An Investigation Into The Mathematical and Physical Origins of The Fine-Structure Constant


 The fine-structure constant, α, unites fundamental aspects of electromagnetism, quantum physics, and relativity. As such, it is one of the most important constants in nature. However, why it has the value of approximately 1/137 has been a mystery since it was first identified more than 100 years ago. To date, it is an ad hoc feature of the Standard Model, as it does not appear to be derivable within that body of work — being determined solely by experimentation. This report presents a mathematical formula for α that results in an exact match with the currently accepted value of the constant. The formula requires that a simple corrective term be applied to the value of one of the factors in the suggested equation. Notably, this corrective term, at approximately 0.023, is similar in value to the electron anomalous magnetic moment value, at approximately 0.0023, which is the corrective term that needs to be applied to the g-factor in the equation for the electron spin magnetic moment. In addition, it is shown that the corrective term for the proposed equation for α can be derived from the anomalous magnetic moment values of the electron, muon, and tau particle — values that have been well established through theory and/or experimentation. This supports the notion that the corrective term for the α formula is also a real and natural quantity. The quantum mechanical origins of the lepton anomalous magnetic moment values suggest that there might be a quantum mechanical origin to the corrective term for α as well. This possibility, as well as a broader physical interpretation of the value of α, is explored.


61
The fine-structure constant, α, also called Sommerfeld's constant, and the  where ge is the electron spin g-factor, a dimensionless value, and µB is the Bohr 92 magneton, a unit of magnetic moment.
where αe is 0.00115965218128(18) and is referred to as the electron's anomalous 101 magnetic moment. The anomaly arises from quantum effects at the particle level that 102 cause the value of ge to slightly exceed 2. The full value of ge can be formulated well 103 through perturbative quantum field theory techniques, thus far matching up to 10 104 significant digits of the experimentally determined value [3, 4, 5, 6]. 105 The principal idea here is that one of the factors in the equation for µz (specifically ge) 106 requires the addition of a small corrective, or anomalous, term -2 times 107 0.00115965218128(18), or 0.00231930436256(35)to obtain the true value of ge and 108 thereby µz. 109 A similar situation appears to arise in the setting of α. That is, as there is an anomalous 110 value associated with the electron's magnetic field that must be accounted for to 111 calculate the accurate value of the spin, there also appears to be an anomalous value 112 associated with the electron's electric field that must be accounted for in the calculation 113 of α. The concept of electric field lines can help in initial steps to identify the anomalous 114 electric field value. 115 The electric force between two electronsor, for ease, an electron and positron, If the field surrounding each particle were divided into an odd number of sectors, the 140 non-participating field lines in the blind spot could be relegated to a single region within 141 the larger field, with an equal portion of the electric field on either side of it. Dividing 142 the field into 3 sectors would be the minimum needed for this purpose, but with this, 143 the blind spot would have to take up nearly a third of the electric field, likely 144 overcompensating for the area. Whatever the best value happens to be, there is also 145 the question of whether it should be considered an ad hoc construct or whether the 146 division is something fundamental to the electric field. Only the latter would be of 147 benefit in understanding the nature of α. 148 To help identify an appropriate value to segregate the field lines in the blind spot from The fact that two mathematical constants (to an approximation) naturally arose from 189 the choice of 9 (and ultimately 8) sectors per particle also supported the use of 9 for the 190 number of electric field sectors surrounding the particles. field sector value will be referred to as the "s-factor," Se. The proposed general equation 211 for α is thus: The absolute value is used because in electric repulsion the sectors are, in a sense, 214 "missing" (leading to their having a negative value). In electric repulsion, two electrons The question remains, however, as to whether Se, and thereby equation (5), is actually 228 fundamental in nature, given the apparent ad hoc decision to divide the electric field 229 into 9 sectors to account for the blind spot in the field. It appears that this question can 230 be answered by way of the anomalous value, (Se)α. 231 As the value of Se is regarded above as a completely arbitrary choice into which to divide 232 the electric field, (Se)α would be an equally arbitrary value, as it directly stems from that 233 choice. As such, it would be highly improbable for (Se)α to have any connection to 234 fundamental constants in nature, being much more likely to have nothing to do with 235 them. However, (Se)α can be derived, to several significant digits, by using the values of 236 the anomalous magnetic moments of the electron, muon, and tau particle (αe, αμ, and 237 ατ, respectively). The result is achieved through the following power series: limitations in probing to sizes that would reveal any internal structure. As such, just as it 325 is suggested that the virtual photon cloud associated with the ground state of the 326 hydrogen atom is an "inside-out" mapping of the electronic structure of the atom, so 327 too might the 8-sector electric field, or equivalent virtual photon cloud, of the electron 328 be an "inside-out" mapping of some finer, as yet identified, electronic structure of the 329 elementary particle. Indeed, the identification of the 8 sectors of the electric field might 330 be a first glimpse into that structure. 331 The question of "Why does nature chose 8 specifically?" cannot be immediately 332 answered for either the lepton or the atom. However, this concept would represent yet 333 another "octet rule" in particle physics: In the model above, a lepton couples with another lepton through 8 electric field sectors (plus the field's anomalous portion). The 335 "Eightfold Way" concerns the organization of hadrons, and the currently established 336 "Octet Rule" concerns the 8-zoned valence shell of at least the main group elements, 337 even when there are many more electrons between the nucleus and the valence shell. 338 Thus, while there are always exceptions, a "theme of 8" appears to be carried through 339 from leptons, to hadrons, to some lepton-hadron interactions.

