14. Prime number and quantum theory

[Spin energy]

 

Strong, weak force can be expressed in various ways, but as long as I know, it can be expressed as follows using the formula mentioned in previous articles. This period is also the time when various elementary particles are generated from light.

 

(1)  $1/\beta^2+\gamma^2=\theta^2, \theta=2GM/rc^2$

 

$\beta^2=(1 \pm \sqrt{1-4/\theta^2})/2$

 

If we set spin is $\beta^2$ and $\theta=2$ then, the spin becomes $1/2$. $\theta$ symbolizes the angle of kinetic energy ratio, $\beta^2$. Therefore we can treat the spin as an energy. We can be curious about why the inside of the radical must be $0$, which is because the elementary particles are not composed of an infinite number, but a finite number.¹ Additionally, kinetic energy and potential energy are usually calculated separately, but in this state, they are combined in the form of $\beta^2\theta = \theta/\gamma^2=\beta\gamma=1$. At this , the role of $1/\beta, \gamma$ is changed.

 

If we look at the case where the inside of the radical is negative in equation (1), we can see that this can also be expressed as the square of a rational number.

 

If $\theta^2=2b^2/(2a-1)$ ($a, b$: prime number including $1$),

 

$\beta^2=1/2 \pm ci/2b$, ($b^2+c^2+2=4a$)²

 

There are countless prime numbers $a$, $b$, and $c$ satisfying this equation.³  One thing to remember here is that it is necessary to consider the case where the square of a prime number is expressed as the product of the complex conjugate number, that is, $N^2=NN^*=|N|^2$.⁴

 

In case $\theta^2=|\theta|^2=1$, $\theta^2$ can be obtained from the equations of gravitational potential.⁵

 

$1/\sin^2\theta-1+\sin^2\theta=0$

If $\theta \approx \sin\theta$, $1/\theta^2-1+\theta^2=0$ and $\theta^2=\beta^2=(1 \pm \sqrt{3}i)/2$. These appear to represent elementary particles such as nuclei or quarks.⁶

 

[Wave and particle motion on the curved angle, $\theta$]

 

 The equation (1) can be modified like below.

 

$1/\beta \pm i\gamma= \theta(\cos\delta \pm i\sin\delta)$

 

$(1/\beta \pm i\gamma)dv= \theta(\cos\delta \pm i\sin\delta)cdr/r$

 

$\theta$ represents the curved angle on the surface of sphere, and $cdr/r=dv$ represents energy, force in subatomic field. $\delta$ is regarded as a precession angle.

 

In imaginary part, $\theta(c\sin\delta dr/r)=\gamma dv$ and in real part, $\theta(c\cos\delta dr/r)=dv/\beta$. The first equation becomes the wave function of light, and the second becomes $\beta^2=v^2/c^2 \approx e^\theta$. This shows that gravity, unlike electromagnetism, is expressed as an exponential function of distance⁷. It also indicates that while potential is expressed as an exponential function of distance, kinetic energy is not. Here, the mass $M$ appears to vary depending on whether it is an particle in subatomic field or in gravitational field⁸. This process appears to be applied both to the moment of creating subatomic particles from light, and to the process of changing from subatomic particles to atomic particles.

 

9. Lorentz Effect in the Gravitaional Field

 

Prime_number_and_quantum_mechanics_20231207.pdf

0.20MB

 

 

[Extension to the molecules]  

 

In addition to the strong and weak forces, we can think about other forces in the same way. Because potentials are expressed as $\phi/c^2=1/{\beta\gamma}, \beta\gamma, \beta/\gamma$ ($\phi$: various potentials). I think it can be expanded and applied to intermolecular forces as well, $\phi/c^2=f(\beta, \gamma)$. If $\beta^2$ have a solution in the form of $a+bi$, even in this case, are $a, b$ rational numbers? This molecular behavior may explain many fractal structures seen in nature.

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^{1. If radical values are extended to complex numbers, it may be related to the Riemann hypothesis.}

^{2. If $a$ is replaced with $a^2$ and expanded to a complex number, $a^2$ becomes a complex prime number.}

^{3. Refer to 'Relation between prime number and quantum mechanics'}

^{4. Compex prime number: If $N^m=|N|^m, N=|N|(\cos(2k\pi/m) + i\sin(2k\pi/m)). k, m:$ natural number less or equal than $|N|$.}

^{5. Refer to part4. chapter 1 in the book, 'General Relativistic Quantum Mechanics'.}

^{6. More information can be found in Riemann zeta function and quantum mechanics.}

^{7. If $\theta\cos\delta$ is set to $1$ in the second equation, the gravitational field equation returns again as $\theta=1/\beta$ (<=> $dv/dt=v^2/r$).}

^{8. Lorentz effect in the gravitational field, Space and time energy}

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