Superluminal Electron-Positron Pair


 It is deduced that when an electron and a positron form a stable structure, the dimensionless speed of either of them, α = v/C, where C is speed of light, satisfies the so-called basic equation of α^2 - b α = 1 = 0, where b is the reciprocal of α, the fine structure constant of a hydrogen atom. One of solutions to the basic equation, the superluminal speed, α 1 = b = 137.036, represents a superluminal pair of electron and positron, in which there is the Lorentz force only while neglecting the Coulomb force between the two particles at an ultrahigh speed. Another solution stands for a positronium with a short-lived life time. The superluminal pair of electron and positron or superluminal electron-positron pair consists of an electron and a positron moving at a superluminal speed of and has a stable quantized energy system with quantized energy of E ̃_n=2m_0e υ ̃_n^2=nhV ̃_n where v = nbC with n being an integer, and is able to radiate and absorb rays of electrons and positrons with ultrahigh energy. The superluminal electron-positron pair may possibly be a particle of dark matter. A divided superluminal electron-positron pair on an energy level n can release electrons and positrons, moving at a superluminal speed of v = nbC, which may possibly be particles of dark energy as well. Therefore, we are led to conclude that there might exist quantized superluminal motions of electrons and positrons in the universe.


Introduction
Einstein believed that speed of light is the upper limit of speed of any particle of matter. This is determined by his theory of relativity. However, the ratio of visible matter to the total matter in the universe is less than 5 % while invisible matter takes more than 95 % of the total [1 -11]. As shown in this article, it is reasonably anticipated that there might be superluminal electrons in the invisible matter -dark matter.
Starting from the stable structure of an electron and a positron led to by the deduced basic equation which both the electron and positron obey. The superluminal solution to the basic equation represents a superluminal electron-positron pair (SEPP) that consists of an electron and a positron moving at a superluminal speed. The SEPP may possibly be a particle of dark matter, whereas energy of the SEPP in a free state may be a particle of dark energy. It is correspondingly concluded tat there possibly exist superluminal motions of electrons and positrons in the universe.
Historically，the substance composed of an electron and a positron under the action of the Coulomb interaction is termed a positronium that is a hydrogen-like atom with a life time of nanoseconds [12][13][14][15][16]. Positronium is an old topic investigated and discussed by many theoretical and experimental authors for several tens years since positron was predicted by P. Dirac from relativistic theory of electrons in 1928 and experimentally discovered by C. D. Anderson in 1932 [17].
However, there was no exception that the hydrogen-like substance is unstable and both of particles are undoubtedly moving at a subluminal speed. It is shown below that a stable structure of substance composed of an electron and a positron is able to be formed, when they approach to each other under a certain condition exerted by the Lorentz force only in comparison with the negligible Coulomb force, and is by no means the positronium. In this substance two particles are whirling at a superluminal speed around a momentary center of mass relative to each other. The substance might be a particle of dark matter.

Basic Equation
Based on the fact that a charged particle in motion undergoes the Coulomb and Lorentz forces produced by another moving particle with charge, as shown in Fig. 1, when an electron and a positron move towards but apart from each other by R at the same speed v in opposite direction, it is possible for the two particles to form a stable structure of matter due to interaction under the certain condition. This is because either of the particles may be treated as the center of a momentary circular motion of the other one as shown in Fig. 2. The necessary and sufficient condition leading to this possibility is the force acted on between two particles is the Lorentz force only, which is true when the speed v is as high as superluminal.
In general, the momentary circular motion of the electron in Fig. 1 is simultaneously exerted by the Coulomb and Lorentz forces, obeying the equation of motion of 2 / = 2 /4 0 2 + 0 2 2 /4 2 . (1) According to the Bohr's quantum assumption of angular momentum, where = 2 ℏ is the Plank constant; 0 and 0 are the dieletric constant and magnetic permulbility, respectively. If R is is reduced from both sides in Eq. (1), the rest is the equation of energy relation.
Introducing the dimensionless speed of electron.
where C is the speed of light in vacuum. On account of 0 0 2 = 1. Based on Eqs. (1)-(3), we are led to an important relation that may be called the basic equation.
This equation describes the law which the dimensionless speed follows. In Eq. (4) b is just the reciprocal of the fine structure constant a of a hydrogen atom (ab = 1) which is given by = 2 0 / 2 = 137.0360 (5) There are two solutions to the basic equation, 1 and 2 , which are found to be reciprocal to each other, i. e. they satisfy that Each of the solutions can be expanded to a power law series as shown below Eqs.
Obviously, the two approximate solutions meet the relations where the subscript 1 represents a superluminal and 2 stands for a subluminal motion, respectively (the same below except that explained otherwise). The approximate subluminal solution � 2 is for a hydrogen-like atom, the so-called positronium under the action of the Coulomb force only and its fine structure constant is the dimensionless speed of electron, i.e., � 2 = = 7.297353× 10 −3 and its speed is equal to � 2 = that is clearly subluminal and much less than the speed of light, which has been investigated for many years [12][13][14][15][16] and we do not intend to discuss it in this article. In contrast, the superluminal approximate solution of � stands for a pair of electron and a positron moving at a superluminal speed. They are associated with each other by the strong Lorentz force and thus called the superluminal electron-positron pair (SEPP).

