounding vacuum, one could essentially solve the equation for the case of a particle in a scalar potential which does not change with time. The difficulty is then to translate the respective significance of the time and space variables found in the Schrödinger equation with the concept of space and time as presented in our theory. Similarly, the exact significance in our theory of some Q.M. standard operators such as the energy and momentum operators

is not at all certain. Similarly, while in Q.M. the above operators are essentially "local" operators, our theory does not appear to enjoy necessarily the same characteristic. Actually, the concept of "locality" as it is usually employed in Q.M. does not translate easily in ours. The difficulty in translating one theory into the other arises partly from the fact that in Q.F.T., particles such as electrons are essentially considered point like. Their description is made in terms of a single position operator and a single momentum operator for the whole particle with the resulting Heisenberg uncertainty on both. Similarly, in Q.F.T. , one speaks of a single time co-ordinate t, while we have to deal with three time co-ordinates t_i at every nodes forming a particle (7 for electrons and fundamental quarks).

A FORMAL CONNECTION

Notwithstanding the above, conceptually, it appears simpler to keep our matrix field representation and to translate for the dynamics. Indeed, our theory is essentially static. However to change frame of references or go to our normal observer frame of reference S', one would apply a similarity transformation to each of the nodes

Note that formally, we could demand that slightly different U be applied to the different nodes of the particles under observation, since the nodes occupy a priori different space-time locations. However, such a requirement would not provide for large differences given the extremely small size of the particles considered with respect to the observer. A "global" transformation appears adequate for our limited purpose.

Since the determinant and trace are both invariant under such transformations, the rest mass of the particle will not change. In fact, this is only true if U is a global transformation, i.e. it is the same for all nodes within a given particle. Indeed, if different similarity transformations were required for different nodes, we would have:

In fact, the last two expressions would not make sense since U_A (respectively U_B) would only be applicable to A (respectively B). On the contrary, if U is global, we have:

confirming our statement that the rest energy is invariant under this approximation.

The only similarity transformations that one is normally interested in physics are those corresponding to Lorentz transformations. These usually correspond to rotations in the classical four-dimensional space-time but can be generalised to rotations in systems with any number of co-ordinates. In our case we will consider unitary transformations over R⁶ i.e. orthogonal transformations:

Such transformations preserve distances in R⁶. By virtue of some well-known theorems in algebra, we can always set

where H is a skew-symmetric operator. We also note that H can always be written as

Therefore

Remembering that s ₂ was nothing else than the matrix representation for the complex number "i", we can draw an analogy with the usual polar form representation of unitary operator

where D is a self-adjoint operator. If I insist that S ₂ commute with H, then S ₂H is also a symmetric operator. Since H is real, H is also Self-adjoint. Since H carries all the required information about the behaviour of the internal component of our particle under a change of co-ordinate system, it is a dynamical operator. Note that since S ₂ is also our electric charge operator and it commutes with H, this would make S ₂ a constant of the relative motion of the particle or of the dynamical operator H. The electric charge would then be conserved under the transformation induced by H.

The dynamical self-adjoint or symmetric operator H above should contain all the necessary information about the motion of the particle, it should be a function of the speed at which the particle is moving with respect to the observer. For comparison purposes, consider the familiar relationship between the Heisenberg (A_H) and Schrödinger (A_S) representations of operators in Quantum Mechanics:

where U(t,t₀) is some unitary operator. In particular, in the special case where A_S is independent of time t, we would have, as demonstrated in any basic Q.M. manual,

where H_S is the Hamiltonian for the system. In the more general case, the unitary operator U (t,t₀) would be solution to the following equation:

We established earlier that to go from the photon frame of reference (S_p) to an arbitrary frame of reference (S), we could use in a first approximation the relation:

Written in polar form, we would have:

[AB] = Exp ^(-S₂H) [AB]_p Exp (S₂H),

Compare this relation with the results of the last paragraph. It is very tempting to associate our self-adjoint operator H to some appropriately modified form of the Hamiltonian of the Q.M. system that is also self-adjoint and a function of speed alone or momentum in the case of a free particle.

It was not my intention in the above paragraphs to present a formal derivation of the relationship between the formalism of this theory and the formalism of modern Quantum Field theories. I merely wanted to point out that a connection appeared possible and that there were some striking similarities. A lot of work would be required to formalise such a connection and this would bring me far away from my immediate concerns.

