Unifying Fundamental Atoms?

II- The Effect of the Structure

Author: Samir Abuzaid

Affiliation: The Egyptian Philosophical Society

1. Introduction

We continue exploring the consequences of postulating that every natural

body (classical and quantum) is composed in the final analysis from a definite number

of identical indivisible 'fundamental atoms', that are called here 'the final particles'.

Based on our detailed study, three consequences follow consistently in full agreement

with current physics. First, all natural bodies, classical and quantum, become

governed by one fundamental equation. Second, a unified background space-time for

both realms emerges. Third the four basic forces of nature reduce to one fundamental

force field.

In a previous paper we presented the theory of the final particles that describes

the basic properties of such fundamental atoms through the model of a random walk

on a mesh.

Through such a model, we presented analytical and experimental proof for the

first two consequences mentioned above. Namely, we presented the proof that such a

postulate leads to a unified fundamental equation for both of the classical and the

quantum realms, which is Schrödinger equation. In this respect the classical realm

appears in the case that the number of the final particles that constitute the body

approaches infinity. At such a state Schrödinger equation reduces to Laplace equation

that governs the forces of gravity of classical bodies.

On the other hand, we presented analytical and experimental proof that the

postulate of the final particles leads to a 'dynamic' view to the distribution of matter of

the bodies in the universe that can be described geometrically. Such a dynamic

geometric description of the distribution of matter in the universe has been proved to

be fully equivalent to the dynamic geometric description of space-time, for which the

general relativity is the accepted theory. Proof has been established through

calculation of periastron precession of binary pulsar PSR 1913+16, bending of light

2

rays grazing the surface of the sun, and Mercury's perihelion precession, in addition to

presenting the results of comparison to the Post-Newtonian Approximation.

In this paper we continue our work and present the analytical and experimental

proof for the third consequence, which is that the postulate of the final particles leads

to unification of the four basic forces. This is realized through reducing these four

force fields to one fundamental force field which is attraction between the final

particles. Analytical proof is to be presented by introducing the structure of the

'elementary' bosons that emerges from the interactions of the final particles and

deduce its basic properties.

In other words, we prove that this postulate leads to an evolutionary

perspective for the 'elementary' particles of the SM. It simply says that the

'elementary' particles emerge through an evolutionary process as a probabilistic

consequence of the postulate of the final particles. This is proved through formulating

the 'probabilistic' evolutionary structure of the 'elementary' particles of the SM, and

calculating the 'probabilistic' force fields that arise between these 'structured'

'elementary' particles. Probabilistic analysis shows that this view leads to the

appearance of four composite 'bosons': one structureless and three structured. The

structureless particle produces the scalar field of gravity. The other three structured

particles A, B and C produce three vector fields: weak, electromagnetic and strong.

As will be shown in the following, calculations of the relative values of these

forces, through the model of the final particles in low velocities, show that the

weakest force is by of the strong force. The strong force is only

times the electromagnetic force. Finally, the force of Gravity is only

by of the strong force. Analysis of the effects of high velocity proves that these

relative values will converge as the velocity approaches the speed of light.

We show that these results are generally in agreement with the findings of the

SM, with one deviation however, which is the status of the and particles

(responsible for the weak interaction in the SM). Consistency with the SM is restored

through proving that the and particles are composite particles and that the weak

force can be reduced to its constituents, which are the newly postulated A particles.

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1.1 An underlying level of existence

In view of the current efforts for a unified theory of gravity and QM [4], the

view presented here essentially describes an underlying level that aims at unifying

current physics, a position that gains increasing support in current physics [10.

Ch.15]. Current laws of physics remains the same, however, these laws become

rooted and unified on the new underlying level. Hence, both of Newton's law of

universal gravitation that describes interaction of classical bodies and Schrödinger's

equation that describes quantum states become unified when we interpret the mass

in both formulations as composed of a definite number of identical final particles

that extend probabilistically all over the space, representing the underlying level.

Hence, both equations are unified at a deeper level.

Similarly, through such a definition of mass both of the dynamic description

of matter distribution and dynamic description of space-time become unified and

interchangeable. For, if space-time of the universe is distorted in a specific form,

distribution of matter will inevitably take the same form. Hence, if we can define

distortion of distribution of matter through interaction of the final particles (on the

underlying level) then distortion of space-time is included implicitly. Hence, the

assumption of the existence of the final particles unifies background space-time for

both of the classical and the quantum realms on a deeper level.

Finally, proving that the 'elementary' bosons of the SM possess a structure

makes reducing the four basic forces to the forces between the final particles, that

represent the underlying level, a straight forward procedure. For, if we can describe

the structure of such 'elementary' bosons it will be possible in principle to define the

effects of such a structure on the resulting forces between these bosons. Hence, also

the four basic forces are unified at a deeper level of existence.

This means that in the cases where we deal with each of the classical and

quantum realms separately, we can use current laws of GR and QM (and QFT),

separately. However, in the cases where we need to take both of the effects of the

classical and the quantum realms into consideration together, then we will have to

deal with the laws of the unifying underlying level.

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1.2. How we present this work

Our plan is as follows. In sec.2 we review the basic concepts and results

introduced in part1 of this study in order to keep the analysis and results presented in

this paper in context. Sec.3 will be devoted to introducing the analysis of the

underlying structure of the 'elementary' bosons and linking such analysis to the basics

of the SM. In sec.4 based on the analysis of these structures we introduce the

probabilistic method through which we calculate the relative values of the forces

between the 'elementary' bosons. As a proof of such a structure, the results of

calculations are compared to the experimental findings of the SM. In sec.5 we show

how we can make use of the results of this study to formulate the unified relativistic

equation for gravity and electromagnetism as an equivalent formulation to the positive

solution of Dirac equation. Finally, in sec.6 we review our results.

2. The Unified Equation and Unified Space-time1

A fundamental question in physics is what are, if any, the ultimate constituents

of the material world? A possible answer is that all natural bodies are composed of

'identical indivisible atoms'. In general, current physics doesn't eliminate such a

possibility[11, p.13]. Form anther side we know since the advent of the theory of

Quantum Mechanics (QM) that subatomic particles are described by a probabilistic

wavefunction that extends probabilistically all over the space. Hence, if we accept that

elementary subatomic particles are composed of 'fundamental atoms', it becomes

reasonable to postulate that such ' atoms' are endowed with perpetual random motion

all over the space, formulating the wavefunction of the composite particle.

2.1. The postulate of the final particles and definition of mass

In this sense, the total mass of the body becomes distributed probabilistically

all over the space, which leads to a new definition of mass.

1 This section represents a review of part1 of this study.

5

Quantum Leaps bet.

surfaces of equal

probability

2.1.1. Definition of mass

By definition, every natural body (classical or quantum) is composed of a

definite number of identical final particles that extend probabilistically all over the

space. If the mass of each final particle is , then the mass of any specific

body/particle is given by multiplying the two values, hence mass is given by,

This definition of mass is composed of two parts. The first (i.e. the number )

reflects the probabilistic distribution of the total number of the final particles of the

body in space. The second (i.e. ) reflects interaction of the final particles with the

matter of the universe. With respect to the first part, each final particle performs

perpetual random motion all over the space. This 'random motion' is modeled into

random walk on a mesh that starts at the center of the body and extends all over the

space, where it is modeled as reflection at imaginary boundary of the Universe (fig.1).

Fig. 1. Random motion of the final particles of the body are represented by a

random walk process that starts at the center of the body and is reflected at the

boundaries of the universe generating the surfaces of equal probability density

mediated by Quantum leaps. This geometric picture of the mass of a free non-

relativistic body/particle is represented mathematically through Schrödinger

equation.

Boundaries of the Universe

(real or imaginary)

Surfaces of equal

probability density

The body

d

Random motion of

final particles

6

Hence, the average probabilistic number of the final particles at any specific

point of space can be calculated through the laws of steady-state random walk on a

mesh. Therefore, from fig.(1) and the laws of random walk, the probability of the

number of the final particles at any point with distance from the center of mass is

[9].

Accordingly, mass of the body is defined probabilistically. At the center of

mass of the body/particle the probability of finding 'all' the final particles of the

body/particle (i.e. its full mass) is one, hence, mass at this point is defined through eq.

(1) above. However, at any point in space we have to multiply eq.(1) by the

probability of finding the particle at such a point in order to define the effect of the

mass of the body at this point. Hence, we may differentiate between the mass of the

body/particle at its center, which we will refer to simply as mass , and its

probabilistic mass at any specific point in space, which we will refer to here as the

'apparent' mass which is given by , in the case of a free body.

