1. Introduction
We know intimately the term "atom" which
comes from ancient Greek and means "uncuttable" or translated as
"indivisible." In the early 19th century, the scientist John Dalton
introduced the modern definition of an atom to characterize chemical elements.
It was discovered that Dalton's atoms are not actually indivisible about a
century later. An atom consists of three basic types of subatomic particles:
electrons, protons, and neutrons, which occupy the tiny space in an atom.
Protons and neutrons form the nucleus that contains most of an atom's mass.
Electrons are the lightest charged particles in nature and revolve around the
nucleus of an atom. An electron is seemingly indivisible yet. Until today, we
have not split an electron into two or more smaller particles. We only make the
positive and negative electron annihilation. A free neutron is unstable,
decaying into a proton, electron, and neutrino. However, a free proton is
stable, and is composed of two up quarks and one down quark in the modern Standard
Model. Furthermore, whether a quark can be cut into smaller parts or whether
the matter is infinitely divisible.
This paper tries to answer the above questions from
a different perspective. What will happen when light or electromagnetic,
gravity waves and other waves make up the primary particles that constitute the
fundamental elements of matter? The assumptions are derived then:
Light or electromagnetic waves, weak interaction, gravity, and strong interaction are independent waves without rest masses. But their structures are different.
Light and gravity can be described by the wave equation with the field strength Ê and speed c.
- c
Weak
and strong interaction can be described by the 4-dimensional Laplace equation
with field strength Ê' and speed c'.
- d
According to the unified electro-weak theory, light and weak interaction have the same speed cL with spin number +1 or –1.
- e
Gravity and strong interaction have the same speed cG without spin, and cG is constant in a vacuum.
- f
The primary particles, which are electrons, electron neutrinos, and dark neutrinos in this paper, are made by the above four types of waves.
2. The Formation of Primary Particles
To suppose that the birth of primary particles may
divide into the following two parts:
The
light and weak interaction couple together (hereafter referred to as the E-W
couple) when they have the same spin number and the second-order partial
derivatives of their fields with respect to time
are
equal. The gravity and strong interaction couple together too (hereafter referred
to as the G-S couple) when the second-order partial derivatives of their fields
with respect to time
are
equal. So we have
and
Where
Êe,
Êw,
ÊG, and
ÊS are electric, weak
interaction, gravitational, and strong interaction fields.
The E-W couple has no spin. The original spins of the coupled waves convert the electric or weak charge property.
It
makes a primary particle when the two coupled waves attract each other and
shrink to a tiny sphere. One E-W couple and one G-S couple produce an electron
or a positron whose charge property depends on the original spin
of the E-W couple. The dark
neutrinos are composed of two G-S couples. Two E-W couples with different
original spin compress themselves into an electron neutrino. But they cannot
attract each other with the same original spin.
Further, we assume that each field around the
primary particle is time-independent and spherically symmetrical. Thus, in the
spherical coordinate system, we have the uniform Laplace equation for the above
equations (3), (4)
whose general solution is
where Aj and Bj
are constants, are associated Legendre polynomials, j and k
are integers, j = 0, 1, 2, 3, …, k ≤ j, and j is
called the degree of associated Legendre polynomials.
3. The Fields and Binding-Energy
We can now derive the electric, gravitational, weak
interaction, and strong interaction fields Ê based on equation (6) and
existing physical laws and data.
It is reasonable that equation (6) can be
transformed into a pair of conjugate solutions.
Clearly, there is
where the subscripts
a and
b denote electric, gravitational, weak interaction, or strong interaction.
The field Ê may be split into the macroscopic item and the quantum factors , . The degree of associated Legendre polynomials j rules properties of the field Ê because it is not only an exponent of r in the macroscopic item but also impacts forms of the quantum factors.
Compared equation (7) to Gauss's law of electrostatics and Newton's law of gravity, the electric field
Êe is
and the gravitational field
ÊG is
where '–' means attractive interaction, and '+' means repulsive interaction, as usual in this paper,
is the mathematical electric charge, and
is the mathematical mass or the gravitational charge.