THE TENSOR FIELD AND GRAVITATION

In general relativity, it is stipulated that gravitation is the result of a local deformation of space-time caused by the presence of matter. The tensor field represented by the collection of the space-time and matter nodes on a lattice provide us with a mechanism to explain such a local deformation of space-time and how gravitation could then be generated by the presence of matter. The difficulty is then to explain how a given local deformation of space-time generated by the presence of some massive particle affect or interact with another distant deformation of space-time generated by another massive particle. Actually, one of the two particles does not have to be massive since it has been experimentally verified that even photons are deflected in their trajectory by the gravitational field produced by a sufficiently large mass. This is explained by the principle of equivalence of mass and energy: since the massless photons have energy due to their large momentum. Indeed, for a rapidly moving object, the force of gravity also depends on the object's momentum in addition to the object mass.

Considering this difficulty of accounting for the propagation at a distance of the information concerning the presence in the field of a certain massive particle, presence which appears to influence the dynamics of all other material bodies present in the same field, three possible explanations come to mind.

The first possible explanation concerns a process similar to the transmission of the photonic field that appears to provide an explanation for the electromagnetic interaction at a distance between two electrically charged particles. Indeed, consider the colored gluons c ± , the main constituent of particles such as protons and neutrons. All the gluons in immediate contact with the vacuum nodes Z^v , interact with them and produce a virtual gluon in accordance with the usual relation:

X⁺ * Z^v = [ Z^v + g ].

While the g ’s contribute to the photonic radiation field and therefore to the measured electric charge of the particle, Z^v remains as a virtual gluon and does not appear to contribute to any measurable physical quantity. Furthermore, since Z^v has the same mathematical representation as the normal vacuum nodes, it appears that such a virtual gluon would have extreme difficulties to travel within the vacuum lattice. However, one could always consider that such a probability of transmission still remains but at a much reduced speed than what is experienced in the case of the g 's. And the inverse squared law would then be introduced in the same manner as it has been for the electromagnetic force. One could then argue that this explains the relative extreme weakness of the gravitational force compared to the electromagnetic force. In other words, the vacuum lattice would act as a medium of very high relative permittivity as far as the propagation of this virtual gravitational gluon is concerned. The only problem with such an explanation is that the light mesons, p ⁰ and p ± , and the electrons and muons would not cause any gravitational force since their gluonic clouds were assumed to be composed of Z^v" and

Z^v" * Z^v = [ 0 ] , and

S ₂ * Z^v = g = [ 0 ].

Another difficulty of such an explanation would be that the gravitational force produced by a given particle would only depend on the surface gluons and not on the internal ones. Therefore a large proportion of what constitute the mass of that particle would not be contributing to the total gravitational field.

THE LATTICE SCALING FUNCTION

A second possibility has to do with the uniformity in the shape of the lattice. It was assumed from the start, based on the principle of indivisibility of the fundamental charges, the S 's, that the distance between each nodes of the lattice was always the same, no matter what singlet or doublet was at a node. Assume instead that this distance is effectively always one A.U. but that the relative length is actually scaled down or up in accordance with some function of the energy density and flux (or total energy) provided by the interaction of two adjacent nodes. This would result in an actual geometrical deformation of the lattice in addition to the "formal" deformation generated by the different values of the nodes whenever and wherever matter is present. The result would be an actual shrinking of the lattice in areas where heavy mass is present with a resulting pull on the surrounding lattice, as illustrated at figure 60. This pull would necessarily be felt at infinity in all directions but its effect would naturally tend to be negligible or zero at infinity, since there would be a natural tendency for the "non-deformed" vacuum nodes to resist such a pull and the resulting geometrical deformation.

Some general considerations that could apply to a potential expression determining the scaling of the length of a string between two lattice nodes immediately comes to mind:

a. if the total energy of a bound is zero, we have no matter or flux present and the lattice string should be of a nominal basic length that one could arbitrarily fix to be 1;

b. if the energy density of a bound, as provided by the determinant, is zero and the flux (the trace) is 12, the lattice should be of the same length as the spacing between the "empty" space-time vacuum nodes;

c. if the energy density (determinant) is different than 0 and 12, the value for the vacuum, the lattice should be of a dimension smaller than that of the vacuum, even when negative.

Consideration C. above arises from the fact that the electron mass is -6 A.U.M. which would tend to indicate that it would provide a gravitational push rather than a pull contrary to all experimental observations. An equation that describes a variation of the distance between the nodes as a function of the total energy should works. It would provide a shrinking of the lattice not only for the energy values greater than that of the vacuum but also for the negative and smaller energy values, such as those appearing between the various nodes of an electron. The scaling function should have a maximum at the vacuum value. Based on these considerations an apparently acceptable expression for such a scaling function would be:

where E_t = Det + Tr, the total energy between two nodes and k is, a priori, some constant.