The part of the definition of the mass of the body/particle in eq. (1) that is

referred to as , reflects the probabilistic interaction of the final particles of the

body/particle with the other final particles of the universe. The other particles of the

universe can be divided into two kinds: other bodies (that are also composed of final

particles) and free spread final particles.

Interaction of the final particles of the body under consideration with the final

particles of another body/particle is termed in current physics as 'Gravity', whereas

interaction of the final particles of the body/particle with the final particles of the rest

of the mass of the universe (including free final particles) is called 'Inertia'. However,

both phenomena are one and the same phenomenon which expresses interaction of the

different final particles through complete (i.e. non-biased) random motion.

Since the forces that arise from interaction of the final particles of the body

(either with another body or with the rest of the mass of the Universe) depend only on

the number of interactions of the final particles, then by definition the force of gravity

(per unit mass) is equal to the force of inertia (per unit mass). From another side, the

definition, in current physics, of the effects of inertial mass of the body is resistance to

change of its state of motion (according to a factor that is to be defined

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experimentally). Since in our definition inertial mass is the result of the 'total'

interactions of the final particles of the body, then we can state in general that the

effects of the mass of one final particle is also resistance to change its state of

motion.

Therefore, by definition, gravitational mass is given by , inertial

mass is given by and, .

2.2. The unified fundamental equation

Therefore, the theory of the 'final particles' leads in a natural way to the view

that if we assume that the mass in Schrödinger equation is equal to , where is

the mass of the final particle (taken to be unit mass) and is the number of the final

particles of the body/particle, then Schrödinger equation becomes the unified equation

for both of the classical and the quantum realms. Hence, applying the transformation

to Schrödinger equation we get,

And the solution is de Broglie representation of the wave function,

is the amplitude of the wave. and ; is the

wavelength and is its period that are given by,

In the case of classical bodies, Schrödinger equation reduces to Newton's law

of Gravity. For classical bodies the number approaches infinity hence the

characteristic values and approach zero, hence eq. reduces to,

for free particles, which is Laplace equation that governs the phenomenon of gravity.

This result can also be understood visually from fig. (1) above. For in the case

of classical bodies the quantum gap vanishes and the 'spherical' shapes of the surfaces

of equal probability become continuous. In such a case the 'apparent mass' (i.e.

the probabilistic mass) at any specific point at distance from the center of mass is

calculated by dividing the mass by the surface area at such a point which is ,

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i.e. . Given that the final particles (that constitute the mass) are

endowed with the power of attraction, then mass represents a scalar field that is

governed, in the classical case by Laplace equation. Therefore, we can define the

interaction of such a field with any other mass to produce the forces of gravity by

defining the coupling constant experimentally.

2.3. Local dynamic space-time

Since the mass of the body extends probabilistically all over the space over

time, then any expression of change to space-time is intrinsically an expression of

change of the distribution of the mass over time (and vice versa). In other words,

description of the dynamics of space-time (keeping the dynamics of the distribution of

mass with no change) is equivalent to description of the dynamics of the distribution

of mass (keeping the dynamics of space-time with no change).2

This observation leads straightforwardly to the second consequence of the

theory of the final particles, which is the local dynamic space-time. In this conception

of space-time, under the effects of high velocity and high gravitation the distribution

of mass in space becomes dynamic. We show that this conception of space-time is

fully consistent with the dynamic space-time described by the current accepted theory

which is GR. In technical terms, dynamic geometry of space-time can be replaced by

dynamic geometry of the distribution of mass, keeping space-time 'pseudo-classical',

with exactly the same results.

Consider fig. (1) above, which represents the distribution of a free mass/field

that extends probabilistically all over the space. The mass/field is at rest, or moving

with slow velocity, relative to a slow moving reference frame. Since in principle, as

we now know, there is no absolute reference for distances and time intervals (or

alternatively there is no absolute space-time) we will define the term 'slow reference

frame' practically. Hence, the mass/field presented in fig. (1) is moving in reference

either to the Cosmic microwave background radiation, or to earth which is moving

2 This rule applies also to other force fields, namely, the electromagnetic, the weak, and the strong.

9

(with the Sun) with velocity of 370 km/s relative to cosmic microwave background

radiation [6, p. 231].

This conception of space-time is local because it is based on one local

reference frame, which is (or taken to be) a slow moving one. It is dynamic because

the dynamic nature of space-time is compensated through calculating the 'apparent'

increase of the masses of the system (as well as other force fields) under the effects of

high velocities and gravitation. Space-time is 'pseudo-classical' because distances and

time intervals of the system (as well as the characteristic values of the waves of fields)

that occupy specific points on the space-time background in reality vary dynamically

with the evolution of the dynamics of the system.

The basic procedure of the 'local dynamic space-time' theory is fairly simple.

First, for any two point masses we calculate the effects of high velocity and

gravitation on the distribution of the masses of the two bodies in 3-d space. Second,

we treat space and time as if it were classical in the Galilean sense. Third, we use the

calculated values of the masses in 3-d space to calculate the dynamic parameters (i.e.

momentum, forces and energy) and hence deduce the trajectory and levels of energy

as a function of time. For the case of several bodies/particles the effects on the masses

will be evaluated by superposition, which will then require sophisticated techniques.

2.3.1. Lorentz transformations

According to the theory of the final particles, and from the above definition of

mass, motion of the body under high velocity compared to the speed of light leads to

distortion of the mass distribution of the body. The same effect, i.e. distortion of the

distribution of the mass, results from high gravitational fields of other bodies. The

form of the distortion in the case of relativistic velocity along the direction of motion

is identical to the well known formula termed as Lorentz transformation. However, in

the theory of the final particles this formula is extended in two respects. The first is

the effect of the angle between the direction of motion (or the line that joints the

centers of the two masses under mutual gravitational effects) and the axis at which the

effect is calculated. The second is the extension of the formula of Lorentz

transformation to cover the case of high gravitational fields.

01

a- Velocity Lorentz transformation

Hence, if we denote the factor of transformation of mass under high velocity

relative to the speed of light as , then,

is the speed of the final particles, which can be approximated to the speed

of light in vacuum. is the velocity of the body with respect to a slow moving frame

of reference. is the angle between the direction of motion and the axis at which the

effect is calculated. Therefore, approximating to , the transformation factor

along the direction of motion, at which takes the form of the well known

Lorentz transformation factor given by,

b- Acceleration Lorentz transformation

In the case of the effects of gravity, distortion is due to an equivalent velocity

that is given by,

Here, is the square of the average equivalent instantaneous velocity of the

final particles due to high gravitational effects on the body. is the mass of the body

that exerts gravitational effect, and is the distance between the centers of the two

masses. The integration is to be carried out all over the space (i.e., form to ),

however, the values from to – and form to cancel each other out. The

minus sign indicates the direction of the gravitational field.

Accordingly, the apparent increase of mass due to gravity is given as follows.

For two bodies and , the average equivalent instantaneous velocities resulting

from the effect of each body on the other are given by,

00

Since we know from eq. (3b) above that the effects of velocity of the body is

given by velocity Lorentz transformation (along the direction of motion),

Substituting with velocities and

, the effects of the masses of and

on each other are given by,

is the speed of the free individual final particles, which can be approximated,

practically, to the speed of light .

Here, the factors

and

can be termed as the acceleration Lorentz

transformation, which is the result of the average equivalent instantaneous velocity of

the final particles of the body. In the same way as above, if is the angle between the

line that joins the centers of the two masses and one of the coordinate axes (taken

arbitrarily as the axis) then the acceleration Lorentz transformation factor for the

mass distribution is given by,

02

The total distortion of the mass distribution is given by multiplying by

which results . The resulting 'distorted' equal probability surfaces are then used to

describe the interaction with other bodies and to define levels of energy of the body.

2.4. Application to classical bodies

In this section, we apply the procedures of the 'local dynamic space-time', to

the case of approaches infinity, i.e. for the classical ordinary bodies under high

velocity and high gravitation.

In accordance to the above analysis, the force between two compact bodies

moving with high velocities is given by,

According to the above rules, in order to evaluate for the two body

problem we calculate the reciprocal effects of high velocities and gravitational

interactions on each of the two masses and .

Taking first the effects of high velocities and , with respect to a slow

moving reference frame, such as earth. In such a case, we multiply the increase of the

mass of due to its velocity by the increase of mass due to its velocity .

