Richard Feynman, the genius Nobel laureate in physics, once wrote the number "137" on his blackboard and said:
"It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it. (...) You might say the 'hand of God' wrote that number, and 'we don't know how He pushed his pencil.'"
He was talking about the "Fine-Structure Constant (α)."
This dimensionless quantity, which represents the strength of the electromagnetic force, is known through observation to have a value of α≈1/137.035999... However, no physicist had ever been able to theoretically derive "why it takes this specific, non-integer value."
The Intergeometric Theory has finally deciphered the movement of this "hand of God."
The reason the fine-structure constant could not be expressed by a simple mathematical formula is that it is a product of two entirely different elements: a "pure geometric ratio" and the "decay rate of energy consumed by the universe from its birth to the present."
1. The Blueprint of the Universe Was Geometry (Derivation of the 0th-Order Approximation)
First, let's derive the "coupling constant e," the foundation of the fine-structure constant, using only pure geometry.
The electromagnetic force is generated when a "spherical wave (electric field)" spreading in all directions from a charge is mediated by a "plane wave (virtual photon)."
Because it attempts to transmit the information of a "round wave" using a "flat wave," a geometric "deviation (projection loss)" inevitably occurs. This loss can be accurately calculated in the following three steps.
① Direction of Virtual Photons (Coupling Constant × 1/2)
When virtual photons fly through space, their directions are random. Only photons in the "forward direction" toward the target can mediate the force; those in the opposite direction are useless. Consequently, the force is probabilistically halved.
② Tilt of Virtual Photons (Coupling Constant × cos45°)
The polarization (tilt) of a photon is also random, ranging from 0° to 90°. The probability of interaction is determined by the mean square of the projected amplitude components. The probabilistic average from 0° to 90° corresponds to "45°," the geometric representative angle.
③ Average Angle Between Sphere and Plane (Coupling Constant × cos30°)
This is the most beautiful piece of geometry. The surface of a sphere (electric field) and a plane moving in a specific direction (photon) can never be perfectly superimposed, no matter how hard you try. When you rigorously integrate the average solid angle where the normal vector of the spherical wave and the normal vector of the plane wave overlap within 1/8th of a spatial octant, the coefficient "cos30°" astonishingly emerges.
Multiplying all three of these geometric losses yields the "0th-order approximation" of the photon's coupling constant e:
\[e(0th)=\frac{1}{2}\cos{45^\circ}\cos{30^\circ}\]
Substituting this into the formula for the fine-structure constant \(\alpha=e^2/(4π)\), we get \(\alpha\approx1/134.04\).
Using pure geometry alone, a value terrifyingly close to the famous "137" popped out!
However, it slightly deviates from the actual observed value (137.035...). Is this discrepancy a flaw in the theory?
No, this is precisely "the proof that the universe is alive."
2. Discrepancy Between Theory and Observation: Vacuum Energy is "Decreasing"
If this were the newborn universe immediately after inflation, the fine-structure constant would indeed be exactly "1/134.04".
However, as the universe expanded, stars shone, and it evolved, it consumed "vacuum energy."
At the beginning of the universe, the "Higgs vacuum expectation value (\(\nu\))," which gives mass to elementary particles, was exactly twice the Higgs mass (\(M_H\)) to maintain physical symmetry (\(\nu=2M_H\)). (The strict theoretical value of the Higgs mass can be calculated with extreme precision by back-calculating from the electron mass. For details, see [another article].)
However, in the current universe, this vacuum expectation value has decreased by the "cosmic energy decay rate (\(\chi_0\))" as follows:
\[ν=2M_H−\left(\frac{3\chi_0}{81}\right)M_H=\left(2-\frac{3\chi_0}{81}\right)M_H\]
The current vacuum expectation value \(\nu\) is known very accurately from experimental data of another weak force, the "Fermi coupling constant." Comparing this observed value with the equation derives the current universe's decay rate to be \(\chi_0\approx 89.0\%\).
3. The Completion of the "God's Equation"
The decay of vacuum energy means that the density of virtual photons flying through space has also decreased.
Let's apply this current cosmic decay rate \(\chi_0\) to correct the 0th-order approximation formula we found earlier.
\[e=\left(1−\frac{\chi_0}{81}\right)\times\frac{1}{2}\cos{45^\circ}\cos{30^\circ}\]
Substituting this corrected value of e into the fine-structure constant formula \(\alpha=e^2/(4π)\) yields...
\[\alpha=1/137.03599..\]
This perfectly matches the latest experimental observation value: "1 / 137.035999177(21)".
The reason physicists, including Feynman, continually failed to express the fine-structure constant with a simple mathematical formula (combinations of constants like \(\pi\)) is now clear.
The fine-structure constant is not a universal, unchanging mathematical constant. It is a dynamic number created by combining a "pure geometric baseline (approx. 1/134)" with a time-varying environmental parameter: "the current age of the universe (decay rate \(\chi_0\approx 89.0\%\))."
(Why the current universe's \(\chi_0\) is at this specific value will be explained in detail in [another article].)
The blueprint of the universe was drawn with a geometry far more beautiful and dynamic than we could have ever imagined.
