Equation of Motion for the Electron or Proton Cores in Free Space According to the Planck Vacuum Theory

This paper argues that the equation of motion (EOM) for the electron or proton cores is the relativistic energy-momentum equation from the Compton scattering calculations. It is noted that the Compton radii are constants of the motion. 

Introduction

The theoretical foundation [1]–[5] of the PV theory rests upon the unification of the Einstein, Newton, and Coulomb superforces: where the ratio $c^4/G_d$ is the curvature superforce that appears in the Einstein field equations. G is Newton’s gravitational constant, c is the speed of light, $m_{\ast }$ and $r_{\ast }$ are the Planck mass and length respectively [6, p. 1234], and $e_{\ast }$ is the massless bare (or coupling) charge. The Planck time is $t_{\ast }=r_{\ast }/c$ [6, p. 1233]. The fine structure constant is given by the ratio $\alpha \equiv e^2/e_{\ast }^2$, where e is the observed electronic charge magnitude. The ratio $e_{\ast }^2/c$ to the right of the arrow is the spin coefficient for the Planck particle (PP), the proton, and the electron cores, where $\hslash$ is the reduced Planck constant. One of the $e_{\ast }$s in $e_{\ast }^2$ belongs to the PP under consideration and the other to any one of the remaining PPs making up the PV state.

(1) $$\begin{array}{r}\frac{c^4}{G_d}(=\frac{m_{\ast }{c}^2}{r_{\ast }})=\frac{m_{\ast }^{2}G}{r_{\ast }^2}=\frac{e_{\ast }^2}{r_{\ast }^2}\to r_{\ast }{m}_{\ast }c=\frac{e_{\ast }^2}{c}(\hslash )\end{array}$$

Particle Spin

The electron, proton, and PP Dirac cores associated with the PV theory defined above are, respectively. The ± signs in the equations include the antiparticles. Their coupling to the highly energetic PV state is through the spin equations where it appears that particle spin and (3) exist to separate (in magnitude) the masses on the left from the mass $m_{\ast }$ on the right. The spin is generated in the zero-point PV oscillations [7].

(2) $$\begin{array}{r}(\pm e_{\ast },m_e)(\pm e_{\ast },m_p)\text{and}(\pm e_{\ast },m_{\ast })\end{array}$$

(3) $$\begin{array}{r}{r}_e{m}_{e}c=r_p{m}_{p}c=\frac{e_{\ast }^2}{c}=r_{\ast }{m}_{\ast }c\end{array}$$

All of the preceding equations are fixed in the sense that their structure is determined at the high PV energy level. This level is roughly nineteen (proton cores) to twenty-two (electron cores) orders-of-magnitude more energetic than the processes taking place at the electron or proton levels. The masses $m_e$ and $m_p$ are assumed to be created along with their Compton radii $r_e$ and $r_p$ within the PV state.

Conclusions and Comments

The present author has been searching for an EOM for the electron and proton cores in (2) for some time, while the answer is already available from the equations below.

The relativistic energy-momentum equation is [8] where W is the total energy, cP is the total kinetic energy, and P is the relativistic momentum. W and P include particle-spin, which has its source to the right of the arrow in (1). This equation is connected directly to the Einstein General Relativity equations [9].

(4) $$\begin{array}{r}{W}^2=c^2{P}^2+m_e^2{c}^4=c^2{P}^2+c^2(m_{e}c{)}^2\end{array}$$

For the first elementary particle in (2) the solution to (4) can be expressed as which includes the positron (+ sign). This is the EOM for the electron core $(\pm e_{\ast },m_e)$. In the rest frame P is all spin motion so the normalized energy becomes

(5) $$\begin{array}{r}W/m_e{c}^2=\pm {[1+(P/m_{e}c{)}^2]}^{1/2}\end{array}$$

(6) $$\begin{array}{r}W/m_e{c}^2=\pm 2^{1/2}.\end{array}$$

These results also apply to the proton core by replacing $m_e$ with $m_p$ and using the (− sign) for the antiproton.

In closing, (3) leads to the corresponding PV potentials

where the Compton radii are constants of the motion [8]; or

(7) $$\begin{array}{r}\frac{e_{\ast }^2}{r_e}=m_e{c}^2\frac{e_{\ast }^2}{r_p}=m_p{c}^2\text{and}\frac{e_{\ast }^2}{r_{\ast }}=m_{\ast }{c}^2\end{array}$$

in terms of the spin coefficient $e_{\ast }^2/c=\hslash$.

(8) $$\begin{array}{r}\frac{e_{\ast }^2/c}{r_e}=m_{e}c\frac{e_{\ast }^2/c}{r_p}=m_{p}c\text{and}\frac{e_{\ast }^2/c}{r_{\ast }}=m_{\ast }c\end{array}$$