2026-08-10
The fine structure constant as the basis of the logarithmic scale of physical quantities
Could the fine structure constant determine not only the strength of the electromagnetic interaction, but also the natural scale of transitions between physical quantities? If we represent radii, frequencies, energies, and momenta on a logarithmic scale with a base equal to the inverse of the fine structure constant, cumbersome numerical relationships unexpectedly transform into simple integer steps. The classical electron radius, the reduced Compton length, and the Bohr radius are arranged in a single sequence, and the products and ratios of physical quantities transform into linear sums and differences. This allows us to view familiar formulas as manifestations of the general geometry of scales.
Logarithmic scale of physical quantities with a base of \(\alpha_{fs}^{-1}\)
In physics, quantities differing by tens of orders of magnitude are used simultaneously. In conventional notation, their relationships are hidden behind products, ratios, and powers. If we switch to logarithmic coordinates, the product becomes a sum, the ratio becomes a difference, and the power becomes a multiplication by an exponent. Reciprocals will differ only in sign. With a special choice of base, some important ratios also become integers.
For the electron scale system, the natural working base is the reciprocal of the low-energy fine structure constant:
\[\tag{1} \iota =\frac{1}{\alpha_{fs}(0)}\approx137{,}0359991776. \] This base relates the classical electron radius \(r_e\), the reduced Compton length \(\bar\lambda_C\), and the Bohr radius \(a_0\):
\[\tag{2} r_e=\alpha_{fs}\bar\lambda_C, \qquad \bar\lambda_C=\alpha_{fs}a_0, \] \[\tag{3} \frac{\bar\lambda_C}{r_e} =\frac{a_0}{\bar\lambda_C} =\iota . \] Therefore, the base \(\iota \) here is not an arbitrary notation. It is chosen so that two successive transitions between known electron scales correspond to the same unit step.
1. Two Types of Logarithmic Coordinates
It is necessary to distinguish between the absolute tabular coordinate and the normalized level coordinate. The absolute coordinate of the numerical value of \(X\), expressed in a pre-selected unit \(X_{\rm u}\), is defined as
\[\tag{4} L_X=\log_{\iota }\!\left(\frac{X}{X_{\rm u}}\right). \] For example, the unit for radius is \(1\,{\rm m}\), for frequency — \(1\,{\rm s}^{-1}\), and for energy — \(1\,{\rm J}\).
A normalized coordinate compares a value with a reference value of the same dimension. For radius:
\[\tag{5} N_r=\log_{\iota }\!\left(\frac{r}{r_e}\right). \] It is dimensionless, does not depend on the unit of length, and directly numbers the geometric states:
\[\tag{6} N_r(r_e)=0, \qquad N_r(\bar\lambda_C)=1, \qquad N_r(a_0)=2. \] 2. Extended table of physical values of \(M_1\)
Let's set for each level
\[\tag{7} r_N=r_e\iota ^N, \qquad \omega_N=\frac{c}{r_N}, \qquad f_N=\frac{\omega_N}{2\pi}. \] Let's add the period, characteristic energy, and momentum:
\[\tag{8} T_N=\frac{1}{f_N}, \qquad E_N=\hbar\omega_N=\frac{\hbar c}{r_N}, \qquad p_N=\frac{E_N}{c}=\frac{\hbar}{r_N}. \] | Magnitude | Level \(N=0\), \(r_e\) | Level \(N=1\), \(\bar\lambda_C\) | Level \(N=2\), \(a_0\) |
|---|---|---|---|
| Radius \(r\), m | \(2{,}81794032\cdot10^{-15}\) | \(3{,}86159267\cdot10^{-13}\) | \(5{,}29177210\cdot10^{-11}\) |
| Angular frequency \(\omega\), с\(^{-1}\) | \(1{,}06387086\cdot10^{23}\) | \(7{,}763 44072\cdot10^{20}\) | \(5{,}66525641\cdot10^{18}\) |
| Frequency \(f\), Hz | \(1{,}69320306\cdot10^{22}\) | \(1{,}23558997 \cdot10^{20}\) | \(9{,}01653561\cdot10^{17}\) |
| Period \(T\), s | \(5{,}90596619\cdot10^{-23}\) | \(8{,}09329978\cdot10^{-21}\) | \(1{,}10907342\cdot10^{-18}\) |
| Energy \(E=\hbar\omega\), J | \(1{,}12192822\cdot10^{-11}\) | \(8{,}8710579\cdot10^{-14}\) | \(5{,}97441974\cdot10^{-16}\) |
| Geometric energy \(E_g\), eV | \(7{,}00253\cdot10^7\) | \(5{,}10999\cdot10^5\) | \(3{,}72894\cdot10^3\) |
| Projection \(E_{\alpha}=\alpha_{fs}E_g\), eV | \(5{,}10999\cdot10^5\) | \(3{,}72894\cdot10^3\) | \(27{,}2114\) |
| Impulse \(p=\hbar/r\), kg m/s | \(3{,}74234972\cdot10^{-20}\) | \(2{,}73092453\cdot10^{-22}\) | \(1{,}99285192\cdot10^{-24}\) |
The energy values have a transparent meaning:
\[\tag{9} E_{g,0}=\frac{m_ec^2}{\alpha_{fs}}, \qquad E_{g,1}=m_ec^2, \qquad E_{g,2}=\alpha_{fs}m_ec^2. \] Here \(E_g=\hbar c/r\) refers to the original geometric frequency \(\omega_g=c/r\). At the Bohr radius, it gives \(E_{g,2}\approx3{,}72894\,\text{keV}\). The dynamics of an electron in a hydrogen atom contains an additional factor \(\beta=\alpha_{fs}\), since \(v_B=\alpha_{fs}c\). Therefore
\[\tag{9a} \omega_B=\alpha_{fs}\omega_g(a_0) =\frac{\alpha_{fs}c}{a_0}, \qquad E_H=\hbar\omega_B =\alpha_{fs}E_{g,2} =\alpha_{fs}^2m_ec^2. \] After this projection, we obtain \(E_H\approx27.2114\,\text{eV}\) — the Hartree energy. The binding energy of the ground state of hydrogen is \(E_H/2\approx13.6057\,\text{eV}\). Thus, the table must clearly distinguish between the original geometric frequency \(c/r\) and the physically observed dynamic frequency, which contains the projection factor.