Weak interaction has an intensity of a similar magnitude to the electromagnetic force at very short distances (around 10
-18 meters), but this starts to decrease exponentially with increasing distance. Its effective range is about 10
-17 to 10
-16 meters
[1, 2, 3]. All of the above can help us to determine weak interaction field
Êw from equation (7).
Êw is so supposed to equal to
where
is the mathematical weak charge,
Rcw is the critical radius of weak interaction, and
m is an integer greater than 1.
The strong force is a short-range interaction (around 10
-15 meters) similar to the weak force. But its range is more complex than the weak force. At distances comparable to the diameter of a proton, it is approximately 100 times as strong as the electromagnetic force. At smaller distances, however, it becomes weaker. In particle physics, this effect is known as asymptotic freedom
[4, 5, 6]. Moreover, it is supposed that the fields of the four fundamental forces have a unified form in a very tiny range. Hence equation (6) can be translated into the strong interaction field
ÊS
where
is the mathematical strong charge,
RcS1 and
RcS2 are the 1
st and 2
nd critical radius of the strong interaction, and
n is an integer greater than 1.
Further, it is assumed that RcS1 << Rcw < RcS2 and m ≠ n.
Now we turn to determine the energy. The energy density of wave equation (1) is given by
, and the equations (1) and (2) are Lorentz invariance. So we hope the equation of energy is Lorentz invariance, too, and let the binding-energy
Ea-b of the four fields be
Note that
and
are independent of time, and there are
and
When we use light as a measurement medium to determine the energy as usual, we can translate equation (14) into the spherical coordinate system with the optical medium, which is
where
R is the radius at which two fields begin to interact with each other.
The general energy expression can be calculated when equation (7) is substituted into equation (17).
Based on equation (18) and associated with equations (10) to (13), we first compute the self-binding-energy of the four fields. The self-binding-energy of the four fields
Ee-e,
EG-G,
Ew-w, and
ES-S are
and
The binding-energy of the four fields, such as
Ee-G,
Ee-w,
Ee-S, etc. are
and
For the convenience of subsequent calculations, we require the results of our defined energy to be consistent with those of the conventional physical method. Compared equations (19) and (20) to electric and gravitational potential energy formulas, it can easily find that the relations of mathematical electric charge
ê and mathematical mass
to electric charge
q and mass
m are
and
where
k is the Coulomb constant, and
G is the gravitational constant.
4. The Structures of Primary Particles
A primary particle looks like a tiny spheroidal balloon with two envelopes (
Figure 1). Each of them is made by an E-W couple or a G-S couple. The envelopes can characterize as:
The whole biding-energy of the coupled waves concentrates on the envelopes.
The macroscopic items of combined field strengths of the two coupled waves are equal on the envelopes. Outside the envelopes, the coupled waves become two independent static fields. But there are no fields inside the envelopes.
The size of the envelopes, which means the size of a primary particle, too, depends on the critical radius of weak or strong interaction.
The two envelopes have the same inherent frequency νin, although this is not mathematically required.
The degree of associated Legendre polynomials j is the same on the two envelopes.
Behaviors of the two envelopes obey the Self-Conjugate Mechanism, which requires that one occupies the surface of and the other must take up , or they are conjugate to each other.
Hence, the biding-energy of a primary particle
Epri can be generally described as
based on equation (17)
Epri is clearly equivalent to the rest mass.
Evidently, the total energy of a primary particle comprises the biding-energy or the rest mass and the energy in static fields.
4.1. An Electron Neutrino
An electron neutrino is composed of two E-W couples with different original spin. In order to explore its structure, these assumptions should be adopted:
Its radius re_ν is equal to the critical radius of weak interaction Rcw.
The charges in equations (10) and (12) are equal and minimal for an electron neutrino, which means when and are the mathematical electric charge and the mathematical weak charge of an electron neutrino.
Integrating equations (10), (12), and the above characters, we have two field equations on envelopes of an electron neutrino
ÊE, e_ν
where '–' only indicates that two E-W couples are attracted to each other on the envelopes.
