Fermions: Dirac Internal Exchange Frequencies

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 ${\psi }_1=\left(\begin{array}{c}{U}_1\\ D_1\end{array}\right)$ ${\psi }_2=\left(\begin{array}{c}{U}_2\\ D_2\end{array}\right)$ ${\psi }_3=\left(\begin{array}{c}{U}_3\\ D_3\end{array}\right)$ ${\psi }_4=\left(\begin{array}{c}{U}_4\\ D_4\end{array}\right)$ where ( ${\psi }_1,{\psi }_4$ ) are coupled via linear combinations to yield spin-up and spin-down fermions. Likewise, ( ${\psi }_2,{\psi }_3$ ) 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.

• Fermions
• Internal Frequency
• Exchange Energy
• Coupling

1. Dirac Equation with Real Wave Functions

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

$\left(i\hslash {\gamma }^{\mu }{\partial }_{\mu }-mc\right)\Psi =0$ (1)

One may separate the Dirac operator $i\hslash {\gamma }^{\mu }{\partial }_{\mu }-mc$ and the complex wave function $\Psi$ into their real and imaginary parts [3] [4]

$\Psi =\left(\begin{array}{c}\begin{array}{c}{\psi }_1\\ {\psi }_2\end{array}\\ \begin{array}{c}{\psi }_3\\ {\psi }_4\end{array}\end{array}\right)$

With ${\psi }_1=\left(\begin{array}{c}{U}_1\\ D_1\end{array}\right)$ ${\psi }_2=\left(\begin{array}{c}{U}_2\\ D_2\end{array}\right)$ ${\psi }_3=\left(\begin{array}{c}{U}_3\\ D_3\end{array}\right)$ ${\psi }_4=\left(\begin{array}{c}{U}_4\\ D_4\end{array}\right)$ 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 ( $p_y,p_z=0\to {\partial }_y,{\partial }_z=0$),

the Dirac equations take the form:

$-\frac{m{c}^2}{\hslash }{\Psi }_1=+{\partial }_t{\Psi }_4-c{\sigma }_x{\partial }_x{\Psi }_4$ (2)

$-\frac{m{c}^2}{\hslash }{\Psi }_4=-{\partial }_t{\Psi }_1-c{\sigma }_x{\partial }_x{\Psi }_1$ (3)

$-\frac{m{c}^2}{\hslash }{\Psi }_2=+{\partial }_t{\Psi }_3+c{\sigma }_x{\partial }_x{\Psi }_3$ (4)

$-\frac{m{c}^2}{\hslash }{\Psi }_3=-{\partial }_t{\Psi }_2+c{\sigma }_x{\partial }_x{\Psi }_2$ (5)

This shows, that, ${\Psi }_1$ is coupled with ${\Psi }_4$ and ${\Psi }_2$ is coupled with ${\Psi }_3$. 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 ${\Psi }_i$ is a 2-vector with real components. Thus, the Dirac Equation is actually 8 equations of real components with coupled pairs $\left({\Psi }_1,{\Psi }_4\right)$, and $\left({\Psi }_2,{\Psi }_3\right)$.

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

$\left({\left(\frac{m{c}^2}{\hslash }\right)}^2+{\partial }_t^2\right)U_1+c{\partial }_x{\partial }_t{U}_1=-\frac{m{c}^3}{\hslash }{\partial }_x{U}_4$ (6)

$\left({\left(\frac{m{c}^2}{\hslash }\right)}^2+{\partial }_t^2\right)D_1-c{\partial }_x{\partial }_t{D}_1=+\frac{m{c}^3}{\hslash }{\partial }_x{D}_4$ (7)

$\left({\left(\frac{m{c}^2}{\hslash }\right)}^2+{\partial }_t^2\right)U_4-c{\partial }_x{\partial }_t{U}_4=+\frac{m{c}^3}{\hslash }{\partial }_x{U}_1$ (8)

$\left({\left(\frac{m{c}^2}{\hslash }\right)}^2+{\partial }_t^2\right)D_4+c{\partial }_x{\partial }_t{D}_4=-\frac{m{c}^3}{\hslash }{\partial }_x{D}_1$ (9)

$\left({\left(\frac{m{c}^2}{\hslash }\right)}^2+{\partial }_t^2\right)U_2+c{\partial }_x{\partial }_t{U}_2=-\frac{m{c}^3}{\hslash }{\partial }_x{U}_3$ (10)