Hence, the factor is given by,

On the other hand, for the case of two compact bodies of masses and ,

we multiply the increase of each mass under the effect of the other, hence the factor

is given by,

And consequently, for the combined general case, the factor is given by,

03

2.4.1.Comparison to the P.N. Approximation

Calculations of [1] of the acceleration of body under the effects of

using the Post-Newtonian Approximation, is based on harmonic coordinates Hence,

we need to transform eq. (13) into harmonic coordinates in order to perform

comparison to their calculations.

On such a basis, the final formulation of the acceleration of a body under

the gravitational effect of body , with rest masses and respectively, the two

bodies are moving with and respectively with respect to our chosen slow

moving frame of reference (and and with respect to the center of mass), the two

bodies can be compact bodies and can be moving with high velocities compared to the

speed of light, and is the distance between the centers of the two bodies at rest

relative to our chosen slow moving reference, with transformation to harmonic

coordinates, is given by,

is the universal gravitational constant, and is the speed of light in vacuum.

The post-Newtonian approximation is based on the assumption that gravitational

fields inside and around bodies are weak and that characteristic motions of matter are

slow compared to the speed of light. This means that one can characterize the system

in question by a small parameter , where

04

Where , and denote the characteristic velocity, mass, and size or

separation within the system; and are the characteristic pressure and density

within the bodies; and are Newton’s gravitational constant and the speed of light,

respectively [13, p. 5940].

Therefore, we take the ratio as a reasonable translation of the

term slow velocities compared to the speed of light, for which , hence

we set for our comparison the value of up to .

In our comparison we will first compare the two formulae for the case of non-

relativistic velocities with weak gravitational fields. Afterwards we take the case of

relativistic velocities with classical gravitational fields. Finally, we make the

comparison for the two cases together.

We first take the case of the Sun-Mercury two body system, with non-

relativistic velocities. We insert the following values: (the closest

distance between the Sun and Mercury), for which

(close to the mass of the Sun), , for which

(close to the mass of mercury), with m/s, and

m/s, we get ratio of full agreement (up to 11 digits). Accurate

Calculation of the value of the perihelion precession of Mercury using this formula

gives 43.1137 arc sec/century, while the observed value is 43 arc sec/century [5,

p.69].

If we take the case of the binary pulsar PSR B1913+16 at its periastron (i.e.

the closest distance between the two stars), with non-relativistic velocities, we insert

the following values: , for which

and , for which , we get a

ratio of agreement 0.999997. Calculations of the periastron precession of the binary

pulsar PSR B1913+16 using the above formula gives 4.108678 degrees/year. The

calculated values based on measurements is 4.226595 degrees/year [12, p. 26].

If we decrease the distance between the two stars in the above binary system

to , for which

, we get ratio of

agreement of . If we further decrease the distance up to the limit defined above

05

for 3 we take for which

taking and

taking we get agreement of .

In the case of relativistic velocities we will perform the comparison for

velocities up to 0.175 of the speed of light for which is about 0.03, taking much

smaller values for the ratio .

Therefore we take , , for which

, , for which . Now we take

(velocity of ) m/s, for which , and

m/s, for which . Applying these values for the two formalisms (i.e.

the formalism given in eq. (14) above and the P.N. App. Given by [1]) we get a ratio

of agreement of .

For lower values of velocity the agreement increases rapidly, for example for

maximum ratio of , we get agreement of , and at

, we get agreement of and so on.

For the case of bending of light rays grazing the surface of the Sun, calculations

using the above formula gives ''. This value is approximately equal

to the calculations based on the Schwarzschild solution of GTR, which is ''

and is confirmed experimentally using radio signals. The latest results (1991) by

Robertson et al., using Very Long Baseline Interferometry (VLBI), verified this

prediction to accuracy of [8, p. 249] 4.

Finally, applying the maximum values of for both of velocity and

gravitational fields of , we get agreement of , as expected, and the agreement

increases steadily as we lower this value. We note here that at the limit of application

of the Post-Newtonian Approximation its results start to deviate gradually by one or

two digits from the exact solution. Hence, the factor of agreement of 0.97 suggests

3 Here we assume two compact bodies that are represented by point masses, hence distance from the

center is outside the body, in the near zone for which the P.N. Approximation is applicable.

4 See detailed calculations in the aforementioned previous paper.

06

that the values calculated through eq. (14) above represent the exact solution for the

problem.

In view of the above results, it becomes legitimate to consider the calculations

based on eq. (13) and (14) as the exact calculations for the forces and acceleration

between two compact bodies moving with low relativistic velocities. For, it proves the

basic principle presented here, which is that we can present an exact formula for the

forces of two bodies relative to some chosen slow moving frame of reference. In

addition, in these formulae there is no limit for the ratios , , and

, which makes possible for us to calculate the motion of any binary system

including the motion of two extremely compact bodies 'Black Holes' moving with

very high velocities compared to the speed of light.

Therefore, these results prove the formula of the correcting factor

given

above for the case of the Classical realm. And if we accept this factor for the classical

case, then we should also accept such a factor for the Quantum case.

2.5.The case of a structureless quantum particle

Here, we apply such procedures to the high velocity and high gravitational

effects on the fundamental mathematical representation of structureless quantum

particles, which is Schrödinger equation.

Therefore, we first calculate the factor which describes the effects of high

gravity and velocity on the 'classical' mass of the body, i.e., its distorting effects on its

random distribution all over the space. And then we calculate the quantum effects on

such distorted distribution through the 'transformed' Schrödinger eq.(2) in order to

define the resulting 'probabilistic' distribution of the wavefunction. From above, the

factor of transformation of the mass distribution is given by5,

5 For simplicity of the equations we will use the symbol for .

07

This shows that we can calculate the values of this factor along the three

Euclidean coordinates ( , hence we can define the vector . On

the other hand according to our rules the factor applies to the distribution of the

mass of the body/particle in space. Therefore, the mass in the equation becomes

now a 3-vector in Euclidean space despite that it has the same values in all directions

in the free static case. Hence, Schrödinger equation takes the form,

From another side, the characteristic values of the wave given in equation (2c),

takes the form,

Taking and , the solution is given by,

is to be calculated for the case under investigation. For the general case of

two isolated particles is given by,

, and , are now vectors in the 3-space that are to be calculated

through equations (3 & 5) above. is the mass of Body1, is the mass of

Body2, is the distance between the centers of masses of the two bodies. The

equation is embedded in a flat local non-Minkowskian space-time. Velocities and

are to be measured relative to a slow moving reference frame such as earth.

3. Evolution of the Structure

At this point we have proved that the postulate of the final particles leads to a

unified conception for the ontology of both of the classical and the quantum realms, a

unified equation of motion for both realms, and a unified background space-time,

which is 'local dynamic space-time'. A full theory of space-time has to deal with the

08

subatomic particles taking into consideration the effects of the other three basic

forces, the weak, the strong and electromagnetism. Therefore, in this section we

introduce the effects of the structure of the 'elementary' particles of the SM and prove

that the postulate of the final particles unifies the four basic forces of nature and

reduces them to one fundamental force.

In order to perform this task we need first to have a more clear idea about the

structure of such particles and its relation to the three basic forces.

3.1. An Outline of the Underlying Structure

As mentioned above, the postulate of the final particles leads logically to the

view that the current 'elementary' particles of the SM possess a structure. since these

final particles are endowed with the power of attraction and extend probabilistically

all over the space, we expect appearance of all sorts of formations.

From the probabilistic point of view, some formations are much more

probable than others. The most probable formation for those final particles is

structureless formations. We expect that those structureless formations can take

probabilistically any number of final particles and consequently any value of mass.

However, from the probabilistic point of view, also, there exists a specific 'critical'

mass after which it is more expected that these critical masses will combine together

forming the first structured particles.

The simplest structure is composed of two structureless particles. However,

this structure is expected to collapse after a short period of time due to mutual

attraction. The next in simplicity is formation of three structureless particles along one

axis where the structure is preserved through the process of random spin around the

central particle.

Therefore, the most simple structured particle is that composed of three

structureless particles along one axis with perpetual random spin, call it the A particle.

Logically we expect combination of these A particles to form more complex particles.

If the A particles combine in parallel it will not form new 'structured' particles, but it

will formulate condensates of A particles. However, if they combine on two

perpendicular axes we get a new structured particle that is composed of six

structureless particles, call it the B particle. If the B particles combine together it will

09

not form a new structured particle because of its formation on 90 degrees. However, if

three A particles combine along three axes with 60 degrees angle, they will form a

new structured particle, call it the C particle (see fig.2-4).