Parameters of Electron in a Superluminal Motion
When only the Lorentz force is exerted on an electron-positron pair while the Coulomb force is negligible, which is true at as high as a superluminal speed from Eq. (1), the dimensionless speed of two particles in motion described by Eq. (4) is simplified to 2 − α = 0，and hereby, the approximate solution is so simple that � 1 = = ，According to Eq.(10), the speed � 1 =bC = constant. It means that the electron and positron have been coupled to be one entity due to the Lorentz force, making a circular motion at the superluminal speed of � 1 =bC around the momentary center of mass of the other particle, resulting in a SEPP. Based on the reciprocal relation, Eq. (6), between subluminal and superluminal speeds and the fact that the speed of light C is a singularity of mass for a subluminal motion from Einsteins relativity, it is reasonably believed that for a superluminal motion the speed of light C is also the singularity of mass, i.e., the mass approaches to infinity when speed tends to the speed of light C and the superluminal mass-speed relation can thus be written as where α > 1. Eq. (14) is applicable to all the superluminal solutions, including those on energy levels, which are reciprocal to the subluminal solutions, and also indicates that when a particle is in a superluminal motion of α, its mass is equal to the one in the subluminal motion of 1/α. where Eq. (15-3) is obtained from Bohrs quantization of angular momentum of the electron or positron in its momentary circular motion, but it is also easy to get from Eq. (1) in which drop off the the last term -the contribution of the Lorentz force on the right side hand of the equation. If so, it is verified that the Bohr quantization condition of angular momentum remains valid for a superluminal motion; � 1, is the sum of kinetic and potential energies of electron. The absolute value of potential energy equals double kinetic energy and � 1, is therefore negative, implying that the potential is prevailing. The double absolute value of � 1, is the binding energy of electron and positron, which is greater than the binding energy of atomic nuclei by four orders of magnitude.
Moreover, according to Eq. (8), all the perturbative terms expressed in terms of b show that the existence of the Coulomb force is to vary parameters of electron slightly. Therefore, in analogy to the fine structure constant a of a hydrogen atom, its reciprocal b = 1/a can be viewed as the fine structure constant of the SEPP as well as the dimensionless speed of electron in motion ( 1 α = b).

Structure Picture of a Superluminal Electron-Positron Pair
Assuming that the charge of an electron is uniformly and symmetrically distributed over a sphere, the mass energy of the electron itself, 0 2 = 2 /8 ε 0 = , where W e is the energy of the electrostatic field of the electron with a radius of =1.408971x10 -15 m, which is calculated from the above relation. Noting that in the superluminal approximate solution the radii of R e and are identical, compared with Eq. (15-3), it is obtained that located on the two ends of the connecting line � 1 and make a momentary circular motion with a radius of the connecting line � 1 around the center of mass of the other . This is a circular motion of two point masses connected to be like a dumbbell due to the Lorentz force.