$\left({\left(\frac{m{c}^2}{\hslash }\right)}^2+{\partial }_t^2\right)D_2-c{\partial }_x{\partial }_t{D}_2=+\frac{m{c}^3}{\hslash }{\partial }_x{D}_3$ (11)

$\left({\left(\frac{m{c}^2}{\hslash }\right)}^2+{\partial }_t^2\right)U_3+c{\partial }_x{\partial }_t{U}_3=-\frac{m{c}^3}{\hslash }{\partial }_x{U}_2$ (12)

$\left({\left(\frac{m{c}^2}{\hslash }\right)}^2+{\partial }_t^2\right)D_3-c{\partial }_x{\partial }_t{D}_3=+\frac{m{c}^3}{\hslash }{\partial }_x{D}_2$ (13)

These 8 real components equations demonstrate the existence of coupled pairs: ( $U_1\iff U_4$), ( $D_1\iff D_4$), ( $U_2\iff U_3$) and ( $D_2\iff D_3$).

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:

${\Psi }_1=\left(\begin{array}{c}{U}_1\\ D_1\end{array}\right)=\left(\begin{array}{c}\begin{array}{c}\cos \left(px-\left({\omega }_0+cp\right)t\right)\\ \sin \left(px-\left({\omega }_0-cp\right)t\right)\end{array}\end{array}\right)$

${\Psi }_4=\left(\begin{array}{c}{U}_4\\ D_4\end{array}\right)=\left(\begin{array}{c}\begin{array}{c}\sin \left(px+\left({\omega }_0+cp\right)t\right)\\ \cos \left(px+\left({\omega }_0-cp\right)t\right)\end{array}\end{array}\right)$

${\Psi }_2=\left(\begin{array}{c}{U}_2\\ D_2\end{array}\right)=\left(\begin{array}{c}\begin{array}{c}\cos \left(px-\left({\omega }_0+cp\right)t\right)\\ \sin \left(px-\left({\omega }_0-cp\right)t\right)\end{array}\end{array}\right)$

${\Psi }_3=\left(\begin{array}{c}{U}_3\\ D_3\end{array}\right)=\left(\begin{array}{c}\begin{array}{c}\sin \left(px+\left({\omega }_0-cp\right)t\right)\\ \cos \left(px+\left({\omega }_0+cp\right)t\right)\end{array}\end{array}\right)$

Here $p\equiv \frac{p_x}{\hslash }$, 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 ${\partial }_x$ terms ( ${\partial }_x=0$).

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

$\left[{\partial }_t^2+{\left(\frac{m{c}^2}{\hslash }\right)}^2\right]{\Psi }_i=0$ (14)

And likewise for all 8 components $U_i$ and $D_i$:

$\left[{\partial }_t^2+{\left(\frac{m{c}^2}{\hslash }\right)}^2\right]U_i=0$ (15)

$\left[{\partial }_t^2+{\left(\frac{m{c}^2}{\hslash }\right)}^2\right]D_i=0$ (16)

Solving this equation by setting $\Psi =\cos \left(\omega t\right)$ or $\Psi =\sin \left(\omega t\right)$ shows that all components of the fermion at its rest frame, are oscillating at a rate given by ${\omega }_0=\frac{m{c}^2}{\hslash }$.

Inserting the values in MKS units one obtains, for all fermions ${\omega }_0=1.36554\times {10}^{32}\times m$.

For an electron $m=0.511$ MeV/c² and thus ${\omega }_0\approx 7.76\times {10}^{11}$ GHz. (=0.512 MeV). For a proton, $m=939$ MeV/c² and ${\omega }_0\approx 7.296\times {10}^6$ GHz and for the τ quark $m=173$ GeV/c² and ${\omega }_0\approx 2.63\times {10}^{17}$ GHz ( Table 1 ).

Table 1

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

particle	${\omega }_0$	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 ${\omega }_0$ from the same total fermion mass m, and not from U's or D's individual masses (if any).

As shown earlier, each ${\Psi }_i$ is a 2-vector with real components.

