Theory Of The Origin Of The Electron

Theory of the Origin of the Electron Z.Todorovic Faculty of Sciences, Prishtina University, Serbia Quantum-mechanical theory of the possible origin of the electron has been derived with reliable working formulae, which describe the concrete physical characteristics of the electron. PACS: Electron, virtual electron, photon

Since the discovery of the electron dated in 1897 by J. J. Thomson, the first experimentally result was the ratio of the electric charge of an electron to its mass, which we shall use in our consideration. Physicists solve numerous problems in the area of experimental and theoretical physics. considering electron as a point-particle with electric charge e, Einstein’s energy mc2, impulse p = mv , Bohr΄s magnetic moment ( µ B = e / 2m0 ) and spin of /2 [1,2] . The goal of this consideration is to explain the possible origin of the electron and derive reliable working formulae, which can be applied to concrete physical characteristics of the electron. We postulate that a creation of the bare electron represents a process concerning a photon and virtual electron, where virtual electron is a part of the virtual pair: electron-positron. We are going to construct a quantum-mechanical theory, according to measurements by Purdue University physicist’s et.al [ 3] . According to experimental data they stated following [ 3] : “An electron may not be exactly what we’ve thought it was since its discovery 100 years ago…the tiny particle may not be a simple negative point charge, as scientists often describe it. All charges (bare electrons) are surrounded by clouds of virtual photons, which spend part of their existence dissociated into fermion-antifermion pairs.Surrounding the electron’s core is a fuzzy “cloud” of virtual particles, which wink in and out of existence in pairs. One particle in the pair is positively charged, the other negatively charged. The cloud consists of polarized pairs, which essentially cancel each other out so that they do not “add” any net electric charge to the electron. The cloud plays a key role in how we perceive the electromagnetic force from the electron.” On the basis of such experimentally founded picture of the electron, our obligation is to form quantum-mechanical theory, which would be to describe that experimentally facts concerning all physical characteristics of the electron. The crucial question is: where is origin of the electron charge from? The answer on that question is in the following consideration. We think that photons with energy of W ≥ 2m0c 2 carries the charges and appropriate masses for electron and positron within the transverse components, but in an inactive form. Particularly we point out on the Thomson΄s reciprocal ratio ( m0 / e ) , which is contained within these transverse components. We emphasize that formulae for B and E contain all needed information relate to the latent new possible particle into aspect of the electron and positron.Namelly, these formulae in detail read: B = ( m0 / e )ω 0 = ( m0 / e ) m0 c 2 / ( / 2) ; E = ( m0 / e ) c m0 c 2 / ( / 2 ) , or

[(

2

(

+

)]

[(

[(

)]

)]

[(

)]

) [(

)]

what is the same B = ( 2m0 / 2e ) 2 m0c / 2( / 2 ) ; E = ( 2m0 / 2e ) c 2m0 c / 2( / 2 ) . As well, we ephasize, that at the start, as the first step of the disintegration process of that photon into pair electron-positron, previously appear two subphotons, with the appropriate transverse components: B = ( m0 / e ) m0 c 2 / ( / 2 ) ; E − = m0 / e − c m0 c 2 / ( / 2 ) and

[(

)]

) [(

2

[(

)]

)]

(

B = ( m0 / e ) m0c / ( / 2 ) ; E = m0 / e c m0c / ( / 2 ) . We see that these transverse components in the form of the electric fields E  predict by themselves as possible sources of the charges e − or e + , mass at the standstill m0 , 2