P=3x

Fig. 2. Type (A) particle is composed of 3 S particles

with probability of biased motion along the axis of

alignment of 3 times the free random motion. In

comparison to the Standard Model it is the boson that

composes the neutrino, we call it zaidon'.

Fig. 4. Type (C) particle is composed of 9 S particles with

probability of biased motion along any of the three axis of

alignment of 5 times the free random motion. In comparison

to the Standard Model it represents the gluon, which

composes quarks.

P = 5 x

Probabilistic

rotation (spin1)

P=25/6 x

Fig. 3. Type (B) particle is composed of 6 S particles

with probability of biased motion along any of the

two axis of alignment of 25/6 times the free random

motion. In comparison to the Standard Model it

represents the photon, which composes electrons.

Probabilistic

rotation (spin1)

Probabilistic

rotation (spin1)

21

This forms the first most probable formation of the underlying structures of

'elementary' particles of the SM. Deeper scrutiny shows that it is very improbable to

have other forms of first level structures (like formation along three perpendicular

axes, or formation along more than three axes in one plane). This first level of

formation composes what we know today in the SM as elementary bosons. In sum we

have four bosons, one structureless and three structured. Each type of these bosons

can combine with itself to form a condensate of its own formation.

From fig.2-4, it becomes evident that probabilities of random motion of

particles A, B and C are no longer equal in all directions. The random motion of those

particles has become biased toward specific directions. Moreover, the probability

function of those biased random motion can be computed. Since according to the

theory of the final particles the forces arise from interaction of the S particles of the

boson, the forces of interaction between any two of those 'elementary' particles will be

proportional to the number of the probabilistic interaction. And since in principle we

can calculate the biased probability function, then in principle we can calculate the

number of probabilistic interactions. Consequently, we can calculate the forces that

arise between any two of those particles relative to the force of gravity (which results

from unbiased interaction).

Since it becomes highly improbable to continue the process of formation

through direct combination of the A, B and C particles, probabilistically we expect the

process of structure formation to continue on a second level.

condensed 'bosons'

of the same type

Fig.5. The second level of composite particles is

composed of doublets of condensed particles of

first level. In comparison to the Standard Model it

represents a 'fermion' that is composed of doublets

of condensed bosons.

condensed 'bosons'

of the same type

Probabilistic

rotation (spin1/2)

20

In the second level we expect that any two particles of types A, B and C (or its

condensates) at a specific critical mass can combine together indirectly through the

balance of probabilistic spin to form another level. So, we have combinations of

of particles A, B or C. This level of combination represents what we call in the SM

fermions (see fig.5 ).

3.1.1. Spin

Spin of 'elementary' bosons and fermions is essentially a form of biased

random motion toward the direction of 'rotation'. Since random motion of these

particles is modeled as Random Walk on a grid, then we should assume that the grid

is composed of nodal points that are of equal distances from each other. This equal

distance of the grid represents the minimum distance between the critical masses of

the S particles. After a great number of biased random walks of the S particles the

boson will generate the characteristics of real rotation about the center of mass of the

boson. Therefore, the A boson performs probabilistic rotation around one of its S

particles at its center of mass where the two other S particles probabilistically 'rotate'

around it at the minimum distance of the grid. The other two elementary bosons B and

C perform the same rotation because it is composed of the A bosons.

The second level of elementary particles, i.e. fermions, is composed of

condensate 'elementary' bosons. The two bosons condensates become bound in

accordance to fig. 5 above, with rotation about its center of mass. Hence, the bound

state of the two bosons will rotate at half the distance between its composing bosons,

i.e. it should possess half the spin of its bosons.

3.1.2. Wavefunction representation

According to the above we have an evolutionary picture for the elementary

bosons and fermions. The free final particles combine into a critical mass forming the

first 'structureless' S particles. The S particle produces a scalar field and is represented

by Schrödinger equation. Three S particles will generate the first A boson which is

endowed with perpetual probabilistic spin. Two A bosons combine to generate the B

bosons. Three A Bosons combine to form the C boson. Each of the three bosons

produces its own field.

22

Two condensate bosons of any of the three types get bound together to form

each type's fermions, with probabilistic rotation at half the minimum distance of the

grid. The fermion is then to be represented by two combined wavefunctions. The two

functions together describe the distribution of the composing bosons of the fermion

(as well as the distribution of the composing S particles), through probabilistic

superposition (fig.5).

The A bosons are represented by a wavefunction, call it . The B boson is

represented by a superposition of two superposed orthogonal functions. The C

boson is represented by three superposed functions with symmetric configuration.

These configurations combine into the collective function of the fermion that is

composed of bosons with spin half.

Under interaction each function will be distorted due to the effects of an

external field. The external field is a vector field that possesses a direction. Under the

effects of the double spin of the fermion and its composing bosons, the effects of the

external field on the functions has to be defined in accordance to double spin with

respect to the three perpendicular axes, through matrix representation of the combined

functions.

In addition, from the above structure it becomes clear why bosons can

occupy the same point in space-time to formulate boson condensates and fermions

can't. Bosons occupy space in 2-D6 and therefore can combine in parallel to make

condensates, as such it doesn't conflict with each other during rotation. On the other

hand fermions can't combine in parallel because it possess probabilistic rotation (spin)

on three axes and occupy space in 3-D, hence it will be in conflict with other fermions

on the same point in space-time.

This whole picture of construction is fully logical. It says simply that

probabilistically this is to a high degree the most probable formation for the structured

particles. However, such a picture should be consistent with the experimental findings

of the SM. A simple look shows that there is one deviation from the elements of the

6 We assume extremely small size of the S particles compared to the minimum distance of the grid.

23

SM, which is the existence of particle A. This requires the appearance of a well

defined model for the structure of such particles that preserves consistency with the

SM and makes possible to present a mathematical proof of such consistency.

3.2. The Zaidon Model

In the SM, apart from gravity, we have three types of fundamental forces (the

weak, the electromagnetic and the strong). Four elementary particles 'carry' these

forces ( and for the weak, photons for the electromagnetic and gluons for the

strong). These are called the elementary Bosons. On the other hand there are fermions

that are divided into leptons and quarks, each comes in three generations, and each is

composed of two types. Added to this picture, is the newly discovered particle the

Higgs Boson [11, ch.2].

Here we concentrate on the appearance of the fundamental forces due to

biased random motion, so we will concentrate on the appearance of the 'Bosons' that

are responsible for these forces. According to the theory of final particles, we have

only three elementary particles that are responsible for three types of biased random

motion and hence three types of interactions (or forces) through Particles A, B, and C.

As will be shown in the following, Particle B represents the photon, and Particle C

represents the gluon. Deviation from the SM is in the particle that is responsible for

the appearance of type (A) forces which represent the weak forces in SM.

3.2.1. The Status of the and particles

The picture presented above for the appearance of 'elementary' bosons and

fermions is an evolutionary picture. This picture mandated constructing a new

'elementary' boson that has no counterpart in the SM, namely the A particle. As we

have seen this particle is essential for construction of the other two 'elementary'

bosons, namely photons and gluons. These three particles as will be shown in the

following lead to the appearance of three forces that arise from alignment along one,

two and three axes. In the following it will be shown that the ratio of the A force to

the other two, namely electromagnetism and the strong is fully consistent with the

value of the weak force in the SM. This particle is called here the zaidon, and

zaidons formulate a neutrino, in the same way as photons formulate an electron

and gluons formulate a quark.

24

In other words, the zaidon is now an integral part of the picture introduced by

the SM, and its role is to explain the appearance of the weak force in a way that is

fully consistent with the appearance of the electromagnetic and the strong forces.

Since through the SM and experimental evidence the weak force is explained through

the massive particles and particles (and through the process of symmetry

breaking), then in order for the zaidon model to succeed it has to explain the

appearance of these three particles.

According to [2, ch.7], there are two types of fermionic currents in weak

interactions: charged currents (CC), mediated by the charged vector bosons and

, and neutral currents (NC) mediated by the . Weak interaction effects can be

observed in decays and in collisions only when they are not hidden by the presence of

strong or electromagnetic forces. Purely weak probes are neutrinos. We know three

types of processes: (1) Leptonic processes only leptons are present, both in the initial

and final state. (2) Semileptonic processes Both hadrons and leptons are present. An

important example is the beta decay. (3) Non-leptonic processes only hadrons are

present both in the initial and in the final state, still the process is weak.