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

$\left({\left(\frac{m{c}^2}{\hslash }\right)}^2+{\partial }_t^2\right)U_1=\epsilon U_1$ (17)

$\left({\left(\frac{m{c}^2}{\hslash }\right)}^2+{\partial }_t^2\right)D_1=\epsilon D_1$ (18)

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

$\left({\left(\frac{m{c}^2}{\hslash }\right)}^2+{\partial }_t^2\right)U_4=\delta U_4$ (19)

$\left({\left(\frac{m{c}^2}{\hslash }\right)}^2+{\partial }_t^2\right)D_4=\delta D_4$ (20)

Therefore,

$c{\partial }_x{\partial }_t{U}_1=-\frac{m{c}^3}{\hslash }{\partial }_x{U}_4+\epsilon U_1$ (21)

$-c{\partial }_x{\partial }_t{U}_4=+\frac{m{c}^3}{\hslash }{\partial }_x{U}_1+\epsilon U_4$ (22)

$-c{\partial }_x{\partial }_t{D}_1=+\frac{m{c}^3}{\hslash }{\partial }_x{D}_4+\delta D_1$ (23)

$c{\partial }_x{\partial }_t{D}_4=-\frac{m{c}^3}{\hslash }{\partial }_x{D}_1+\delta D_4$ (24)

This makes the components $U_1$ coupled to $U_4$ and same for $D_1$ to $D_4$.

One may write

${\partial }_t{U}_1=-\frac{m{c}^2}{\hslash }{U}_4+\frac{\epsilon }{c}{\displaystyle \int U_1\text{d}x}\cong -\frac{m{c}^2}{\hslash }{U}_4+\frac{\epsilon }{c}\Delta x\langle U_1\rangle$ (25)

${\partial }_t{U}_4=-\frac{m{c}^2}{\hslash }{U}_1+\frac{\epsilon }{c}{\displaystyle \int U_4\text{d}x}\cong -\frac{m{c}^2}{\hslash }{U}_1+\frac{\epsilon }{c}\Delta x\langle U_4\rangle$ (26)

${\partial }_t{D}_1=-\frac{m{c}^2}{\hslash }{D}_4+\frac{\epsilon }{c}{\displaystyle \int D_1\text{d}x}\cong -\frac{m{c}^2}{\hslash }{D}_4+\frac{\epsilon }{c}\Delta x\langle D_1\rangle$ (27)

${\partial }_t{D}_4=-\frac{m{c}^2}{\hslash }{D}_1+\frac{\epsilon }{c}{\displaystyle \int D_4\text{d}x}\cong -\frac{m{c}^2}{\hslash }{D}_1+\frac{\epsilon }{c}\Delta x\langle D_4\rangle$ (28)

where the integration in x is over the negligible size of the fermion, and where $\langle \rangle$ 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):

${\partial }_t{U}_1=-\frac{m{c}^2}{\hslash }{U}_4=0$ (29)

${\partial }_t{U}_4=-\frac{m{c}^2}{\hslash }{U}_1=0$ (30)

${\partial }_t{D}_1=-\frac{m{c}^2}{\hslash }{D}_4=0$ (31)

${\partial }_t{U}_4=-\frac{m{c}^2}{\hslash }{D}_1=0$ (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:

$\Delta U_1={\partial }_t{U}_1\Delta t=-\frac{m{c}^2}{\hslash }{U}_4\Delta t=-{\omega }_0{U}_4\Delta t$ (33)

Internal fluctuations of $U_1$ and $U_4$ at rate ${\omega }_0$ 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 $\frac{m{c}^2}{\hslash }$ 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:

$E_k<\frac{1}{2}m{\left({\partial }_t{U}_1\right)}^2=\frac{1}{2}\frac{m^3}{{\hslash }^2}{U}_4^2=2.12\times {10}^{-4}\text{eV}\times U_4^2$

If $U_4=\text{cos}\left({\omega }_{0}t\right)$ then the time average of $U_4^2$ is $\langle U_4^2\rangle =\frac{1}{2{\omega }_0}$.

So, the average kinetic energy transfer between the coupled strings in the electron, $E_k$ is less than 2.73 × 10⁻²⁵ eV/sec. For a proton the upper limit is 9.2 × 10⁻¹⁹ eV/sec and 3.12 × 10⁻¹⁴ 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

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 ${\psi }_1=\left(\begin{array}{c}{U}_1\\ D_1\end{array}\right)$, ${\psi }_2=\left(\begin{array}{c}{U}_2\\ D_2\end{array}\right)$, ${\psi }_3=\left(\begin{array}{c}{U}_3\\ D_3\end{array}\right)$, and ${\psi }_4=\left(\begin{array}{c}{U}_4\\ D_4\end{array}\right)$ where ( ${\psi }_1,{\psi }_4$) are coupled via linear combinations to yield spin-up and spin-down fermions. Likewise, ( ${\psi }_2,{\psi }_3$) 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.

References

1. Dirac, P.A.M. (1928) The Quantum Theory of the Electron. Proceedings of the Royal Society of London. Series A, 117, 610-624. >https://doi.org/10.1098/rspa.1928.0023

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