+

2

energy m0 c 2 at the standstill, spin

/ 2 of that new particle, and everything that probably, because of the speed c contained within formula of E , should rotate inside the velocity interval ( 0, c ) within of some sort area. From the proces, by which photon transforms intoa pair electron-positron, whose minimal energy amounts to 2m0 c 2 = ω 0 , appear two equal parts of the subphotons, each with energiy m0c 2 = ( / 2 )ω 0 ready for a creation two new particles at the standstill;one particle of them would be an electron and the other a positron. From that energetic relation one finds a frequency ω 0 = 2m0 c 2 / = c / X 0 . ( m0 -mass of the electron at the state of rest; c- the light velocity in the vacuum). We have the following case:at the beginning moment one photon separates into two equal halves of subphotons;within a certain subphoton releases a charge e − from one of its transverse field components, and within the another subphoton releases a charge e + from its appropriate transverse field omponent.The released charge e − and e − from the virtual electron, where the virtual electron is the part of the virtual pair:fermion- antifermion(electron-positron),  together take part in the interaction with the magnetic field B of the other transversal component of the first subphoton. As well , on the same way, . the releasing charge e + and e + from the virtual positron, where the virtual positron is the part of the virtual pair:fermion-antifermion (electron-positron), together take part in the interaction with  the magnetic field B of the other transversal component of the second subphoton.. We particularly underline that the

(

)

net electric charge of the integrated photon 2m0 c 2 = ω 0 is zero, and that state is the same for the sum of both

(

)

subphotons, because e − and e + cancel each other out e − + e + = 0 .The masses at rest of the integrated photon and

separated subphotons are equal zero.The released charges e − and e + do not mean the whole charged particles like electron or positron; they mean simple the bare charges only.    As we know, the electromagnetic impulse of the photon is definet by relation: π = (1 / 4π ) E ∧ H dV

∫(

)

At the moment of its disintegration, the mentioned photon splits into two subphotons; momentary stops to move with     previous rectlinear direction;mutuality between E and H ,given by relation E ∧ H , violates by itself .The  electromagnetic impulse π at that moment becomes zero, and the consequences of that facts are:     a) A vector product of transverse components is: zero ( E ∧ H = 0 );so, from this relation follows that E and B must be colinear (parallel or anti-parallel)  b)The transverse components of electrical fields E by themselves are placed in a direction along z-axis  c) The bare charges e − and e + release by themselves from E , and a character of their motion predicts possible  circular motion.around former transverse components B . In our further consideration we consider that mutual position between both initial direction of the photon motion and the virtual electron΄s spin is by right angle. The proces for forming bare electron will be described in details. At the initial (starting) moment separated subphoton behaves like a particle with mass m f = 0 , spin / 2 with releasing charge e − , which is in the interaction with  electromagnetic vector potential A , gives impulse

( )

( )

  m    1   p f = e − A = e−  B ∧ r  = − 0 (ω0 ∧ r ) 2 2      ∇A = 0; B = Bez

(1)

 . As well, charge of the virtual electron e − is in the interaction with the same field A , and its impulse is:   m   pve = e − A = − 0 ( ω 0 ∧ r ) 2

( )

(2)

Virtual electron at the standstill possesses mas m0 . As well, the initial spin of the subphoton transform on the same   way, like B , and it becomes parallel with both B .and the virtual electron΄s spin We shall regard the simplest case when the spin of a subphoton is opposite with respect to the virtual electron spin It is clear that total mass of both particles is m∗ = m f + m0 = m0 .(One particle of them is a subphoton and other is athe virtual electron) So, we think that the motion of such pair (consistedf of separated subphoton and virtual electron) charged by 2e − , we might describe, in the cilindrical coordinate system, by the following Schrődinger's equation [ 4,5,7] :

( )

i

[ (

   p − p f + pve ∂Ψ = EΨ = ∂t 2m∗

)]

2

(3)

Ψ

∂ ∂ ∂  1     , ; A = B ∧ r ; ∇A = 0 . where are: p = −i ∇; ∇ =  , 2  ∂r r∂ϕ ∂z  The solution of equation (3) is the wave function Ψ in the following form:  r2  exp − − iω 0t  2  4X0 ; Ψ ( r,ϕ , z, t ) = R ( r ) Φ (ϕ ) Ζ( z ) exp( − iω 0t ) = 4πr0 X 0  r2   exp − 4 X 02  ∞ ∗  1  R( r ) = ; R Rrdr = 1; Φ (ϕ ) =  X0  2π 0