These forms of weak interaction are explained in the zaidon model presented

here through treating the and particles as composite. According to this view

these particles represent a transition state between the original state of decaying of a

particle and the final stable state.

As such, the particles are composed of one photon and one zaidon

(decomposition of a gluon). A gluon as shown above is composed of three zaidons,

hence gluons can decompose into photons (that formulate electrons or

positrons, which are composed of zaidons), plus a neutrino (or antineutrino),

which is composed of zaidons. This is then, a transition state between a quark

(which is composed of gluons) that is in a state of decay and a final stable state of

an electron or a positron (which is composed of photons) plus a neutrino or anti-

neutrino, such as in beta decay.

Similarly, two particles are composed of four zaidons (decomposition of two

photons, hence the name comes from the 'Z' particle plus the 'on', like the phot-on and

the glu-on). An example is neutrino–electron scattering [2, ch.7], in which the

25

current is a transition state between a muon neutrino (which is composed of

zaidons) and an electron in a state of decay (which is composed of photons, or

zaidons) and a final stable state of also muon neutrino and an electron.

Having this picture in mind, we need to describe how the three basic forces

emerge from the first and second level structures of 'elementary' particles7.

4. Emergence of the four basic forces

According to the theory we present, matter is composed of random final

particles that are endowed with two basic intrinsic features: 1) perpetual random

motion all over the space, 2) the power of attraction between any two final particles

upon direct probabilistic contact. The perpetual random motion all over the space

makes possible for bodies that are separated in space to interact without the need of

the unphysical postulate of action-at-a-distance. Interaction between two bodies takes

place probabilistically in direct interaction between the final particles that compose

each of the two bodies involved in interaction. This form of interaction generates the

forces of gravity. The final particles, as shown above, compose the structureless S

particles. These in turn compose three types of structured particles (particles A, B and

C), that in turn formulate three types of fermions. Therefore, these three types of

bosons will play the role of the final particles for structureless particles, and interact

through its biased random motion to generate three types of forces that bind fermions.

As such, each of the three fermions is a particle/field at the same time and produces

its force field that interacts with other force fields that are produced by other fermions

to generate the forces between them.

On the other hand, in the SM the three elementary bosons 'carry' the forces

between fermions. According to [11, ch.2] the received view is as follows. The

concept of a force evolved from the idea of objects exerting force upon each other

into the concept of a field. The field has an independent existence, it contains energy.

7 Here we leave aside the question of the appearance of the heavy masses of the and particles

form the extremely small masses of the zaidons and photons. For, explanation of such masses is part of

the problem of the appearance of the different masses of second level particles such as neutrinos,

electrons, and quarks, a task that is left for other writings.

26

Quantum mechanics made this process even more explicit. Electromagnetic waves

consist of photons. So the field idea was replaced by particles. One imagines that the

charged source, the proton, continuously emits photons that then move out and later

return. An electron passing by the proton might intercept such a photon, absorbing its

momentum and energy and thus changing course. In this view the concept of a force

does not make any sense. Instead we have interactions, protons or electrons emitting

or absorbing photons. What we thought of as a force has become the exchange of a

particle.

Comparison shows that these two concepts of interaction are fully consistent

even though the phenomena are described in a different way. In both cases the forces

between the two fermions are 'carried' by the intermediate bosons, but in our theory

now we know how the fermions can become the source of the bosons, they are

composed of them.

On the other hand, since fermions are composed of bosons, and since bosons

perform perpetual random motion all over the space (because it is composed of final

particles), then the composing bosons of the two fermions that are in a state of

interaction will interact all over the space producing the forces that bind them.

Therefore, the force that arises from interaction of two specific bosons is the sum of

the forces between their composing final particles, and the forces that bind two

fermions, in turn, is the sum of the forces between their composing bosons. This

makes description of the forces between two bosons (and consequently between two

fermions) fully consistent with the description of the gravitational forces between two

classical bodies.

Therefore, if we define the force between any two final particles as , and if

we can calculate the average number of interactions of the composing final particles

of the two bosons, we can calculate the unit force between any two bosons. Since

currently we don't know the value of , we can use instead our knowledge of the

value of the universal gravitational constant . The factor makes possible for us to

calculate the 'gravitational' forces between any two classical (or quantum) particles.

Hence, if through our knowledge of the structure of the elementary bosons we can

calculate the relative value of the number of interactions between any two specific

27

bosons through its composing S particles, then we can calculate the relative forces

between them. Consequently we become able to calculate the value of the three basic

forces independently from experiment.

4.1. Interaction probability functions

According to the above, the three elementary bosons of the SM are

constructed out of a specific number with specific construction of one structureless

particle, the S particle. Each S particle performs random motion with equal

probability in every direction. Modeling the random motion into a random walk on a

3D grid, the S particle performs random walk with 1/6 probability in six directions

[9].

Our aim, in order to calculate the forces of interactions for each of the three

bosons, is to calculate the probabilistic number of interactions between the composing

S particles (and hence between the composing final particles) of the two bosons that

are in a state of interaction. This value has to be calculated under one fundamental

condition, which is that the biased random walk of the boson has to generate the

characteristics of the unbiased random motion after a great number of steps8. We have

three types of formations, as mentioned above, that formulate three types of

elementary bosons.

The first elementary particle is the zaidon, which is constructed out of three S

particles aligned along one axis. The zaidon as a collection of 3 random S particles

performs biased random walk with probability of matched walk along the axis of

alignment of 3 times the probability of the single particle, which is 1/6 (fig.2).

The second elementary particle is the photon which is constructed out of two

zaidons in a symmetrical configuration, i.e. perpendicular to each other. Along any of

its four directions in a random walk we have a probability of matched walk of 4 times

8 This condition reduces to the fundamental postulate of the final particles, which is that it is endowed

with full perpetual random motion. Modeling such a condition on the RW process translates that each

final particle is allowed to move on the grid in one specific direction at a time with the condition that

after a great number of random walks the particle will generate the characteristics of full random

motion. Since the structured particles are composed of those final particles then it will fulfill the same

condition.

28

the probability of one particle, in addition to probabilistic contribution from the other

two particles which is 1/6. Hence, the total value is 25/6 times the probability of one S

particle (fig.3).

The third type of elementary particles is the gluon, which is constructed out of

three zaidons in a symmetric form, i.e. along three axes with equal angles. The

probability along any of the three axes is 5 times the probability of one S particle with

no contribution from the other four particles. The reason is that the other four

particles, due to its position, lie outside the 3D grid and hence it will not contribute to

the matched random walk which generates the forces of attraction (fig.4). For,

matched random walk requires matched motion of two S particles along two adjacent

points on the grid, a case that is not fulfilled for the two inclined axes.

Therefore, the probabilistic number of interactions between any two bosons is

given by calculating the matched random walks between its composing S particles.

This gives the probabilistic number of interactions between the two bosons along one

step of random walk. In order to have the probabilistic number of interactions for a

unit volume of space we calculate the matched random walks in one unit of space of

the grid. Since one unit of space of the grid space is composed of 3 by 3 by 3 points,

i.e. 27 points, then the laws of probability tell us that we have to raise the probabilistic

number of interactions calculated before for one step to the power of 27.

4.1.1. Analysis of the problem of probabilistic matched interaction

In the free unbiased random walk we take equal probability of 1/6 for each one

walk on a 3D grid. If we have two walkers that start at the same point, the transition

probability for both walkers to another point is given by multiplying the two values of

probability, i.e. 1/36. Call this 'matched' random walk. Under specific conditions the

random walk can become 'biased' toward specific directions through a specific

probability function. In such a case we can superimpose such biased probability

function onto the free probability function and treat it as a branching probability (i.e.

as if it follows in time).

Therefore, if we have biased random walk for a specific direction with

probability 1, then the final probability for that direction is 1 multiplied by 1/6, hence

29

the total probability of biased walk becomes 1/6. If we have biased random walk with

probability 1/6, then the total probability of biased walk is 1/36.

Now we apply this simple rule to particle B above (which represents the

photon, as a component of an electron), as an example. The B particles, due to its

configuration as well as due to its combination into fermions with double spin,

perform biased random motion. Under interaction, due to its configuration the particle

becomes forced to perform matched random walk along two perpendicular axes on

the plane of rotation, i.e. with probability 1. Therefore the probability of matched

random walk along these two directions becomes 1/6 for each direction (we have two

directions for each axis). For the third axis, the particle performs free random walk

with probability 1/6 for the two remaining directions. Therefore, the probability of the

random walk becomes 1/36 and the probability of matched random walk becomes

1/36 by 1/36.