Based on equation (31) and associated equations (19), (21), and (24), we can easily compute the biding-energy of an electron neutrino
Ee_ν.
Combining the envelopes' characters b. with equations (10), (12), and (32), the fields around an electron neutrino
Êe_ν can be directly written as
4.2. Dark Neutrinos
Two G-S couples make a dark neutrino. However, the strong interaction field has two critical radii, so there are two types of dark neutrinos, and they are named Dark I and Dark II. Similar to
Section 4.1, it is assumed that:
The sizes of Dark I and II are equal to the 1st and 2nd critical radius of strong interaction.
Dark I and II have the same mathematical mass and mathematical strong charge.
The mathematical mass and the mathematical strong charge are equal, i.e., . is minimal.
Replicating the process of the previous section, we have the fields of a Dark I on the envelopes
ÊE, D_νI
where '–' only means that two G-S couples are attracted to each other on the envelopes.
Based on equation (31) and associated equations (20), (22), and (27), we can compute the biding-energy of a Dark I
ED_νI
To get the fields of a Dark II on the envelopes
ÊE, D_νII and the biding-energy of a Dark II
ED_νII, we imitate the last process and have
and
Following the computations of the particle external field in the previous section, we can obtain the fields around a Dark I
ÊD_νI
and the fields around a Dark II
ÊD_νII
Comparing equations (36) with (38) reveals that, as far as measurements, namely energy, are concerned, there is little difference between Dark I and Dark II. However, their volumes are significant differences in the microscopic domain. There should only be Dark IIs in most cases following the principle of energy minimization.
4.3. An Electron or A Positron
Electrons and positrons have the same structure. We will not distinguish significantly between electrons and positrons during the subsequent descriptions and computations. One E-W couple and one G-S couple attract each other to form an electron or a positron, so its structure is the most complex in primary particles. Following the assumptions about dark neutrinos, it is supposed that:
The radius of an electron re equals the critical radius of weak interaction Rcw, although there are three critical radii for weak and strong interactions.
The mathematical electric charge and the mathematical weak charge are equal, i.e., .
The mathematical strong charge are minimal, which means .
Referring to the way we did in the previous sections, we can obtain the field of an electron on the envelopes
ÊE, e and the biding-energy of an electron
Ee.
and
According to the previous assumptions
RcS1 <<
Rcw <
RcS2,
, and
, we have
and
Now we directly give the result for fields around an electron
Êe.
Next, the examination of equations (41) and (42) reveals that the 2
nd critical radius of strong interaction
RcS2 should be the geometric characterization parameter of a G-S envelope rather than the 1
st critical radius of strong interaction
RcS1. It is further assumed that
Rcw and
RcS2 are proportional to the wavelengths of the E-W and the G-S couple, respectively, i.e.,
,
. Thus, we can directly write with the envelopes characters d.
which shows that the gravity speed
cG is faster than the light speed
cL when the above equation compares with equation (42).
5. The Interactions Between Two Primary Particles
Imagine that two static primary particles are initially rested on each other and then separated by a repulsive force for a distance of
l. Particles I and II have the potential and kinetic energy in the separated state (
Figure 2). At the initial state, particles I and II are equivalent in that they share a common center of the sphere (
Figure 2) because there are no fields in envelopes of primary particles.
Based on the law of conservation of energy, the initial energy of particles I and II is equal to the sum of the potential energy and kinetic energy (including magnetic energy for electrons) after their separation. Therefore, referring to equation (17), the potential energy of the two primary particles
can be defined as
where
and
are the external fields of primary particles I and II,
l is the distance between the two particles (
Figure 2),
Rc is the larger of the two particles’ radii, and
converts to kinetic energy (including magnetic energy for electrons).
