Fermions: Dirac Internal Exchange Frequencies
Doron Kwiatorcid
Mazkeret Batyia, Israel.
DOI: 10.4236/jhepgc.2025.111001   PDF    HTML   XML   27 Downloads   122 Views  

Abstract

Using real fields instead of complex ones, it was recently claimed, that all fermions are made of pairs of coupled fields (strings) with an internal tension related to mutual attraction forces, related to Planck’s constant. The solution to Dirac equation gives four, real, 2-vectors solutions ψ 1 =( U 1 D 1 ) ψ 2 =( U 2 D 2 ) ψ 3 =( U 3 D 3 ) ψ 4 =( U 4 D 4 ) where ( ψ 1 , ψ 4 ) are coupled via linear combinations to yield spin-up and spin-down fermions. Likewise, ( ψ 2 , ψ 3 ) are coupled via linear combinations to represent spin-up and spin-down anti-fermions. Here, a deeper investigation of the free fermion internal frequency is discussed, hinting to an exchange interaction between the two components of which a fermion is made of. An upper limit estimate is given to the strength of this interaction.

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Kwiat, D. (2025) Fermions: Dirac Internal Exchange Frequencies. Journal of High Energy Physics, Gravitation and Cosmology, 11, 1-7. doi: 10.4236/jhepgc.2025.111001.

1. Dirac Equation with Real Wave Functions

The relativistic Dirac Equation, describing a free Fermion of mass m is given by [1] [2]:

( i γ μ μ mc )Ψ=0 (1)

One may separate the Dirac operator i γ μ μ mc and the complex wave function Ψ into their real and imaginary parts [3] [4]

Ψ=( ψ 1 ψ 2 ψ 3 ψ 4 )

With ψ 1 =( U 1 D 1 ) ψ 2 =( U 2 D 2 ) ψ 3 =( U 3 D 3 ) ψ 4 =( U 4 D 4 ) all real components.

As will be shown, these 4 components represent two fermions and two anti-fermions. Each pair is the source for two opposing spin states.

After some work and boosting to a system moving with the particle along the +x axis ( p y , p z =0 y , z =0 ),

the Dirac equations take the form:

m c 2 Ψ 1 =+ t Ψ 4 c σ x x Ψ 4 (2)

m c 2 Ψ 4 = t Ψ 1 c σ x x Ψ 1 (3)

m c 2 Ψ 2 =+ t Ψ 3 +c σ x x Ψ 3 (4)

m c 2 Ψ 3 = t Ψ 2 +c σ x x Ψ 2 (5)

This shows, that, Ψ 1 is coupled with Ψ 4 and Ψ 2 is coupled with Ψ 3 . As will be shown later, linear combinations of these represent spin-up and spin-down fermion and anti-fermion.

2. Solution with 8 Real Components

Each Ψ i is a 2-vector with real components. Thus, the Dirac Equation is actually 8 equations of real components with coupled pairs ( Ψ 1 , Ψ 4 ) , and ( Ψ 2 , Ψ 3 ) .

Applying a time derivative to the first equation of each pair and using the second component of each pair, leads to:

( ( m c 2 ) 2 + t 2 ) U 1 +c x t U 1 = m c 3 x U 4 (6)

( ( m c 2 ) 2 + t 2 ) D 1 c x t D 1 =+ m c 3 x D 4 (7)

( ( m c 2 ) 2 + t 2 ) U 4 c x t U 4 =+ m c 3 x U 1 (8)

( ( m c 2 ) 2 + t 2 ) D 4 +c x t D 4 = m c 3 x D 1 (9)

( ( m c 2 ) 2 + t 2 ) U 2 +c x t U 2 = m c 3 x U 3 (10)

( ( m c 2 ) 2 + t 2 ) D 2 c x t D 2 =+ m c 3 x D 3 (11)

( ( m c 2 ) 2 + t 2 ) U 3 +c x t U 3 = m c 3 x U 2 (12)

( ( m c 2 ) 2 + t 2 ) D 3 c x t D 3 =+ m c 3 x D 2   (13)

These 8 real components equations demonstrate the existence of coupled pairs: ( U 1 U 4 ), ( D 1 D 4 ), ( U 2 U 3 ) and ( D 2 D 3 ).