The term 'biased' here means that the random motion of the particle is subject

to a specific condition that controls its free random motion. In the case of the particle

B, each of the S particles that compose it (and consequently the whole B particle) due

to its specific configuration is forced to perform a specific sequence of steps that

guarantees two conditions: 1) matching with another particle that exists

probabilistically at the same point in space; and 2) generates the full pattern of a free

unbiased random motion after a great number of walks.

In the case of the B particle these two conditions are fulfilled if the B particle

collectively performs a sequence of steps of the form followed by free

walk along the axis, assuming that the S particles are aligned on the axes.

Since the two bosons that are in a state of interaction spin with the same rate

the values of the forces of interaction will not be affected, nevertheless the direction

of the resulting force can be affected. If they spin in the same direction relative to

each other the force will be opposite to that in the case of spinning in opposite

directions.

The second step toward calculating the matched probabilistic number of

interactions function is to calculate the probabilistic contribution of the S particles that

compose the B particle. From above we have two types of probability. The first is the

31

direct matched random walk with probability of 1/6. The second is the sub-probability

of branching 'matched' random walk with probability of 1/36 by 1/36. Since the B

particle is composed of six S particles, then the net effect that produces probabilistic

'matched' random walk is given by multiplying the above random walk probabilities

by the contribution of these S particles.

From fig. (3) above, the net contribution of the S particles for the B particles is

25/6 for the matched walk along the two perpendicular axes. For the third direction

(perpendicular to the plane of rotation) we have contribution of 2 particles at the

center with probability for each, and two particles at the peripheral with 1/6

contribution for each. The contribution of the particles for the third axis has then to be

divided by 4 directions in order to calculate its effects on one direction of the two

basic axes of matched random walk9. Similar calculations have to be performed for

the contribution of the composing S particles for particles A and C.

4.1.2. Calculations of the probabilistic number of matched walks

Applying the rules above for the three bosons we get the following results.

a- Probabilistic number of Interactions of the zaidons

- Probabilistic number of interactions of two zaidons

Dividing this value by the probability of matched walk for the case of free

non-biased random walk (i.e., the case of Gravity) which is 1/36 we find that the

number of the matched walks in the case of the zaidon is times the number

of that of Gravity.

- Probabilistic number of interactions of a photon and a zaidon

9 We make our calculations for one direction because it is more simple and intuitive. However, we can

also sum up the probabilities for the six directions and divide the results by 6, for which we get the

same relative probabilities calculate here.

30

For interaction of a photon with a zaidon we have interaction of 3 S particles

along the axis of matched interaction. For the other two axes we have at the center 2 S

particles from the photon against one from the zaidon. Hence, we get,

- Probabilistic number of interactions of a gluon and a zaidon

For interaction of a gluon with a zaidon we have interaction of 3 S particles

along the axis of matched interaction. For the other two axes we have at the center 3 S

particles from the gluon against one from the zaidon. Hence, we get,

b- Probabilistic number of Interactions of the photons

- Probabilistic number of interactions of two photons

Dividing this value by the probability of f free random walk, which is

we find that the number of the matched walks in the case of the photon is

times the number of that of Gravity.

- Probabilistic number of interactions of a gluon with a photon

For the case of interaction of a photon with a gluon only 4 S particles perform

matched random walk, along two axes. Along the third axis at the center we have 3 S

particles from the gluon against 2 particles from the photon. Hence we get,

Dividing this value by 1/36, we find that the number of the matched walks in

the case of the gluon- photon interaction is times that of Gravity.

c- Probabilistic number of Interactions of two gluons

32

Dividing this value by the probability of matched motion for free random

walk, which is 1/36, we find that the number of the matched walks in the case of two

gluons is times the number of that of Gravity.

4.2. Calculation of the relative values of the basic forces

As explained above, these calculations of the relative probabilities of the

matched biased random walks for the different cases pointed out above reflect in a

direct manner the relative values of these three types of forces with respect to the case

of Gravity, for each step of the random walk on a 3D grid. On the other hand, our

knowledge of the probabilistic matched interaction of any two bosons relative to the

values of matched interaction of two S particles is directly equivalent to the relative

values of the forces between the two bosons to the forces of gravity.

The relative values of the forces, as pointed out above, can't be calculated on

the basis of one step of random walk, because the forces in reality appear as a result of

interaction in 3D space. Hence, our comparison for the relative forces is to be made

over a unit space of the grid that is composed of points, i.e. 27 points. In

order to calculate the total value of the probability of generating the force of

interaction we have to raise the probability value to the power of 27.

Hence, the relative values of the forces are then given by raising the relative

values of the matched probabilities calculated above to the power of 2710

.

a- Interactions of zaidons

- Interaction of two zaidons

Consequently, the relative value of the force of type (A) that results from

matched random motion of two zaidons with respect to Gravity is given by,

10 In the following analysis of the relative forces we deal with elementary bosons leaving detailed

treatment of the construction of their corresponding fermions to a future work. In this context for

simplicity we call the family of elementary fermions termed electron, muon and tauon and their

antiparticles collectively as 'electrons', just like quarks and neutrinos.

33

This force results from interaction of two neutrinos (of any of its six types).

- Interaction of a photon with a zaidon

This force results from interaction of an electron (of any of its six types) with

a neutrino (of any of its six types).

- interaction of a gluon with a zaidon

This force results from interaction of a quark (of any of its twelve types) with

a neutrino (of any of its six types).

b- Interaction of photons

- Interaction of two photons

The relative force that results from matched motion of two photons with

respect to Gravity is given by,

This force results from interaction of two electrons (of any of its six types).

From detailed analysis of the matched random walk of the two photons, if the two

photons under interaction are rotating with opposite directions the resulting force is

negative (attraction), and if the two photons are rotating on the same direction we get

the same force with positive direction (repulsion).

- Interaction of a photon with a gluon

The relative value of the resulting force is given by,

This force results from interaction of an electron (of any of its six types) with

a quark (of any of its twelve types). Similarly, if the two bosons are spinning in

34

different directions the force is attraction, and if they spin in the same direction the

force is repulsion.

c- Interaction of two gluons

The relative force that results from matched motion of two gluons with respect

to Gravity is given by,

This force results from interaction of two quarks (of any of its twelve types).

Therefore, the relative value of the interaction of a photon and a gluon to the

interaction of two photons is given by,

Hence, the force of interaction between a gluon and a photon is about one

third the force of interaction between two photons. Photons possess two perpendicular

axes of interaction, whereas gluons possess three axes. In case that the two rotate

probabilistically in the same direction, i.e. with zero relative spin, they encounter

matched interaction once in each full revolution. Hence, the force of interaction is

approximately 1/3 of that between the two photons.

However, if the two rotate probabilistically in opposite directions, their

relative spin becomes doubled the normal boson spin. In such a case analysis shows

that the two will encounter matched interaction twice in each full revolution.

Therefore, the force of interaction becomes doubled, i.e. it becomes about 2/3 of the

force of two photons with negative direction.

d- The direction of forces

All forces between the four bosons described above are forces of attraction

between its composing final particles. However, as shown above, the S particles don't

perform rotation, whereas Bosons A, B and C perform perpetual random rotation

(spin). This results in a difference in the direction of the force between the two cases.

For the case of the S particles, which don't perform spin, its free random

motion doesn't collectively exert an effect on the direction of attraction forces. Hence,

35

what remains is the direction of the random motion of the S particles between its two

sources, i.e. the bodies in interaction. Hence, the forces between the two bodies arise

from its composing final particles and possess a probabilistic 'scalar' magnitude and

directed along the line that joins the two structureless bodies (or bosons) that

represent the sources of the two S particles. This represents a scalar field of attraction.

For the case of two bosons, A, B and C, at the state of interaction, the two

bosons perform matched random motion and together exert combined rotation. This

combined rotation gives the force that arise between their matched random motion a

direction that depends on the direction of motion of the two bosons in space-time.

This generates vector fields.

The two bosons can be probabilistically rotating on the same direction or on

opposite directions. The forces of interaction are the same in both cases, however the

direction of the force will be reversed. So, if the forces in the first case are considered

positive, then the forces in the second case is considered negative, or vice-a-versa.

