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