2. Comment to Guynn’s ApproachGuynn’s pioneering relation for Sommerfeld’s fine-structure constant α [3] can be rewritten in a form that indicates a nice reciprocity relation using the galactic rotation velocity vg due to Thomas precession [7]
α = 2 π c | v g | ( 1 φ ′ | v g | + φ ′ ⋅ k 2 | v g | ) (3)
With φ ′ = ( 2 − 2 1 / 3 ) 3 / 2 = 0.63667394565092 ≈ 2 π = 0.636619772 (4)
where k 2 ≡ m / s is a dimension-preserving factor [3].
Such reciprocity relations, frequently found in nature, point again to the golden mean dominance of physical science and life in general [8] [9].
When using the approximation (4) and choosing β g = v g c = − 0.000739437964740 [3], the fine-structure constant can be estimated simply as
α ≈ π 2 | β g | = 0.00729760191 (5)
However, the difference to the experimentally estimated value is only 0.000000249. Alterations of “fundamental” constants recommended by the IRT
theory [10] are quite well in this reliability range [7]. Therefore, we cannot exclude that the conjecture α | β g | = π 2 is correct.
Again a paradigmatic reciprocity relation can be formulated using α − 1 = 137.03599 ⋯ [8]
π ⋅ | β g | ≈ 1 π ⋅ α − 1 (6)
This is the real mystery behind number 137, if any mystery can be seen at all. It may be considered as a signature of matter-wave duality and galactic entanglement. Schwinger’s intuitive α/π is cutting edge [11], but QED is not.
Since Sommerfeld had investigated the spectrum of hydrogen and assigned the speed of the electron in the first Bohr orbit as fine-structure constant β 1 = α [1], this “constant” has been found to be more universal and connected to rotating entities “from particle scale to galactic scale” [3]. Therefore, it is recommended to replace the designation fine-structure constant simply by Sommerfeld constant.
Some other approximate relations for α have been applied in Chapter 4.
Another approximation for | β g | using the golden mean φ = 5 − 1 2 is [12]
| β g | ≈ φ 6 24 ⋅ π = 0.000739116 ⋯ (7)
Also the elementary charge e can be approximated using the galactic velocity vg
e ≈ − π ⋅ 2 ϵ 0 h | v g | = − 1.602243 × 10 − 19 C (8)
The exact CODATA value is =1.602176634 × 10−19 C.
With ϵ 0 h c = 1.32621132174 × 10 − 18 C as a calibration constant, Equation (8) can be recast into
e ≈ − π ⋅ 2 | β g | ⋅ ϵ 0 h c (9)
Using Guynn’s vg, the mass of the electron can be approximated in the same way giving (see Appendix A2)
m e ≈ π ⋅ 3 ⋅ k 1 c 4 | β g | = 9.1101587 ⋯ × 10 − 31 kg (10)
where k 1 ≡ kg ⋅ m 4 ⋅ s − 4 is again a dimension-preserving factor [3].
The concise CODATA value is m e = 9.1093837015 ( 28 ) × 10 − 31 kg .
The quotient e/me then delivers
e m e ≈ − ϵ 0 h c k 1 ⋅ 2 3 ⋅ | β g | 3 2 ⋅ c 4 = − 1.7587436 ⋯ × 10 11 C ⋅ kg − 1 (11)
compared to the CODATA value of e m e = − 1.75882001070 ( 53 ) × 10 11 C ⋅ kg − 1 .
One could more precisely adapt these approximations by a small variation of the involved “fundamental” physical constants.
Guynn’s convincing formula for the mass me of the electron used, besides vg, the maximum vm of the difference velocity (see Chapter 3).
m e = 2 3 k 1 | v g | v m c 2 = 9.10938356006879 × 10 − 31 kg (12)
In our approximate approach for me in Equation (10) we used the simple relation (23) given in Chapter 3.
With respect to the importance of the circle constant π an excerpt from reference [8] is given:
… the area A enclosed by a circle of radius 1 yields
A = π = 4 ∫ 0 1 1 − x 2 d x , (13)
where π is Archimedes’ constant, the well-known circle constant. One obtains the circumference C by using the reciprocal of the integrand
C = 2 π = 4 ∫ 0 1 1 1 − x 2 d x . (14)
This connection between the boundary and the enclosed area is of fundamental importance. It may be thought of as a geometrical analog to the more general matter-wave duality…
The Lorentz integral angular limit in Guynn’s approach is equivalent to relation (13) [3]
θ L max = ∫ 0 1 γ ( x ) d x = ∫ 0 1 1 1 − x 2 d x = π 2 (15)
The value of φ ′ in relation (4) is φ ′ ≈ θ L max − 1 .