Equation (47) must still hold certainly when two primary particles move in the opposite mode of
Figure 2, i.e., two rested on each other particles are attracted at the initial distance
l and approach each other until they come together. It is therefore assumed that the Self-Conjugate Mechanism remains between two interacting primary particles. According to existing physics knowledge, the Self-Conjugate Mechanism makes two sets of fields around one particle conjugate to two sets of fields around another depending on the rotation of two particles. In other words, two particles have achieved the Self-Conjugate after they rotate one cycle with angular velocity
ω. There are Self-Conjugate lines in pairs that are further presumed at
for particle I and
for particle II in their respective spherical coordinate systems, i.e., these lines are in the opposite position while
rI =
rII =
r,
θI =
θII =
θ (Refer to
Figure 2). So when we take the
rI-coordinate system of the particle I as a reference (Refer to
Figure 2), in equation (47), the item
, and the item
where
tc is the time of conjugation of two primary particles, and
η = 0, 1, 2, ……,
η. Hence, equation (47) still holds, just with a minus sign difference. Here, the minus sign only indicates that energy is gathering in this process, contrary to equation (47). Later, we continue to use equation (47) without distinguishing whether the energy is spreading or gathering in the motion of two primary particles.
Thus we can follow the results from the previous chapters when there is the potential energy between two primary particles, and the distance of
l remains constant. There are only two conjugate forms between the two particles because each of the two particles comprises two coupled waves. Hence, equation (47) can be translated into
where the subscripts
a to
d, and
w to
z denote electric, gravitational, weak interaction, or strong interaction.
According to Newtonian mechanics, the work done is the same as the potential energy when primary particles I and II move relative to each other. Reversing the Newtonian mechanics definition of work, i.e.,
, the force between two primary particles
is therefore
Two Self-Conjugate primary particles have no initial phase difference in zenith and azimuthal angle under the requirements of the assumption of Self-Conjugation lines. From equation (48), they have potential energy or force when they rotate in the same direction, while they have zero potential energy or rest when they do in opposite directions. Therefore, the force mode has two forms, all right-handed or up rotation and all left-handed or down rotation, which shows each primary particle has spin values of .
5.1. Two Particles of the Same Type
Start by computing the interaction between two electron neutrinos. Combining equations (19), (21), and (49), we have the potential energy
Ee_ν^e_ν
Since two equations (34) have attractive and repulsive states under the Self-Conjugate Mechanism, from equation (50), the force of two electron neutrinos
Fe_ν^e_ν is
where the sign '–' or '+' depends on the Self-Conjugate forms between two electron neutrinos, and the "–" or "+" should be random.
Association equation (49) with equations (20), (22), (27), and (39), we can compute the potential energy between two Dark Is
ED_νI^D_νI
and the force between two Dark Is
FD_νI^D_νI
Duplicating the last process yields the potential energy between two Dark IIs
ED_νII^D_νII
and the force between two Dark IIs
FD_νII^D_νII
The potential energy between two electrons
Ee^e has two forms because an electron is composed of one E-W couple and one G-S couple. Same as electron neutrinos, the two forms should be random and rely on the Self-Conjugate forms between two electrons. Combining equations (19) to (28) and (49), we can compute the two forms of the potential energy. One is
Ee^eI when the two E-W couples are conjugate, and the two G-S couples are conjugate.
Another is
Ee^eII when the two E-W couples are conjugate to the two G-S couples.
Derivation of the last two equations can yield the forces between two electrons in both forms that are
and
plus
in equations (57) to (60).
Comparing equations (57) and (58), (59) and (60) shows that the potential energy and force between two electrons are very different in the two Self-Conjugate forms, with the smaller one close to zero.
5.2. Two Particles of the Different Type
Similar to the last section, this section still starts by computing the interaction of an electron neutrino with another primary particle. Combining equations (23) to (28), (49), and (50), the potential energy between an electron neutrino and a Dark I
Ee_ν^D_νI is
and the force between an electron neutrino and a Dark I
Fe_ν^D_νI is
plus
in above two equations.
Repeating the previous processes gives the potential energy between an electron neutrino and a Dark II
Ee_ν^D_νII
the force between an electron neutrino and a Dark I
Fe_ν^D_νII
the potential energy between an electron neutrino and an electron
Ee_ν^e
and the force between an electron neutrino and an electro
Fe_ν^e