Types of α Values
If all 18 sectors were involved between the two interacting particles, α -1 would equal 355 about 153 (from 18eπ) instead of about 137 (from 16eπ)-indeed, at a superficial level, 356 153 seems as "unusual" a number as 137. The different scenarios lead to three types of 357 α values, a basic value (αB) associated with the value 18, encompassing each interacting 358 particle's full electric field; the true, corrected value of α; and a reduced value (αR) that 359 takes the blind spot per field into account but at a gross level ( The basic value could not really exist, as it would be physically impossible for all 18 363 sectors to be involved in the coupling in any meaningful way. For this to happen, the 364 field would have to be highly contorted to allow the sectors of field lines on the far side 365 of each particle to play a part in the interaction.  426 Thus, as noted above, Se can be written as Se = 8 + 2(Se)α, similar to the equation for ge, 427 where ge = 2 + 2αe. 428 As with the full value of ge, the full value of Se can also be formulated perturbatively. In 429 the case of ge, the perturbative method is applied to quantum field theory, specifically 430 QED. The perturbative formula for the QED contribution to ge is as follows: 431 (12) 432 The formal power series of α/π corresponds to quantum corrections as determined 433 through Feynman diagrams, which in turn correspond to real quantum activity at the 434 particle level. The coefficients of the formula (Ci) have been calculated to (α/π) 5  The class of quantum activities that would yield a correction on Se is currently not clear. 440 However, Se can be mathematically formulated in a similar way to ge by using αR (again, 441 1/16eπ, see (3/π) 3 (3/π) 3 (3/π) 3 (3/π) 3

C5
(1/π) 5 (1/π) 7 (1/π) 9 (1/π) 11 In all, the perturbative treatment for Se shown above in conjunction with equation (5)   524 represents, for the first time in history, a full mathematical expression for α that 525 involves mathematical constants, leads to an exact match with the established value of 526 the constant, has the potential to calculate α to an indefinite number of decimal places, 527 and is linked to a physical aspect of the electric field. This physical aspect concerns field 528 geometry at a gross level (accounting for a blind spot in the field), but likely also 529 quantum activity within the greater electromagnetic field. 530

531
As alternate mathematical expressions for calculating a quantity can often be 532 informative, the perturbative solution above was converted into other forms: 1) an 533 alternate perturbative series, with a leading term slightly higher than the value of 4 used 534 above, 2) a related expansion series with non-integer exponents, and 3) a generalized 535 continued fraction. Each offers additional insight into α that might prove useful in 536 further analysis of the constant, both from a mathematical and physical perspective.

538
In addition to the above, the full value of Se can be formulated as follows: 539 2 = 0 0 + 1 + 2 2 + 3 3 + 4 4 + 5 5 + ⋯, There are several noteworthy issues concerning the fraction, the first of which is its 597 regular structure, providing a straightforward representation of the full value of Se. 598 Indeed, the possibility exists that the fraction has a regular structure that extends 599 indefinitely, as the generalized continued fraction of pi does, although a more precise 600 experimental value of α would be needed to be sure. 601 For example, the partial denominators could continue as multiples of 2 or even powers

660
Although not a traditional generalized continued fraction, this expression also leads to a 661 value for α that is a 12-significant-digit (exact) match with the current CODATA value of 662 the constant. The point here is simply that something "nonclassical" might be happening 663 at a physical level in relation to the area of the fraction following 2 6 (or a lower point), 664 likely quantum mechanical activity. This would be consistent with the fact that many 665 experiments agree on the first 9 significant digits of α, which again, the portion of the 666 fraction down to 2 6 will lead to. However, they tend to differ slightly beyond this point.

Conclusion
The nature of α has been a mystery since its discovery more than 100 years ago, and 672 there have been numerous attempts to identify the mathematical basis of the constant. 673 This study presents a full mathematical formula for α that leads to an exact match with 674 the 2018 CODATA value of the constant, and that, importantly, is connected to a 675 physical aspect of the electric field. As such, it is likely connected to quantum 676 mechanical activity also, as the overall electromagnetic field is quantized in nature. 677 At the heart of the mathematics is the idea of a dimensionless anomalous electric field 678 value of about 0.023 associated with the electron. This is particularly notable, as the 679 electron is also associated with a dimensionless anomalous magnetic field value of 680 about 0.023/10, which has been well established through perturbative methods applied 681 to QED and through experimentation. In fact, the anomalous value for the electric field The author declares no competing interests. 747 The author has no relevant financial or non-financial interests to disclose. 748 The author has no conflicts of interest to declare that are relevant to the content of this 749 article. 750 The author certifies that he has no affiliations with or involvement in any organization or 751 entity with any financial interest or non-financial interest in the subject matter or 752 materials discussed in this manuscript. 753 The author has no financial or proprietary interests in any material discussed in this 754 article. 755 The data that support the findings of this study are openly available in the 2018 CODATA 756 Recommended Values of the Fundamental Physical Constants, reference number 1.     (3/π) 3 (3/π) 3 (3/π) 3 (3/π) 3