$\alpha =\frac{R}{r}\frac{{\stackrel{\xaf}{\lambda}}_{P}}{{\stackrel{\xaf}{\lambda}}_{e}}\approx 13.535\times \frac{2.10309\times {10}^{-16}}{3.86159\times {10}^{-13}}\approx \mathrm{0.0073715.}$ (7)

Still, this differs somewhat from the fine structure constant (0.0072973525664 CODATA 2014); the number is too large. However, the approximation used to calculate the radius of the sphere-packed electrons making up the proton mass will actually slightly overestimate the radius of the sphere-packed sphere. This is because the sphere-packed sphere’s outer surface is not smooth but rather jagged. We could measure the average radius of the sphere-packed spheres by measuring the radius from the inside radius and the outside radius and divide by two (see Figure 1).

Figure 1 illustrates how we account for the sphere-packed sphere’s jagged surface, namely the diameter of the average of the blue line and the green line. To find the green line, we can use the Pythagorean theorem to discern the distance, as shown in the lower part of Figure 1. If one properly adjusts for the jagged surface of the hypothetical sphere-packed sphere, it then seems that the radius of the large sphere should be very close to 13.4012 relative to the electron’s radius (the contracted radius ratio).

$\begin{array}{c}{R}_{R}\approx \frac{\sqrt[3]{\frac{N}{\text{\pi}}}\sqrt[6]{18}+\left(\sqrt[3]{\frac{N}{\text{\pi}}}\sqrt[6]{18}-1+\left(\sqrt{3}-1\right)\right)}{2}=\frac{2\sqrt[3]{\frac{N}{\text{\pi}}}\sqrt[6]{18}+\sqrt{3}-2}{2}\\ =\sqrt[3]{\frac{N}{\text{\pi}}}\sqrt[6]{18}+\sqrt{\frac{3}{4}}-1=13.4012\end{array}$ (8)

Figure 1. The figure illustrates the contracted radius of a sphere (here we only see a cross-section of the sphere). As a sphere-packed sphere’s surface must be jagged, a good approximation for the radius is found by taking half the average of the black-lined diameter and the green-lined diameter. To find the green-lined diameter, we need to use the Pythagorean theorem, as illustrated in the subfigure below. The contracted proton radius can, in the same way, be seen as 1836 sphere-packed spheres. The green-lined diameter is equal to the black-lined diameter minus $2r-2\left(\sqrt{3}-1\right)r\approx 0.54r$ , where r is the radius of the small spheres, which, based on recent developments in mathematical atomism, must be $r={l}_{p}$ , which is the Planck length.

And from this, we can calculate the fine structure constant by dividing the contracted radius ratio by the mass ratio:

$\alpha =\frac{{R}_{R}}{\frac{{m}_{P}}{{m}_{e}}}\approx \frac{13.4012}{1836.152}\approx \mathrm{0.00729854.}$ (9)

This also means that the fine structure constant can be represented by the contracted radius ratio multiplied by the ratio of the reduced Compton wave-lengths. The calculated value is extremely close compared to

${\alpha}_{c}=0.0072973525664$ , which is the fine structure constant given by CODATA

2014. The difference between the two numbers is close to $\frac{{\alpha}_{c}-\alpha}{{\alpha}_{c}}=0.0161\%$ .

We do not claim that this is what the fine structure constant must represent, but again it is interesting that this is a dimensionless number.

Alternatively, we could have used the classical electron radius ${r}_{e}=\frac{1}{4\text{\pi}{\epsilon}_{0}}\frac{{e}^{2}}{{m}_{e}{c}^{2}}=\alpha {\stackrel{\xaf}{\lambda}}_{e}\approx 2.81794\times {10}^{-15}$ . The classical electron radius2 divided by the reduced Compton wavelength of the proton3 is:

$\text{Radius}\text{\hspace{0.17em}}\text{ratio}=\frac{\alpha {\stackrel{\xaf}{\lambda}}_{e}}{{\stackrel{\xaf}{\lambda}}_{P}}\approx \mathrm{13.39905249,}$ (10)

and the fine structure constant is given by

$\alpha \approx \frac{\frac{\alpha {\stackrel{\xaf}{\lambda}}_{e}}{{\stackrel{\xaf}{\lambda}}_{P}}}{\frac{{m}_{P}}{{m}_{e}}}=\frac{\frac{\alpha {\stackrel{\xaf}{\lambda}}_{e}}{{\stackrel{\xaf}{\lambda}}_{P}}}{\frac{\frac{\hslash}{{\stackrel{\xaf}{\lambda}}_{P}}\frac{1}{c}}{\frac{\hslash}{{\stackrel{\xaf}{\lambda}}_{e}}\frac{1}{c}}}\approx \frac{13.39905249}{1836.152}=\alpha \times 1\approx \mathrm{0.007297353.}$ (11)

^{2}The previous analysis also means we can write the classical electron radius as
${r}_{e}=\frac{R}{r}{\stackrel{\xaf}{\lambda}}_{P}\approx 13.4012\times {\stackrel{\xaf}{\lambda}}_{P}$ .

^{3}Here using CODATA (divided by
$2\text{\pi}$ to get the reduced form):
${\stackrel{\xaf}{\lambda}}_{P}\approx 2.103089101\times {10}^{-16}$ and
${\stackrel{\xaf}{\lambda}}_{e}\approx 3.861592677\times {10}^{-13}$ .