Note that this is essentially the sum of three normal distributions centred at E_t = 0, 12 and -12. The reason for the choice of these three values is that, technically, our theory contains three possible energy levels for three different definitions of the vacuum as given by:

(S ₀+S ₁)*(S ₀+S ₁) = (-S ₀-S ₁)*(-S ₀-S ₁) = 2(S ₀+S ₁) ® [0|12],

(S ₀+S ₁)*(S ₀-S ₁) = (S ₀+S ₁)*(-S ₀+S ₁) = 0 ® [0|0],

(S ₀+S ₁)*(-S ₀-S ₁) = -2(S ₀+S ₁) ® [0|-12].

While we selected the first one for our choice of the vacuum, it remains that the two other values might play a role in the scaling of the geometry of space-time. The combination of multiple distributions is also the only alternative to permit negative and smaller-than-12 energy bounds to be shorter than the vacuum bounds and therefore ensure that all particles will be subjected or will generate gravity pull. It also ensure that, say, an energy value of -6 does not cause more or less scaling than one of +6. This is illustrated at figure 61 on a graph showing the behaviour of our distribution F with respect to the total energy. Note that the graph does not go to zero between the three peaks, its appearance is the result of the scale used to plot the graph. It is also interesting to note that the mass of the electron (-6) and the quarks (+6) are respectively at the two local minima provided by that distribution, as such they represent fundamental particle size. Note also that this concept of local extrema for the energy distribution of the bound length has its equivalence in the unified theory of electro-dynamics and weak interactions where it is associated with the symmetry breaking mechanism.

SOME FUNDAMENTAL CONSTANTS

The idea is now to determine the value of the constant k in the above expression for the length of each bound. Note that the units for F are cm/sec, since the length of each bound in the vacuum lattice is given in terms of a unit of space length by unit of time length. We know that the mass of an electron is -6 AUM while the mass of a proton is 11016 AUM; the ratio of both masses is 1836. Looking at the respective representations of both particles, one can write the following relation:

where the numbers in brackets and the coefficients correspond respectively to the total energy of each bound of the particles and the number of time each value appears.

Solving this equation for k, using appropriate computerised mathematical tools, one gets

Now substituting this value for k in F and converting the various units into the CGS system, one notice that to get the proper cm/sec units for F , one could set the following value for k:

where G = 6.6732 x 10^-8 dyne-cm²/gm²,the gravitational constant,

h = 6.626196 x 10^-27 erg-sec, Plank's constant,

c = 2.9979250 x 10¹⁰ cm/sec, the speed of light,

L = a fundamental unit of length in cm/sec.

This last expression for k allows one to determine the value of the fundamental unit of length l :

It should be noted that the expression in F for the energy squared need also be converted in gm²cm⁴/sec⁴. This value for the fundamental length is of the same order of magnitude than the so-called Plank's length

And it is generally believed that at such lengths, the topology of space-time may fluctuate. This is in essence what has just been demonstrated. It is remarkable that all most fundamental constants of nature would be so connected by a simple expression like F . Especially since it was originally intended just as a tool to illustrate the possibility that gravitational forces are a result of the local scaling down of the fundamental lattice wherever something else than the vacuum is present.

While the above definition of F might appear arbitrary, it should be noted that it has a very similar form to the action used in most modern lattice gauge theory. Normally, in quantum field theory, a field is characterised by a set of vacuum expectation values of certain observable quantities:

The action S(j ) in the exponential is the integral over a Lagrangian density.

In the case of a self-interacting massless scalar field, as stated by C. Itzykson , it is the sum of a kinetic and potential term involving a coupling constant:

For a massive scalar field, one gets, after appropriate Fourier, transform j _n ® j (k).

The similarities between these relations and our scaling function are striking. Both are functions of a quadratic value for the energy density and both involve a summation (or integral) over all possible lattice "strings". The major difference lies in the fact that Lattice Gauge theories assume values for the mass of fundamental particles while our theory specifically provide values for these masses and an explanation of what constitute the mass of a particle.

Assuming the above value for L provides the proper order of magnitude for the lattice size, the true value for the radius of fundamental particles would be of the same order of magnitude. One might wonders then why the classical values for the same particles are reputed to be quite larger:

r_e = 2.81794092 x 10^-13 cm.

r_muon . r_e

r_baryon = order of 10^-13 cm.

r_Bohr = 0.529177249 x 10^-8 cm.