∫

  e2  X 0 =  ; r0 = 2  2m0 c 4πε 0m0c   ( 0 ≤ r < ∞;0 ≤ ϕ ≤ 2π ;0 ≤ z ≤ 2r0 )

∫∫∫ Ψ Ψrdrdϕdz = 1; ∗

2π  1  ; Φ ∗Φ dϕ = 1; Ζ( z ) =   2r  0 0 

∫

 2 r0 ∗  :; Ζ Ζdz = 1   0

∫

(4)

The energy of the system:subphoton-virtual electron at the standstill is defined by the following relations: i

(

)

∂Ψ = ( ω 0 ) Ψ = 2m0 c 2 Ψ ∂t W =

∫∫∫ ( )

 Ψ ∗ HΨ rdrdϕdz =

∫∫∫

2      ∂2 1 ∂  1  2 +  + m0ω02 r 2  Ψ rdrdϕdz = 2m0 c 2 Ψ∗ − r ∂r  2   2m0  ∂r  

(5)

We remind that the law of energy holds, because m0 c 2 was the energy of the subphoton which is taken part in the reaction, and the same value belongs to the virtual electron. The reason why we define the transverse component by formula B = ( m0 / e )ω 0 , and that it is naturally built within othe subphoton, we can see regarding density of the electromagnetic energy of that subphoton 1 1 2 1 2 m0 c 2 m0 c 2  1 ϖ f = ε0E 2 + B = B = =  ε 0 µ0 = 2  2 2 2 µ0 µ0 4πr0 X 0 V0  c 

(6)

A parameter V0 from (6) transforms to the probability current density for the considered system (3,4) Ψ ∗Ψ =

 r2  1  exp − 2  V0  2X0 

(7)

A relation (6) describe a homogenous distribution of the electromagnetic energy density , until relation (7) predicts a change such distribution.into unhomogenous one.Such disturbance of the density means possible appearance a primary particle by a transformation into new one. Separated electromagnetic subphoton becomes localized within the space embraced by Compton wave length,defined by relation (7).Here there is no any rectlinear motion in accordance with the former photon΄s motion in one direction;now, its former rectlinear motion transforms into circular motion.So, ;the bare released charge e − becomes localized within the space embraced by Compton wave length, and smeared by the Gauss΄s distribution.. The subphoton transforming itself into new particle-bare electron gets, in that case, definite mass m0 , and apparently, it is at the standstill with the same distribution as e − . Our aim is to calculate, by using Ψ function (4), physical characteristics of new particle- bare electron.First of all, we may look in a rotor of the magnetic field of a pair which is charged by 2e −

( )

 ∇ ∧ H 2e −

1 er r ∂ = ∂r 0

 eϕ ∂ ∂ϕ 0

1 ez r ∂ ∂ = − H 2e− ∂z  ∂r H 2e −

(

)

  2 eA ∗  eϕ = J 2 e − = ( − 2e ) m Ψ Ψ;  0

(8)

 1      m  A = B ∧ r ; ∇A = 0; B = Bez = 0 ω 0 ez 2 e Using the average value of the magnetic field H 2 e − ,defined by relation (8), we may compute the effective value of the electrical current of the bare electron on the basis of known law of the electrodynamics (Ampere΄s law): ∞

I e− =

∫

  2 r0   j e − dσ = H 2 e − e z dze z

∫

0

0



2 r0 ∞

=

∫ ∫ R ( H 0

0

2e−

 R rdr dz = ( − e )ν 0 

)

  ω c e2 ν 0 = 0 ; ω 0 =  ; r = 0 2   2π X0 4πε 0 m0 c   where r0 is well known physical quantity as classical radius of electron, and electrical current density of the bare electron is equal to:

(9)

 r2  exp − 2  2 eA  2X 0 je − = ( − e ) m0 4πr0 X 02   dσ = ( drdz ) eϕ

   

(10)