Stability of a Superluminal Electron-Positron Pair
The electron and positron in a SEPP is apart from each other by � 1 . We are now discussing the stability of the solution near � 1 . At an arbitrary place R near � 1 , both the electron and positron have kinetic and potential energies. Nevertheless, the potential energy has two parts: the Coulomb potential and the equivalent one from the Lorentz force, as can be seen below. Neglecting the slight change of speed � 1 =bC and noting that μ 0 ε 0 c 2 = 1, either of the electron and positron is exerted by the Lorentz force 0 2 � 1 2 /4 2 = 2 2 /4 0 2 that is equal to a force produced by the equivalent electrostatic potential energy − 2 2 /4 0 ; Thus, the total energy of two particles in the SEPP is given by � 1, ( ) = 1 + 1 = 2 /(4 2 0 2 ) − 2 2 2 /4 0 .
Making a derivative with respect to R and letting it be zero, the location at which energy reaches its extreme value is given by Since the second-order derivative of E 1,e (R) at � 1 is greater than 0, the potential energy reaches its minimum of � 1, � � 1 � = − 2 = − 0 � 1 2 , i. e., the potential energy of the electron and positron possess the minimum value when they are apart from each other by � 1 and thus the SEPP is stable.
What is the effect of the Coulomb force neglected? According to Eq. (9), neglecting all the terms of orders higher than 4 (inclusive of), the increase of speed is ∆ � 2 = 3 . It means that the role of the Lorentz force is to expedite particles in the subluminal approximate solution. In contrast, in the superluminal approximate solution the existence of the Coulomb force arouses a change of speed,but it is a decrease. Because two approximate solutions satisfy 1 � 2 � = 2 , i. e., where the Lorentz force is approximately expressed by 1, = 0 2 � 1, 2 /4 � 1, 2 = 2 2 /4 0 � 1, 2 when it is in motion at the speed of � 1 ，and the Coulomb force is given by 1, = − 2 /4 0 � 1, 2 with the negative sign representing that Coulomb force is outwards, according to the definition of the force sign in this case. Based on Eqs. (18-1) and (18-2), there is ℏ � 1, / � 1, 2 = 2 2 /4 0 � 1, 2 − 2 /4 0 � 1, 2 ，the solution of � 1, can be obtained and the solution of � 1, is deduced from Eq.
Therefore, we are led to conclude that under the influence of the Coulomb force in the superluminal approximate solution the speed of either electron or positron in the SEPP is lightly slowed down, getting the distance between them a little bit larger. This implies that the minimum of speed is � 1, =(1− 2 )� 1 and the maximum of distance is � 1 � 1, =(1+ 2 ) � 1 ，which is equivalent to the spin radius of either of the electron and positron being � , = (1 + 2 ) � .
Obviously, the existence of the Coulomb force in a SEPP not only does not squeeze the electron and positron, arousing their annihilation, but favors the stability of the solution only if the Coulomb force does not exceed a certain value.

Equations and Solutions
In general, Eqs.

Superluminal Approximate Solutions and Discussion on Energy Levels
When there is the Lorentz force only, the basic equation (21) where = 1,2,3 ⋯ ⋯. According (23-4), � = 2 � 1 , i.e. the energy on the n-th level is n 2 times the energy of the ground state: the higher the level, the larger the energy. Because two constituent systems of potential energy are stable, the system of quantized energy of the charged quantum pair is stable as well. The energy on an energy level may also be written as � = , 2 ，where , = 2 2 2 0 is the equivalent energy mass of the quantum pair on the n-th level; ( For example, for k = 2 and l = 1, if the radiation is a ray of electrons or positrons, the frequency of e =3 � 1 = 3.480446x10 24 Hz，reaching 10 24 z of the order of magnitude , which is minimum of the emitted energy.

Possibility of Existence of Quantized Superluminal Motion of Electrons or Positrons
Why is it possible for an electron or a positron to make a superluminal motion at a speed of (2) the binding force of an electron-positron pair is the electrostatic attractive force of 2 2 /4 0 2 , which is unable to make the electron or positron to move at a superluminal speed.
Because the electrostatic attractive force appears to be the Lorentz force of 2 2 /4 0 2 = 0 2 � 1 2 /4 2 (due to 0 = 1/ 0 2 ) that acts on the electron and positron, leading them to move at a superluminal speed of � 1 = . Although the defined speed of light plays an important role, yet it affects rather bC than arbitrary superluminal speed.
(3)Why must the superluminal speed of an electron or a positron be bC or nbC, instead of an arbitrary value?
As the charge of an electron or a positron could be assumed to be uniformly and symmetrically