Notwithstanding these classical values, most modern theories and experiments assign a somewhat much smaller dimension to the leptons (<10^-16cm.). In fact many theories consider the electron to be essentially point like. Our lattice dimension L would tend to justify such treatment. It appears impossible to build instruments that could resolve the fundamental size of the lattice since all instruments would necessarily be larger. The classical measurements above could be the simple result of the fact that one actually measures the extent of the deformation in the space-time geometry rather than the particle itself. This would indicate that the presence of matter noticeably affect the geometry of space-time at a distance from the centre of the particle which is of an order of magnitude of 10²⁰ times the true size of that particle.

THE FUNDAMENTAL PARTICLES

There are four different fundamental particles the mass of which has an absolute value of 6. Consider all the possible pairs of such particle, i.e. (e^-, u), (e^-, d₁), (e^-, d₂), (u, d₁), (u, d₂) and (d₁, d₂), where d₁ and d₂ indicate the two possible configurations of the down quark. In a similar fashion than what was done for the proton and electron, we can write the following six equations in terms of the scaling function F :

4F (-5) + 2F (7) - 6F (1) = 0,

2F (-5) + 0F (7) - 2F (1) = 0,

F (-5) - F (7) + 0F (1) = 0,

2F (-5) + 2F (7) - 4F (1) = 0,

-F (-5) - F (7) + 2F (1) = 0, and

-3F (-5) - 3F (7) + 6F (1) = 0.

The amazing thing is that one cannot find three linearly independent equations. This implies that the only possible solutions for the constant k of the scaling function F for all these equations to be satisfied are the trivial solutions k=0 and k ® ¥ . Therefore

This would be true even if we were to prescribe that k was also a function of the energy. In fact, this would be true no matter what form the scaling function would take.

Is this an indication that our approach does not make sense or is it a manifestation of some strange fundamental property of the fundamental particles? While the prescription L ® ¥ does not appear to make sense, L ® 0 could be an indication that indeed the fundamental particles should be considered as really point like or as virtual singularity. You will also recall that our scaling function had two local minima at -6 and +6. Is this an indication that any particle for which all bounds energy values lies between the peaks would necessary behave as if it was really a virtual dipole of minimal length and of total energy ± 6. The scaling function for the fundamental particles would then apply only once and we would have for the four fundamental particles a single value of F (6). In that case all six equations given above would make sense since they would all be written as

On that basis, the relation for the pair proton-electron can be re-evaluated using:

yielding a new value for l of

that is not much different than the previous one.

From figure 60, it should be obvious that the gravitational field so described would not be isotropic at least within the immediate neighbourhood of a single elementary particle such as a proton. However, it is possible that at a reasonable radial distance r from the particle the lattice deformation could be considered essentially isotropic. It is believed that such a slight anisotropy would not be measurable with present technology given the weakness of the gravitational force and should essentially cancel out when a large amount of matter is being considered due to the randomness in the orientation of the individual protons and neutrons.

Nothing was said about the inverse squared law dependence of gravitational forces. It was not our intention or our pretence in this last article to completely cover or answer all questions relating to the introduction of gravitation in this theory. Our intention was merely to point out some possibilities and to show that the formalism allows the possibility of unifying the concept of gravitation with those commonly encountered in modern theories of particle physics. You will note again that the word unification must be used with caution in this theory, since it has been clearly demonstrated that the mechanism of the electromagnetic and gravitational forces appears fundamentally to be quite different. In particular, as was discussed in an earlier article, this fundamental difference might have something to do with the fact that the two fundamental entities responsible for the structure of space time, S ₀ and S ₁ , form together a subgroup of the total group of eight (or four) fundamental charges.

In all the above exercise of building fundamental, elementary and complex particles, you will have noticed that I never made used of the fundamental charges S ₁ and S ₃. In particular, I did not used the heavy gluon z =s to build heavy gluonic clouds. The reasons are quite simple. I stated that commutativity was an essential characteristic of the constituents of any stable gluonic cloud. The z =s do not commute between each other and consequently cannot form stable matter particles such as protons and neutrons. Therefore it would be impossible to have stars or other similar large cosmic objects formed partly or exclusively of that kind of matter. However, these Afree" gluons could possibly accumulate and condense in large unstable chargeless clouds in space or near the known galaxies and contribute substantially to the overall mass of the universe. Such an eventuality could help explain the observed speed of rotation of the galaxies that is not consistent with their observable mass.