These equations show, that every fermion is composed of 4 real fields which are coupled in a yet to be explored manner.

The solutions are described in the following:

Ψ 1 =( U 1 D 1 )=( cos( px( ω 0 +cp )t ) sin( px( ω 0 cp )t ) )

Ψ 4 =( U 4 D 4 )=( sin( px+( ω 0 +cp )t ) cos( px+( ω 0 cp )t ) )

Ψ 2 =( U 2 D 2 )=( cos( px( ω 0 +cp )t ) sin( px( ω 0 cp )t ) )

Ψ 3 =( U 3 D 3 )=( sin( px+( ω 0 cp )t ) cos( px+( ω 0 +cp )t ) ) 

Here p p x , where p x is the x component of the momentum.

In order to better understand the nature of the Equations (6)-(13), one can boost the coordinates to the fermion’s rest frame along a random axis, say the x-axis.

When boosted to the fermion’s rest frame, where p x =0 , one can omit all x terms ( x =0 ).

Equations (2)-(5) then read (for all four wave functions):

[ t 2 + ( m c 2 ) 2 ] Ψ i =0 (14)

And likewise for all 8 components U i and D i :

[ t 2 + ( m c 2 ) 2 ] U i =0 (15)

[ t 2 + ( m c 2 ) 2 ] D i =0 (16)

Solving this equation by setting Ψ=cos( ωt ) or Ψ=sin( ωt ) shows that all components of the fermion at its rest frame, are oscillating at a rate given by ω 0 = m c 2 .

Inserting the values in MKS units one obtains, for all fermions ω 0 =1.36554× 10 32 ×m .

For an electron m=0.511 MeV/c2 and thus ω 0 7.76× 10 11 GHz. (=0.512 MeV). For a proton, m=939 MeV/c2 and ω 0 7.296× 10 6 GHz and for the τ quark m=173 GeV/c2 and ω 0  2.63× 10 17 GHz (Table 1).

Table 1. Masses and equivalent electromagnetic range Properties of electron, proton and tau-quark.

particle

ω 0

EM range

Mass

electron

7.77 × 1011 GHz

Soft X-ray

0.512 MeV

Proton

14.28 × 1014 GHz

Gamma

939 MeV

τ Quark

2.63 × 1017 GHz

Gamma

173 GeV

3. A Model for the Fermion Internal Structure

Previously [3] it was shown that the description of Quantum Mechanics can be done by use of real fields and operators. The Schrodinger equation was then shown to be describing a double string model with particle exchange between the two strings. It was also shown, that all known quantum mechanical unique phenomena, such as interference and entanglement, can be explained based on real fields approach [3] [4].

The same argumentation was applied to Dirac equation, demonstrating a double string coupling.

Dirac equation shows us, that the internal frequency of each component of the fermion equals the rest mass of that fermion.

Even though a fermion is assumed to be made of 2 components, U and D, each one inherits its oscillation frequency ω 0 from the same total fermion mass m, and not from U's or D's individual masses (if any).

As shown earlier, each Ψ i is a 2-vector with real components.

We will assume next that for a reference frame near static with respect to the fermion

( ( m c 2 ) 2 + t 2 ) U 1 =ε U 1 (17)

( ( m c 2 ) 2 + t 2 ) D 1 =ε D 1 (18)

even in the non-boosted system. Same is true for

( ( m c 2 ) 2 + t 2 ) U 4 =δ U 4 (19)

( ( m c 2 ) 2 + t 2 ) D 4 =δ D 4 (20)

Therefore,

c x t U 1 = m c 3 x U 4 +ε U 1 (21)

c x t U 4 =+ m c 3 x U 1 +ε U 4 (22)

c x t D 1 =+ m c 3 x D 4 +δ D 1 (23)

c x t D 4 = m c 3 x D 1 +δ D 4 (24)

This makes the components U 1 coupled to U 4 and same for D 1 to D 4 .

One may write

t U 1 = m c 2 U 4 + ε c U 1 dx m c 2 U 4 + ε c Δx U 1 (25)

t U 4 = m c 2 U 1 + ε c U 4 dx m c 2 U 1 + ε c Δx U 4 (26)

t D 1 = m c 2 D 4 + ε c D 1 dx m c 2 D 4 + ε c Δx D 1 (27)

t D 4 = m c 2 D 1 + ε c D 4 dx m c 2 D 1 + ε c Δx D 4 (28)

where the integration in x is over the negligible size of the fermion, and where represents the average amplitude over that extent.