The first case, i.e. forces of repulsion, realize in interaction between two fermions, or

between two anti-fermions (or between a lepton and a down quark, or between an

anti-lepton and an up quark). The second case, i.e. forces of attraction, realize in the

case of interaction between a fermion and anti-fermion (or between a lepton and an up

quark, or between an anti-lepton and a down quark). This rule applies also for the case

of interaction between two gluons, however, with much more complicated sequence

of interaction, which resulted in the theory of chromodynamics of the SM11

.

4.3. Comparison to the Standard Model

These relative values of basic forces lead to the well known classification of

the three fundamental forces calculated through experimental procedures and

mathematical formulations of the SM. The force due to the zaidons is the weakest

11 In very general terms, the gluon has to preserve the rule of producing a full random motion after a

great number of random walks. Hence, it can't perform three consecutive positive random walks, it has

either to perform two to the positive direction and one to the negative, or the contrary. So, when

performing matched random walk with a photon, if we take positive direction for the photon it will

encounter positive direction either once (for the same direction of spin), or twice (for the opposite

direction of spin). Notice here the resemblance of the three axes of random walk to the three colors,

and the conditions of matched motion with the SU(3) group.

36

followed by the force due to the photons and the strongest is the force due to the

gluons. The weakest force between two zaidons is by of the strong force

between two gluons. On the other hand, the strong force between two gluons is only

times the force between two photons. Finally, the force of Gravity is only

by of the strong force between two gluons. In addition, gluons interact

with photons with approximately 0.3335 or 0.667 of the force between two photons.

The charges of interaction are reversed in the case of replacement of the top by

bottom quarks as well as for the cases of replacements by the anti-electrons. The

weak charge is neutral with respect to both electrons and quarks. These results are

generally in excellent agreement with the findings of the SM [11, ch.2].

4.3.1.The effects of high velocity and high gravitation

The above relative values of the basic four forces of nature are calculated for

the case of low velocities and low gravitational effects. Since the effects of high

gravitation are treated here as velocity effects, we will discuss here the effects of high

velocities meaning that the same analysis applies also to high gravitation.

According to the above, at high velocities the mass of the body, classical and

quantum, increases in accordance to the Lorentz transformation velocity factor

which is given above by eq. (3). According to the above analysis the effect of the

increase of mass is increase of the number of interactions with other bodies.

Therefore, we will get maximum increase of the masses of the S particles that

compose the A, B and C bosons in the direction of motion, and zero for the directions

perpendicular to it. As such we apply this effect to the three bosons, A,B and C.

For the boson A, i.e. the zaidon, matched motion is along one axis, hence if

the direction of high velocity coincides with such axis the number of interactions will

increase according to the factor . If the direction of motion is on the plane

perpendicular to such axis, the effect of high motion will be extremely low.

The B boson, which is the photon, performs matched random motion on two

axes. Hence if the direction of high velocity is on the plane of its matched random

walk it will affect one of the two axes at a time. Hence, the photon will be affected by

only half the effects of the increase of high velocity. If the direction of motion is on

the plane perpendicular to such axis, the effect of high velocity will be limited.

37

The C boson, which is the gluon, performs matched random motion on three

axes. Hence if the direction of high velocity is on the plane of its matched random

walk it will affect one of the three axes at a time. Hence, the photon will be affected

by only one third of the effects of the increase of the number of interactions due to

high velocity. If the direction of motion is on the plane perpendicular to such axis, the

effect of high velocity will be limited.

This leads to reducing the relative values of the basic forces under high

velocities. In the case that the velocities of the bosons (under interaction) approach the

speed of light, the four forces will converge approaching a relative value that is close

to unity.

In the following as an example we will calculate the relative values for the

basic forces for velocity of 0.95 the speed of light. We get the following results.

The factor

Taking the unit of attraction between two S particles in low velocities, and

assuming that one of the two bosons is moving with high velocity (if the two are

moving with velocity we have to use ), the force along the direction of motion

will be,

And in the other two directions,

Applying the rules above, for the case of gravity, for a free particle, this factor

affects one axis of the probabilistic matched random motion, hence,

Assuming that the effects of high velocity is along the plane of rotation of the

other three bosons, and applying the rules above to eq. (19) we get the following

results.

- Interaction of two zaidons

38

For two zaidons the factor affects the axis of the matched random walk at a

time, therefore,

- Interaction of two photons

For two photons, the factor affects one of the two axes of matched random

walk at a time, therefore,

- Interaction of two gluons

For two gluons, the factor affects one of the three axes of matched random

walk at a time, therefore,

Hence, the relative forces with respect to gravity will be,

For two zaidons,

For two photons,

And for two gluons,

Hence at 0.95 the speed of light, the strong force becomes only the

force of gravity, about 10 times the electromagnetic, 120 times the weak. The

39

electromagnetic becomes 11 times the weak force. At the extreme case where the

velocity approaches the speed of light the values of the four forces will converge and

their relative values will be close to unity12

. These results are generally consistent

with the findings of the SM under high energies [11, ch.2].

5. The case of the electromagnetic effects

Our knowledge of the relative values of the basic forces, the construction and

patterns of random walk of the 'elementary' bosons and fermions in addition to our

knowledge of the effects of high velocities and high gravitation makes possible for us,

in principle, to deduce equations of motion and interaction of elementary particles in

all cases of motion and interaction.

According to the theory presented here, we aim to define the distribution of

the matter of the body in space-time under its own velocity and under external force

fields. Such a distribution makes possible for us to calculate the forces that develop

between the body and other bodies in space-time, and therefore we become able to

define its motion under these external fields. Given such a distribution, definition of

the forces that develop is to be performed through calculating the number of each of

the four types of bosons that are involved in interaction as well as its direction of

motion in space-time and multiply it by its unit forces of interaction calculated above.

5.1. Analysis of the problem of the electromagnetic interaction

In the case of electromagnetic forces, interaction takes place between two

electrons13

or between an electron and a quark. According to the analysis above, such

an interaction takes place between the composing photons (or photons and gluons)

that are involved in interaction.

12 Note that these calculations are not exact and serve only to illustrate the effects of high velocity. The

exact calculations require including the effects of high velocity on the direction perpendicular to the

plane of rotation of the boson and taking the average of the three directions, which will not change the

final result, which is that the four basic forces converge under high velocities.

13 As mentioned before, the term electron is used here to mean any member of the family 'electron,

muon and tauon' and their antiparticles.

41

Therefore, in order to define the electromagnetic effects on the motion and

distribution in space-time of a specific quantum particle that is composed of electrons

(and/or quarks) we need to: 1) define the number of the composing photons of the

electron (and the gluons of the quark) in each of the two bodies involved in

interaction. 2) describe the effects of the probabilistic rotation (i.e. spin) combined

with matched random motion of the two coupled photons/gluons (that constitute the

electron/quark) on its distribution in space-time. 3) from the above information we

define the field of the body that exerts electromagnetic effects on the body under

study.

Given the above, we multiply the numbers of photons/gluons (of the field and

the particle) by each other (to generate the total number of interactions) and multiply

the result by the unit force of interaction calculated probabilistically above. This

defines the electromagnetic effects on the particle in space-time.

Since we are now describing interaction between two bodies/fields at the same

time, and since the effects of the forces of interaction are linear, then it becomes, in

principle, possible to define the electromagnetic effects of any number of bodies

through superposition. In other words, the electromagnetic field is now already

quantized with no need for more procedures.

The first requirement, which is definition of the number of the photons/gluons

can't be fulfilled currently. Instead, it can be replaced by our experimental knowledge

of the value of the charge of the electrons and the quarks. For, the charge of the

electron is no more than the result of multiplication of the number of its composing

photons by the force of interaction between two photons, multiplied by a constant to

be defined experimentally.

The second requirement can be fulfilled through calculating the combined

effects of spin and matched random motion on the distribution of the functions of the

particles in interaction. The photon, as shown above, is described through two

orthogonal wavefunctions, which represents its symmetrical configuration on two

axes. Under interaction with an external field the wavefunction takes the form,

40

Where, is the charge of the particle under interaction, and is the field

potential of the particle that exerts electromagnetic forces.

The photons (and the gluons) perform probabilistic spin 1, and the fermion as

a whole performs 1/2 spin. In addition, the process of interaction requires alternative

matched random walks that restore full random motion after a great number of steps.

This is fulfilled if the photon performs alternating positive and negative random walks

along the two directions of matched random walk. Then we need to take the effects of

spin and matched random walk on eq. (23) above into consideration.