In this article we first defined an algebra, containing zero-divisors, with which it was possible to construct a six-dimensional space-time lattice. Singlets and doublets were defined, using the fundamental S matrices of the algebra, and were used to represents the most fundamental physical elements such as fractional electric charges, photons, neutrinos and other gluons. These fundamental matter constituents were then used to build composite representations of the known elementary particles: quarks, leptons, mesons and baryons. A charge and mass quantization scheme was proposed and used to discriminate between physical and unphysical representations. It was seen that the mass of these alleged representations did not only provide an extremely good theoretical value for the mass of all elementary and other known particles but also provided a simple justification for their various decay modes.

But the most fundamental consequence of the theory presented here is that matter would simply be the result of purely mathematical interactions between the nodes of the lattice that are different from those representing the vacuum. And such a representation automatically accounts for the natural forces in nature. Gravitation would be generated by the local deformation in the cubic geometry of the space-time lattice, whenever matter is present. The other forces would be intertwined in the formalism of the theory and are essentially indistinguishable, except for the electromagnetic force which make itself known by interacting at a distance via the intermediary of the radiative photonic field generated by the presence of electric and magnetic charges. The respective ranges of the forces are irremediably determined by the theory. It is infinite for the gravitational and electromagnetic forces and short range (not further than the adjacent nodes) for the so-called strong and weak forces. This last one cannot really be distinguished in the theory by other means than the relative strength of the interactions and the type of gluons (light or heavy) mediating or generating them.

There are of course loose ends in this theory. Many areas need more rigour in the mathematical formulation. For example, the introduction of the charge quantization rule could possibly be improved upon. Some other areas need to be explored in greater details and depth. Considering the undeniable successes of Q.E.D and Q.C.D. in calculating some physical quantities such as the magnetic moment of the electron with an unprecedented accuracy in experimental and theoretical physics, there is a need to better connect this theory with these great achievements of modern day’s physics. However, this theory is the first one that can claim to offer a reasonable explanation for the fundamental nature of mass, charge and spin, quantities that are normally inserted quite arbitrarily in conventional quantum field theories. It is also the first one which can claim to offer an easy and accurate theoretical prediction of the mass of most known elementary particles, even if there appears to be more different kinds of particles or representations of particles than what is currently reported or assumed. It is also possible that some still unknown exclusion principle will permit to exclude some unphysical particle representations. Could a merge of classical SU(n) symmetry rules and the theory presented in this paper accomplish such a feat? Assuming, of course, that this would be desirable.

It is of course tempting to regard this theory simply as a curiosity which happens to offer coincidentally some striking predictions on the mass and other fundamental properties of most known particles. Some might even regard this theory as a product of "numerology". I have evidently not answered all possible objections to the premises on which the theory is based nor have I been able, in this first compressed article, to offer a possible answer to all modern days physics questions. However, it is remarkable that such a simple formalism and two rather simple axioms allow one to progress so far in the explanation of the origin and potentially "true" nature of mass and charge. These two basic concepts are not explained in current theories, including general relativity. The best that is apparently achieved in attempting to do so is to introduce another field, the Higgs field, and hope that such a field would give all known particles their mass. But what generates the Higgs field itself? Answers would probably be provided at the cost of introducing more arbitrary parameters and even more exotic fields. In that direction, super-string formalism appears to offer some hope but at the cost of multiplying by two the total number of fundamental particles and gluons. In my theory, there is only one parameter. It is the fundamental size of the lattice, which appears to be related directly to the three most fundamental constants of physics h, c and G. In fact one could consider that the experimental determination of l , the fundamental length, would irremediably fix those three fundamental constants and the mass scale of all known particles.

It is true that the relative success of this lattice theory entails that quarks and electrons are not as fundamental as one would have expected or hoped for. Also, they would be all in the same class or family since they are formed of fundamental singlets in their non-excited states. This last result could be more attractive than one might think. After all, the latest development in super-string theory, allegedly the only serious candidate for leading us toward a final theory of everything, is believed to be consistent only when a concept of duality is introduced. That concept seems to infer that quarks are indeed formed of more basic building blocks or solitons that could them-selves be formed of quarks. Furthermore, this non-singular nature of the quarks and electrons, as represented in our theory, appears to explain some non-chiral behaviour of the weak forces.

It would certainly be exciting to investigate the consequences of the suggested models of protons, neutrons and electrons on the representations of naturally occurring matter such as atoms and molecules. In particular, a crucial test would be the ability of the theory to explain the matter formation processes inside stars and how current unresolved problems, such as the solar neutrinos problem, could possibly be explained. Preliminary investigations appear to be promising.

Lately, some authors have claimed that we were near the end of theoretical physics. I, for one, believe that there are still a lot to do in physics and we are not close to seeing the end of it all. For a starter, where does the original mathematical singularity, the primordial ZERO, come from? Or was it simply conceived?