In this case, it is the electron’s radius divided by the proton’s radius, while in the above analysis it was the proton’s contracted radius divided by the electron’s contracted radius (according to the atomism model.). We believe that the classical electron radius likely does not exist in a physical sense; it is just an imaginary unit that has the fine structure constant embedded. On the other hand, the contracted radius ratio is something that possibly exists if the depth of reality is atomism.

When it comes to the relationship between the classical electron radius and the radius of the proton or neutron and their mass ratio, Koshy has, in a recent piece [1] , suggested a similar relationship as a possible interpretation of the fine structure constant. However, we believe that the Haug atomist model has greater explanatory power, in terms of what is gained from Einstein’s special relativity mathematical result that is obtained when using Einstein-Poincaré synchronized clocks. This theory also seems to make more sense when the diameter of the indivisible particle is the Planck length and its mass is the Planck mass. Moreover, a series of infinity problems are elegantly removed via the Haug model.

On its own, this result could be seen as nothing more than numerology. Yet the result of the Haug model is particularly interesting when seen in light of recent developments in mathematical atomism. Haug’s mathematical atomism model is very simple and thus far has been shown to yield the same mathematical result as Einstein’s special relativity theory when using Einstein-Poincaré synchronized clocks.

Interestingly, If the fine structure constant truly could be represented by the radius ratio divided by the mass ratio, then it would not be affected by relativistic effects.

3. Proton Mass Divided by Higgs Mass

The 2014 CODATA-recommended proton mass is 1.672621898 × 10^{−27} kg. That is equal to about 938.2721137 MeV/c^{2}. On 4 July 2012, CMS announced the discovery of a previously unknown boson with mass 125.3 ± 0.6 MeV/c^{2}, see [13] . There is still considerable uncertainty about the mass of the Higgs boson [14] . For a moment, assume the Higgs mass is approximately 128,577.056 MeV/c^{2}. In this case, the proton mass divided by the Higgs mass would be close to identical to the fine structure constant, and it would be a dimensionless constant:

$\alpha \approx \frac{{m}_{P}}{{m}_{H}}\approx \frac{938.2721137}{128577.056}\approx 0.007297353,$ (12)

while a Higgs mass of 125.3 GeV would give a fine structure constant of

$\alpha \approx \frac{{m}_{P}}{{m}_{H}}\approx \frac{938.2721137}{125300}\approx \mathrm{0.007488205.}$ (13)

Still, this suggested value of the Higgs boson seems too far away from what it should in order to be related to the fine structure constant. Moreover, it is also in relation to electrons where the fine structure constant seems to be most important.

4. Summary

The fine structure constant plays an important role in modern physics. Yet there continues to be a mystery concerning exactly what it represents and why it has the mystical value it does. In this paper, we have suggested two new possibilities for what the fine structure constant may represent. It could be related to what we would call the contracted radius ratio of the electron versus the proton divided by the mass ratio, an idea closely related to the work of Koshy [1] . The contracted radius ratio is given from sphere packing of Planck diameter spheres and adjusting for this sphere-packed sphere’s jagged surface. This new ratio seems to be extremely close to the fine structure constant given by CODATA. Alternatively, we have suggested that the fine structure constant could be related to the Higgs mass over the proton mass, but this suggestion seems to give a fine structure constant considerably distant from the one given by CODATA.

We have in this paper not concluded what the fine structure constant truly represents. But we believe that the speculative idea that spins off from atomism deserves further investigation.

Acknowledgements

Thanks to Richard Whitehead and Victoria Terces for assisting with manuscript editing. Thanks to Thijs van den Berg for a useful discussion on the wilmott.com forum in relation to circle geometry.

Conflicts of Interest

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

Cite this paper

Haug, E.G. (2019) The Mystery behind the Fine Structure Constant Contracted Radius Ratio Divided by the Mass Ratio? A Possible Atomist Interpretation. Journal of High Energy Physics, Gravitation and Cosmology, 5, 899-906. https://doi.org/10.4236/jhepgc.2019.53045

References

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Appendix: Radius of Spheres Constructed from a Large Number of Small Spheres

Assuming small spheres with radius r, the volume of such a sphere is

$V=\frac{4}{3}\text{\pi}{r}^{3}.$

When we pack the Planck spheres as densely as possible, they will occupy a volume of

${V}_{t}=\frac{\frac{4}{3}\text{\pi}{l}_{p}^{3}}{\frac{\text{\pi}}{3\sqrt{2}}}={l}_{p}^{3}\sqrt{32}.$

The total volume is then $N{V}_{t}$ . This means we need a larger sphere with radius

$N{V}_{t}=\frac{4}{3}\text{\pi}{R}^{3}$

$R=\sqrt[3]{\frac{\frac{3}{4}N{V}_{t}}{\text{\pi}}}$

$R=\sqrt[3]{\frac{\frac{3}{4}N{r}^{3}\sqrt{32}}{\text{\pi}}}$

$R=r\sqrt[3]{\frac{N}{\text{\pi}}}\sqrt[6]{18}.$ (A1)

It is important to be aware that this formula will only be a good approximation for a very large number of spheres. In the case of a proton, we will assume it consists of 1836 spheres, which is a number of spheres where this formula should be quite accurate.