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Fine-Structure Constant as Pure Geometric Number among Physical Background

*Classification:* NA
*Keywords:* fine-structure constant, four-dimensional spherical space, space-time manifold, variable mode effect.
*Language:* English
# Fine-Structure Constant as Pure Geometric Number among Physical Background

## Mei ZH

## ____________________________________________

### ABSTRACT

### V. CONCLUSION

### REFERENCES

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London Journal of Research in Science: Natural and Formal

Volume | Issue |
Compilation

Authored by Mei ZH , NA

According to B. Feng’s theory, the Fine-Structure Constant is theoretically deducted as, , an accurate theoretical value.

Keywords: fine-structure constant, four- dimensional spherical space, space-time manifold, variable mode effect.

Author: College of Chemistry and Molecular Engineering, Qingdao University of Science and Technology, Zhengzhou Road, Qingdao 266042, China.

### INTRODUCTION

Fine-structure constant (α−1) is a well-known natural constant in physics. It was beginning with an experimental discovery of fine spectrum of light (Michelson, 1887) and was first defined and theoretically explained by Sommerfeld in 1916. The measuring method by using Quantum Hall Effect has been recognized as advanced; the recent accurate value recommended by CODATA in 2014 is as 137.035999139(31).

Fine-structure constant is a combined dimensionless constant and appears in many physical occasions. Many physicists pay much attention to it mainly because of its dimensionless. Though Sommerfeld has given it a good explaination, still the physicists continuing their work for searching its deeper reason: wondering why the “hand of God” wrote down such a mystery non-integer number?

### ANALYSIS

Believing it to be a mathematical combination of integer numbers and mathematical constant, the guess works appeared uninterruptedly ever since, as Wyler’s (9/16π3)(π/5!)¼ = 1/137.036, Eddington’s (162 − 16)/2 + 16 = 136, and others as (9/8π4)(π5/245!)¼ = 1/137.036082, (1 − 1/(30 × 127))/137 = 1/137.035967, cos(π/137)/137 = 1/137.036028 and 29cos(π/137)tan(π/(137 × 29))/π = 1/137.035999787 etc. All of these are said to be mathematical games and can’t give out any information of physical meaning.

The actual determined value would depart from its original theoretical one because of the influences from the measuring method, measuring interaction and relativistic effect. Here, the original theoretical value would be more effective than that of actual to us for revealing the essential law of things, like that in dealing with the gas state function; where the ideal gas is more attractive to us.

### DEDUCTION

According to Sommerfeld’s definition, the original theoretical form and value of α−1 is as follows:

According to recent proposed B. Feng’s theory [1], the charge value of an elementary particle can be expressed by other common physical constants. The deduction steps were as following:

Supposing the Universe to be a four-dimensional spherical space (five-dimensional Euclid space), its projection in four-dimensional Euclid space would produce variable mode effect.

Because there is the following relationship between the surface area (S) and volume (V) of the n-dimensional sphere:

, and

So, the area of four-dimensional spherical space (S4) and the volume of four-dimensional Euclid space (V4) is S4 = 8π2R4/3 and V4 = π2r4/2 respectively. Where the radius R and r has a variable mode effect, their relation is as

It shows that the radius variable mode coefficient is.

The energy ratio of four-dimensional spherical space (ES) to four-dimensional Euclid space (EE) is equal to the corresponding area-volume ratio, we have

It shows the energy variable mode coefficient is 64/3.

Though the Universe is a four-dimensional spherical space, and we right live in such a complex space, however, in a condition of low velocity world, we more look like living in a simple four-dimensional Euclidean space. The work energy we do can only belong to the four-dimensional Euclid space’s (EE). In classical physics, the Euclid energy (EE) can be calculated based on Biot-Savart Law and Lorenz force (F) Law of moving charge (e) in magnetic field intensity (B), we has

, (where, l is the charge moving distance in a circle) i.e.

As elementary particles, electron and proton are indeed the confined movement of light (along the time axis) [2, 3], their energy belongs to four-dimensional spherical space (ES). The classical Euclidean energy EE in (3) ought to be converted by (2). Combining (2) and (3), we have. As frequency () has a relation of, we can rewrite ES as

Because energy has the form as E = hv, we have

, ,

However, the measured charge of an electron is 1.6021766208 × 10−19 C. In such a case, append a fitting factor k in (5), and in order to make them equal, let k = 0.9781465420, thus

Substituting (1) with (6), we have

, as, we finally obtained

That’s just the exact value that calculated theoretically in (1.1).

### DISCUSSION

From the expression in (7), one can see a perfect geometric character of α−1. Where 64/3 is a geometric energy variable mode coefficient, 2π the common circumference ratio. And k is also a geometric circumference; outwardly, it plays as a fitting number in (6), however, it comes from a not considered factor in the theoretical deduction of electric charge in (5). The not considered factor would be the curved space coefficient, because, in the process of theoretical deduction in (2)-(5), we treat the space as a plane Euclid’s; nevertheless, the mass density of electron is as high as 1.36512 × 1010 g/cm3, where the curved space factor can’t be neglected. In this way, the adding of k in (6) is necessary and reasonable, as a treating method which has ever been used in our previous work [4] when calculated the mass spectrum of elementary particles, where the same meaning of k was given a fitting number of 0.970624. They look almost to be equal.

From above theoretical deduction process, obviously, it shows us a strong physical background of α−1 rather than a mathematical game.

Another thing is that, to our surprising, the mystery number of α−1 is just the radius of proton which we predicted in literature [5].

α−1 is equal to. It consists of pure geometric parameters of 64/3, 2π and spatial bending factor k.

- Feng B. The topological analysis of material structure. Journal of Wuyi University, 2013, 27(1):42-49.
- Mei ZH, Mei SY. Mass the Confined Movement of Energy. Journal of Modern Physics, 2017, 8:923-925.
- Mei ZH. B. Feng’s Theory (Part Ⅱ): The Origin of Charge and the Unified Field Theory-Going on Kaluza-Klein’s. J Phys Astron, 2018, 6 (1):131-140.
- Mei ZH. B. Feng’s Theory: The Prediction of Mass Spectrum of Elementary Particles and the Confidence of at Least 4-D Space-Time (PartⅠ), Journal of Physics & Astronomy, 2017, 5(3):126-132.
- Mei ZH, Shi J. B. Feng’s Theory (Part Ⅲ): Neutron Model and Nuclei Force Mechanism. Journal of Physics & Astronomy, 2018, 6(1):143-149.

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