Guynn’s famous and stunningly simple relation for the anomalous gyromagnetic factor ge of the electron [3] can also be treated in a different way. The second term of his relation
g e = 5 8 2 3 θ e a − v g c ⋅ 2 2 π α + 3 = 2.00231930436122 (16)
is reformulated giving a function of solely the galactic velocity
− v g c ⋅ 2 2 π α + 3 = 2 ⋅ β g 2 φ ′ + 3 ⋅ | β g | (17)
Using Guynn’s θ e a = arcsin ( v 0 c ) − arcsin ( v 1 c ) = 0.3932960869637 , (18)
The approximation holds θ e a φ ′ = 0.61769389 ≈ 0.6180339887 = φ (19)
respectively θ e a ⋅ π 2 = 0.61778804 ≈ 0.6180339887 = φ (20)
where φ is the golden ratio; v 0 c = 3 2 , v 1 c = 0.6083087004577 [3].
The theoretical background behind the experimental value of electron’s gyromagnetic factor includes among other things the relativistic mass correction, given by the Lorentz transform in accordance with Guynn’s approach [3], whereas the present author already had applied the IRT theory leading to a reduced ge value of g e ≈ 2.00231909 [8] ?
This comment is a Chapter of a paper in preparation about “Golden Ratio Signature Behind Guynn’s Matter/Space Approach”, scanning the different sides of the same coin [13] [14]. The puzzling question is whether frequently observed values have exact golden mean ratio. It has been illustrated in Figure 1 that the maximum of the Hardy-Suleiman relation of e max = φ 5 at β max = φ = 0.6180339887 [15] is clearly related to the maximum of Guynn’s difference velocity curve of
v m ≈ 3 2 φ 5 at β max ≈ 0.6083087 [3]. The number φ 5 can be considered as fundamental, because it is connected to phase transformations from particle dimension to galactic ones [16].
Turning back to Sommerfeld’s constant, an approximation of α using φ 5 can be formulated as
α ≈ π 2 k 1 2 φ 5 c 2 E e = 0.0074376569 ⋯ (21)
where Ee is the rest energy of the electron.
4. Sommerfeld’s Constant and Mozafari’s Coupling ConstantOne can obtain a further approximation of Sommerfeld’s constant by using βm
α ≈ 2 3 ⋅ π ⋅ k 1 k 2 2 β m 2 c 4 E e = 0.0072973391 ⋯ (29)
Remarkably, a reciprocal term connected with this relation resembles Mozafari’s recently published world average value for the interaction coupling constant α s ( m z 2 ) [4] giving
β m 2 3 = 0.117055 ≈ α s = π 10 ⋅ ln ( 10 ) = 0.1168065 (30)
This value was precisely confirmed by measurement and QCD analysis at CERN [20]:
α s ( m z ) = 0.1170 ± 0.0019 (31)
with uncertainties ± 0.0014 (fit) ± 0.0007 (scale) ± 0.0008 (model) ± 0.0001 (param).
One can formulate another numerical relation for the coupling constant αs using relation (23)
α s ≈ 2 3 π 2 = 0.1169956 (32)
Turning to results of the IRT theory and matter—dark matter coupling in disk galaxies [21], one can give a further relation for αs (notice the factor 5 combined with the IRT maximum of em/e0 in relation (24))
α s ≈ 1 5 ⋅ r c r s = 1 5 ( ln ( 3 ) ln ( 2 ) − 1 ) = 0.1169925 (33)
where rc is the core radius of the galaxy, representing the distance from the galaxy center to the core where matter density is one half of the central matter density, and rs is the half-velocity radius.
A golden mean based sketching of the value for αs used a simple reciprocity relation [22]. One cansplitthis relation delivering a term that represents the
inverse circumsphere radius 1 r c i r c = 2 3 + φ of a regular icosahedron of unit edge length
1 5 2 φ − φ 2 = 1 5 φ 2 ⋅ 2 3 + φ = 0.116900 (34)
We see that the grand unification of the sciences, arts and consciousness has made some progress again [6] [23].
, mediated by the circle constant
π, which points to an omnipresent importance of this constant and its intrinsic reciprocity pecularity:
α ≈
π
2|
βg| respectively
. The designation fine-structure constant should be replaced simply by
Sommerfeld’s constant. We present golden mean-based approximations for α as well as for electron’s charge and mass and connect the word average value of interaction coupling constant
αs(
mz) with |
βg|.