24. Probability and Potential in quantum theory

In the previous article¹, nontrivial zeros can be expressed like below.

 

(1)   $\zeta(s)=e^{3i\theta_3}+1=(e^{i\theta_3}+1)(e^{2i\theta_3}-e^{i\theta_3}+1)=0$

 

($s=\frac{1}{2}+\frac{t}{2}i=\frac{1}{2}+\frac{\tan\theta}{2}i$)

 

The equation (1) is similar to $\cos^3\theta+\sin^3\theta$ in field theory of coordinate transformation. The reason for this difference appears to be the mathematical description method. If you calculate it by dividing it into the imaginary part and the real part, it is believed that you will get a matrix expressed by coordinate transformation.

 

 \begin{pmatrix}
\cos^3\theta+\sin^3\theta  & \sin\theta\cos\theta(\cos\theta-\sin\theta) \\
-\sin\theta\cos\theta(\cos\theta-\sin\theta)  & \cos^3\theta+\sin^3\theta \\
\end{pmatrix}

 

There is another condition similar to this, which is thought to represent electromagnetic waves as $\theta=\beta/\gamma$. The difference is that, contrary to what has been mentioned so far, when the potential $\theta$ becomes very large, it is expressed as a complex number. Any number $\beta$ can be expressed in the form $\beta^2+1/(4\theta^2\beta^2)=1$. With this modification, $\beta$ can be expressed as a trigonometric function and the related formulas are simplified. Coincidentally, in physics, it seems to express the boundaries of subatoms or substances.Also, probability and potential seem to have an inverse relationship. Mathematically, the $\zeta(s)$ or its related formulas appear to be a bridge on the border between continuity and discontinuity, probability and determinism. It is also a medium that connects Napier's constant $e$ and the $\pi$.

 

 

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^{1. Refer to About the roots of Riemann zeta function,Potential and Riemann zeta function}