This allows us to neglect the extra terms on the r.h.s of the equations and obtain, even in a non-stationary frame of reference (relative to the fermion):

t U 1 = m c 2 U 4 =0 (29)

t U 4 = m c 2 U 1 =0 (30)

t D 1 = m c 2 D 4 =0 (31)

t U 4 = m c 2 D 1 =0 (32)

These equations indicate to a coupling by exchange mechanism between U 1 and U 4 . When looking at the change in amplitude of U 1 we get:

Δ U 1 = t U 1 Δt= m c 2 U 4 Δt= ω 0 U 4 Δt (33)

Internal fluctuations of U 1 and U 4 at rate ω 0 indicate an energy exchange between the two components. Same argument is valid for the internal fluctuations of D 1 and D 4

So, during the exchange, U 1 gains m c 2 in amplitude per second if U 4 is negative, and loses amplitude if U 4 is positive.

Recall that they oscillate at a rate of 1.24 × 1011 GHz for an electron. This is a picture of two adjacent strings, oscillating in anti-phase, at said frequency (see Figure 1). This ant-phase is a must, in order to keep the particle's momentum zero in the non-x (perpendicular) direction.

We assume that at most, the whole mass participates in the kinetic energy transfer mechanism. This assumption puts an upper limit on E k , the kinetic energy transfer.

An upper limit estimate of the kinetic energy transfer between U 1 and U 4 in the electron (change in amplitude per second is velocity) will then be given by:

E k < 1 2 m ( t U 1 ) 2 = 1 2 m 3 2 U 4 2 =2.12× 10 4 eV× U 4 2

If U 4 =cos( ω 0 t ) then the time average of U 4 2 is U 4 2 = 1 2 ω 0 .

So, the average kinetic energy transfer between the coupled strings in the electron, E k is less than 2.73 × 1025 eV/sec. For a proton the upper limit is 9.2 × 1019 eV/sec and 3.12 × 1014 eV/sec for the tau quark. All cases have values far below our detection capabilities.

Same arguments hold for the kinetic energy transfer between D 1 and D 4 .

Figure 1. Energy transfer model based on anti-phase oscillations between the two components U 1 and U 4 of a fermion.

The interaction may be due to some yet unknown particles, where possible candidates may well be those suggested by Harrari [5] and Hubsch [6].

4. Conclusions

All fermions are made of pairs of coupled fields (strings) with an internal tension related to mutual attraction forces, affected by Planck’s constant. The solution to Dirac equation gives rise to four, real, 2-vector fields ψ 1 =( U 1 D 1 ) , ψ 2 =( U 2 D 2 ) , ψ 3 =( U 3 D 3 ) , and ψ 4 =( U 4 D 4 ) where ( ψ 1 , ψ 4 ) are coupled via linear combinations to yield spin-up and spin-down fermions. Likewise, ( ψ 2 , ψ 3 ) are coupled via linear combinations to represent spin-up and spin-down anti-fermions.

An investigation of the free fermion internal frequency is discussed, hinting to an exchange interaction between the two components of which a fermion is made of. An upper limit estimate is given to the strength of this interaction.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

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[2] Susskind, L. and Friedman, A. (2015) Quantum Mechanics: The Theoretical Minimum. Penguin.
[3] Kwiat, D. (2022) Planck’s Constant—A Result of Two Strings Coupling. Journal of High Energy Physics, Gravitation and Cosmology, 8, 919-926.‏
https://doi.org/10.4236/jhepgc.2022.84062
[4] Kwiat, D. (2024) Elementary Fermions: Strings, Planck Constant, Preons and Hypergluons. Journal of High Energy Physics, Gravitation and Cosmology, 10, 82-100.‏
https://doi.org/10.4236/jhepgc.2024.101008
[5] Harari, H. (1983) The Structure of Leptons and Quarks. Scientific American.
[6] Hubsch, T., Nishino, H. and Pati, J.C. (1985) Do Superstrings Lead to Quarks or to Preons? Physics Letters B, 163, 111-117.
https://doi.org/10.1016/0370-2693(85)90203-5

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