From the discussions above, at the plane of interaction we have two

wavefunctions with matched random walk, and at the third axis we have two

wavefunctions with free random motion. At the plane of rotation (take it arbitrarily as

the plane), the photon performs biased random walk along the two axes, while

fulfilling the condition of approaching full random motion after a great number of

steps. This means that it has to perform alternating positive and negative matched

motion of the form ( ). Random rotation of the photon along such a plane

shifts the direction of the two functions. Keeping the starting point, the random

motion with respect to the axis takes the form ( ).

At the third perpendicular axis, due to its free random walk, we have two

wavefunctions in state of superposition, i.e. two functions multiplied by . However,

in order to fulfill the condition of full random motion, the two functions have to take

opposite sings. Since rotation is on the perpendicular plane it will not affect the

random motion along the axis. Hence, on the -axis we have two functions

multiplied by and – .

Therefore, the net result of the double effect of spin and matched random

walk, in the case of the photon, is that for matching on the -axis we have two

positive functions. For matching on the -axis we have one positive and one negative

function. And for matching along the -axis we have two free superposed opposite

42

functions (multiplied by and – , due to full random motion). In matrix form this

leads to the well known Pauli matrices14

,

The third requirement is to calculate the space-time distribution of forces of

interaction between the photons (that compose the two electrons) or between the

photons and gluons that compose quarks. This is given experimentally through

Maxwell's equations that are used to express the electromagnetic potential where

the electromagnetic force field is given by [3, ch.4],

and are electric and magnetic fields. And Maxwell equations are then

written in terms of the antisymmetric tensor field given by,

As mentioned before, the quantum effects are independent from effects of

random rotation, matched random motion, and high velocities and high gravitation.

Hence, it becomes possible to make use of eq. (23) above which describes the effects

of the electromagnetic field on Schrödinger equation with the above Pauli matrices,

and the effects of high speed to produce the final form for the relativistic equation of

the electron under an external electromagnetic field.

14 This same procedure can be applied to the other two cases, i.e. the zaidon and the gluon. For, the

case of the zaidon we have only one function along the -axis (defined arbitrarily), with free random

walk on the other two directions, which makes a simple case. For, the case of the gluon the situation

becomes complicated and requires more detailed analysis (In the treatment of the SM, the three

functions are divided into eight functions in a state of superposition comprising the eight gluons).

43

5. 2. The Equation of the wavefunction under electromagnetic effects

Applying these procedures we can produce the relativistic equation of a

massive spin electron under the effects of external electromagnetic field as

follows,

Where, is the charge of the particle, is the vector potential, is the scalar

potential, is velocity vector of the particle measured relative to a slow moving

reference frame, and is the velocity of the free final particle which is approximately

equal to the speed of light in vacuum c. Here we assume that the source of external

electromagnetic field is at rest with respect to earth (which is the case in our current

experiments). The factor is applied to both of the mass and the charge in accordance

to the theory of the final particles presented above. So, either we include these effects

on space-time itself, and hence distortion of the distribution of mass and charge will

be taken care of through background space-time itself, or we include those effects

directly on mass and charge keeping space-time flat local non-Minkowskian.

Equation (26) is equivalent to the equation of the positive component of the

wavefunction of the relativistic Dirac equation. Proof is as follows:

Following [7, p. 75-76 ], first we derive a low-energy approximation of the

Dirac equation and apply it to the electron, which has electric charge . Setting

, first we write down the equation in terms of two-component wave

functions. We also make the substitution to include the

electromagnetic interaction. Therefore Dirac equation is written as follows,

44

This is a set of two equations for the two components spinors

In the high-energy limit, where the mass can be neglected, equations for

and are disconnected and represent two independent particles ( , ).

But in the low-energy limit, the difference disappears and . For this

reason we introduce,

With the expectation in the non-relativistic limit. Then the equation

reads

Up to now the equation is exact. At this point we choose the solution

and set

Then we can make the substitution , where now represents the

kinetic energy. Substituting eq. (30) in eq. (29) and applying the non-relativistic

approximation , , eq. (29b) becomes

Substituting this expression in eq. (29a), we obtain [7, p. 76]

Or,

45

The operator on the left hand side represents kinetic energy for the low

energy limit, hence we can write finally taking as the positive component of the

wavefunction of Dirac equation,

Where and are 'non-relativistic kinetic energy' and momentum operators,

is the charge of the particle ( in the case of the electron), is the scalar potential

of the field, and is vector potential.

Multiplying by the Lorentz factor , where we get,

Space-time of eq. (34) is classical, i.e. flat non-Minkowskian, and the equation

up till now is fully equivalent to the positive component of Dirac equation.

Classically, kinetic energy and momentum are defined as

and

, where the mass under motion is approximated to rest mass. We know that

the mass of the particle (i.e. rest mass) increases by the factor under motion with

velocity If we follow observation we can differentiate between rest mass and

moving mass , taking . Hence we can define energy and momentum for

a moving particle with velocity as

and , or alternatively,

, and .

Since kinetic energy and momentum in eq. (34) are approximated to

, and , then energy and momentum under the effects of high

velocity can be defined as and .

Using this definition to classical energy and momentum, under high velocity

(with non-minkowskian space-time), eq. (34) can be written as,

46

Where and are understood as energy and momentum operators for a

particle that is moving with velocity .

Eq. (35), then, tells us that if we don't use the relativistic definition of energy,

(i.e. keeping space-time classical), then we have to compensate this by multiplying

the rest mass of the particle by the factor , and if the equation includes external force

effects (due to the charge q), then the charge also has to be multiplied by the factor

in order to compensate the relativistic effects on the charge (treating charge as a

source of kinetic energy).

In addition, eq. (35) is equivalent to eq. (26) above if is taken to be equal to

. This condition applies in case that motion of the particle is measured relative to a

slow moving reference frame such as earth.

Hence, this derivation presents the proof that if we make a transformation

from Minkowskian space-time to classical space-time for Dirac equation, we have to

multiply both of the charge and mass by the factor in order to preserve the

equation under motion. Therefore, eq. (26) above, which is the unified equation of

gravity and electromagnetism is proved to be equivalent to the positive solution of the

relativistic equation of the electron known as Dirac equation.

If the masses of the two bodies that are in a state of electromagnetic

interaction are known then we can simply use the factor eq. (18d) that describes the

effects of high gravity on the mass of the particle to produce a unified formula for

the effects of high gravity, high velocity and electromagnetism.

6. Conclusion

In this paper we continued our investigation of the consequences of adopting

the fundamental assumption that elementary particles of the Standard Model is

composed of identical indivisible 'atoms' that are endowed with both of attraction and

perpetual random motion all over the space. These 'atoms' are termed here 'the final

particles'.

In a previous paper we presented the proof that this postulate leads to

elevating Schrödinger equation to the status of the unifying fundamental equation for

both of the classical and the quantum realms. In addition, this postulate leads to a

47

unified background space-time for both of the classical and the quantum realms,

which is 'local dynamic space-time'.

In this paper we presented analytical and mathematical proof that such a

postulate leads to unifying the four basic forces of nature. By definition it follows

from the postulate of the final particles that the 'elementary' bosons and fermions of

the SM possess a structure. We showed how such a structure evolves probabilistically

from the postulated final particles and presented the effect of such a structure on

motion and interaction of such particles. Such a structure is presented through

modeling the random motion of the elementary particles into biased random walk

process through which it becomes possible to calculate the probabilities of motion and

interaction of the different particles.

According to this view, the first level of the process of evolution leads to the

existence of one composite structureless boson and three structured bosons that are

endowed with probabilistic spin. The structureless boson generates gravity and the

three structured vector bosons generate three other forces that represent the weak, the

strong and the electromagnetic,. The second level of evolution leads to the appearance

of three basic types of fermions each is constituted from bosons of the same type

that spins probabilistically at half the boson's spin.

The resulting values of the relative forces between the four fundamental forces

calculated through the model of random walk is generally in excellent agreement with

the experimental results of the SM. One deviation from the SM results from such

description, which is related to the status of the and bosons. In our evolutionary

model, these particles are not 'elementary' but composite, they represent a transition

state for the decaying particles. Instead, a new particle that represents an integral step

of the process of evolution, which is the zaidon, is responsible for the weak

interactions and constitutes neutrinos.

As such we proved through analytical and experimental evidence that the

theory of the final particles leads to unification of both of the classical and the

quantum realms on the level of the fundamental equation, the background space-time

and on the level of the four fundamental forces. Hence, if these final particles do exist,

then it follows that physics is unified, or the other way around, if the analytical and

48

experimental proofs presented here are accepted, then these final particles do exist

and physics is unified.

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