By the localizing process the rectlinear motion of a photon transforms into the circular motion, and a spin, in the case  of charged particle,, is defined by rotor of the electromagnetic vector potential A2 e .So, for a pair which is charged by

(2e ) (bare electron-virtual electron) we have: −

1 er r ∂ = ∂r 0

 ∇ ∧ A2 e −

(

)

1 ez r ∂ ∂z 0

 eϕ ∂ ∂ϕ rA2 e −

(11)

  1 ∂ rA2 e − ez = µ0 H 2 e − ez r ∂r If we calculate the quantum-mechanical average value of rA2 e − on the basis of a relationship (11), one obtains the following formula: =

(

∫ [(

)

)]

∞

rA2 e − = R rA2 e − R rdr = 0

(12)

2e

Multiplaying formula (12) by e , the spin of the bare electron one obtains immediatelly Se = e rA2 e − =

(13)

2

which coincides with the value of the subphoton΄s former spin Formula (6) tell us how rectlinear motion of the photon transforms into a circular motion by keeping the same spin΄s value. The photons, naturally, are not having any physical characteristic like a magnetic flux.Since the subphoton change the nature of its motion, as it happens in our case, it may possesses such physical quantity.If we define magnetic flux by the circulation a quantity determined by (12) by a angle of 2π

∫ [ rA

2π

Φ=

2e −

0

]dϕ = π e

(14)

one obtains result which coincides with the quantum of magnetic flux given by BCS Theory [ 6] As well, we may obtain flux (14) as a product of the average value of the magnetic field

µ 0 H 2 e − through the average

value of the area of a current contour (loop) Φ = µ 0 H 2 e − πr 2 = π

(15)

e

where the average value of the contour (loop) area is defined by the following formula: ∞

[( ) ]

πr 2 = ∫ R πr 2 R rdr = π

(

2X0

)

2

(16)

0

Magnetic moment of the bare electron may be calculated as a product of the effective electrical current of the contour (9), which represents one feature of the bare electron, and the average contour (loop) area encircled by that current (16):

[(

µ = I e − πr 2 = [ ( − e )ν 0 ] π 2 X 02

)]

(17) e = µB 2m0 As we know, within a photon there is no any physical quantity like a moment of inertia.Reason is simple;moment of inertia with zero photon΄s mnass at rest gives zeroth value for that moment.However, if we apply the definition of the moment of inertia from classical physics =−

Jm =

∫∫∫ r dm 2

(18)

and introduce a hypothesis that transformed subphoton into new particle possesses mass at rest m0 , then for its differential mass we may write: dm = m0 Ψ ∗ ΨdV

(19)

Inserting dm into (18) moment inertia becomes: Jm =

∫∫∫ r ( m Ψ Ψrdrdϕdz ) = m ( 2

0

∗

0

2X0

)

2

(20)

We see that the value of moment of inertia is in a correlation with the average value of the contour (loop) radius defined by (16). And finally , using the correlation between the energy and the moment of inertia, as per laws of classical physics, we obtain:: W =

1 J mω 02 = m0c 2 2

(21)

The being of energy (21) means the being moment of inertia (20), therefore exists a mass m0 of a new particle produced by the transformation process. The result of formula (21) means that starting zero mass at rest of the subphoton transforms, by the localizing process of that subphoton, into definite mass at rest m 0 , as an experienced physical quantity of the electron. Conclusion There is in physics an experienced fact;at the moment when the nuclear core decays emits an electron into surroundend space.According to laws of quantum mechanic before the core΄s decay electron was not a belomgings as a part of that core, already it is appeared at the moment of the core΄s decay, as a consequence of a process of the interaction with the physical vacuum. As well, similar phenomenon appears in a case of the photon΄s decay.. Within a photon there is not any electrons.But, in the process of its disintegration it disunites into two equal halves with the same energy; one half of them transforms into electron and other into positron.It means that both halves are not absolutely identical. At the moment of the photon disintegration one subphoton becomes polarized by e − , the other one by e + Before the photon΄s disintegration, ,photon was not charged by any quantity of the electricity, already it carrys within itself two transverse components,   mutually placed by right angle E⊥B .At the moment of the photon disintegration, each subphoton holds up both transverse components, but are not any more by rigt angle, already are mutually colinear (parallel, or anti-parallel) ;one of that components, within subphoton, becomes potentially source of a polarization e  for the building future particle. As well, according to mentioned Ampere΄s law (3a), it seems, that the applied physical concept asociated with the classical radius of the electron has not any similarity with the mathematical definition of radius The obtained results to point out why the virtual electron must be taken into the consideration. The answer is in the fact that the electron may not exist in the real world as an isolated particle, already it is always in the link with its virtual electron; virtual electron is the part of the virtual pair electron-positron (fermion-antifermion pair), and electron permanently followed by ithat pair as its »shadow«. The obtained results for spin, magnetic moment, quantum flux and rest energy of electron are in agreement with the experimental data, so the conclusions of this work may be as follows:

(

a)

)

Needed and sufficient condition for disintegration of a photon into a pair electron-positron in spite of energy ≥ 2m0c 2 , is achievement of the zeroth value of its electromagnetic impulse, which is satisfied if the vector product

  of its transverse components becomes zero; the violation E ∧ H = 0 is crucial for the process of the transformation of a photon into new two particle (electron-positron).    b) From the first subphoton with energy m0 c 2 , E − = c ( m0 ) / e − m0 c 2 / ( / 2 ) ; B = ( m0 ) / ( e ) m0c 2 / ( / 2 ) ez = rotA ;

{

−

[

( )(

)

]

[

(

)

]

}

−

releases e from E − , and it (this e means negative polarization or bare charge only) gets in touch with the  charge e − of the virtual electron, and then they together step in a process of the interaction with A .The consequence of such interaction is that the released e − from E − transforms into bare electron at rest which is charged by e − with the Gauss’s distribution which rotates around z-axis, along which is directed magnetic field  B. c)

Simultaneously with the first subphoton, the second subphoton with energy m0 c 2 ,    E + = c ( m0 ) / e + m0 c 2 / ( / 2 ) ; B = ( m0 ) / ( e ) m0 c 2 / ( / 2 ) ez = rotA as well, with the similar process, transforms into positron.

{

[

( )(

)

]

[

(

)

]

}

d) The appearance of the electron mass m0 (formulae (18-21)) represents a consequence of the localizing effect of a photon; when its rectilinear motion with the light velocity stops and then transforms into new circular motion in a certain area. e)

Electron in reality is not one isolated-bare particle only; it may not be a simple negative point charge; bare charge is always followed by virtual electron, where virtual electron is the part of the virtual pair fermion- antifermion from the “cloud”.

f)

Electron may not be an elementary particle, because of the assertion defined by e), already it performs like a complex particle: That fact means that bare electron as a core is surrounded by a cloud of the virtual particles.

g) Electron performs like a current contour (current loop;formulae (15) and (17) f) The “cloud”, consisted of virtual particles in the pairs (fermion-antifermion) which surrounds the electron's core (bare electron) may be to play a key role in how we perceive the electromagnetic force from the electron.

References

[1] H. Dehmelt, Rev.Mod.Phys, 63(1990)525 [ 2] M. Kurepa and D. Filipović, Electron”, p.347, Serbia Academy of Sciences, Belgrade (2000) [ 3] I.Levine et.al. Phys.Rev.Lett. Vol.78, No.3. 424-427 (1997). Purdue news, February 1997. “Research sheds light on electron's structure”. 4 Z Todorović and D.Todorović, Bulg. J. Phys. vol.27. No.1. p.89 (2000)

[ ] [5] Z.Todorović Doctoral Thesis, University of Pristine (1998) [ 6] J.Barden, I.N. Cooper and J.R.Schrieffer, Phys.Rev. Vol.108.No.5, p.1175 (1957)

[ 7] Z.Todorović, University Thought, vol.3, No.2, 91-99 1996(1997).