3. Extended logarithmic table \(M_2\)
Apply the operator \(L_X=\log_{\iota }(X/X_{\rm u})\) to each row of \(M_1\). Then large and small physical values are converted to ordinary decimal coordinates:
| Logarithmic coordinate | \(N=0\) | \(N=1\) | \(N=2\) | Step \(\Delta L\) |
|---|---|---|---|---|
| \(L_r\) | \(-6.809169\) | \(-5.809169\) | \(-4.809169\) | \(+1\) |
| \(L_\omega\) | \(10{,}776168\) | \(9{,}776168\) | \(8{,}776168\) | \(-1\) |
| \(L_f\) | \(10{,}402634\) | \(9{,}402634\) | \(8{,}402634\) | \(-1\) |
| \(L_T\) | \(-10{,}402634\) | \(-9{,}402634\) | \(-8{,}402634\) | \(+1\) |
| \(L_E\) | \(-5{,}124418\) | \(-6{,}124418\) | \(-7{,}124418\) | \(-1\) |
| \(L_{E_\alpha}=L_E-1\) | \(-6{,}124418\) | \(-7{,}124418\) | \(-8{,}124418\) | \(-1\) |
| \(L_p\) | \(-9{,}091417\) | \(-10{,}091417\) | \(-11{,}091417\) | \(-1\) |
Note the rows \(L_f\) and \(L_T\). Reciprocals differ only in sign.
The entire table is specified by a single level number:
\[\tag{10} \begin{aligned} L_r(N)&=L_r(0)+N,\\ L_T(N)&=L_T(0)+N,\\ L_\omega(N)&=L_\omega(0)-N,\\ L_f(N)&=L_f(0)-N,\\ L_E(N)&=L_E(0)-N,\\ L_{E_\alpha}(N)&=L_E(0)-N-1,\\ L_p(N)&=L_p(0)-N. \end{aligned} \] Therefore, the transition to the next scale is a linear translation in the six-dimensional logarithmic space:
\[ \tag{11} \mathbf{L}_N = \mathbf{L}_0 + N \begin{pmatrix} +1\\ -1\\ -1\\ +1\\ -1\\ -1\\ -1 \end{pmatrix}, \qquad \mathbf{L}_N = \begin{pmatrix} L_r\\ L_\omega\\ L_f\\ L_T\\ L_E\\ L_{E_\alpha}\\ L_p \end{pmatrix}. \] This is precisely the main result of the table \(M_2\): a set of related physical quantities does not change independently. One single step \(N\) simultaneously increases the coordinates of the radius and period by one and decreases the coordinates of the frequency, energy, and momentum by one.
4. Constant differences and invariants of \(M_2\)
The relation \(\omega r=c\) in logarithmic form becomes the sum of two coordinates:
\[\tag{12} L_\omega+L_r=L_c=3{,}966999. \] For each of the three columns:
\[\tag{13} 10{,}776168-6{,}809169 =9{,}776168-5{,}809169 =8{,}776168-4{,}809169 =3{,}966999. \] Similarly, from \(fr=c/(2\pi)\), \(Tf=1\), \(E=pc\) and \(E=\hbar\omega\) it follows:
\[\tag{14} L_f+L_r=L_c-L_{2\pi}=3{,}593465, \] \[\tag{15} L_T+L_f=0, \] \[\tag{16} L_E-L_p=L_c=3{,}966999, \] \[\tag{17} L_E-L_\omega=L_\hbar=-15{,}900586. \] The difference between the angular and cyclic frequencies is also constant:
\[\tag{18} L_\omega-L_f=L_{2\pi}=0.373534. \] Thus, familiar physical laws manifest themselves in \(M_2\) as constant sums or distances between rows. The table not only reduces the range of numbers but also makes the linear structure of the relationships visible.
5. Normalized Level Table
Absolute coordinates are useful for numerical calculations, but the geometric structure is most clearly visible after normalizing to the level \(N=0\):
\[\tag{19} \Delta L_X(N)=\log_{\iota }\!\left(\frac{X_N}{X_0}\right). \] | Value | \(N=0\) | \(N=1\) | \(N=2\) |
|---|---|---|---|
| \(r/r_e\) | \(\iota ^0\) | \(\iota ^1\) | \(\iota ^2\) |
| \(\omega/\omega_0\) | \(\iota ^0\) | \(\iota ^{-1}\) | \(\iota ^{-2}\) |
| \(f/f_0\) | \(\iota ^0\) | \(\iota ^{-1}\) | \(\iota ^{-2}\) |
| \(T/T_0\) | \(\iota ^0\) | \(\iota ^1\) | \(\iota ^2\) |
| \(E/E_0\) | \(\iota ^0\) | \(\iota ^{-1}\) | \(\iota ^{-2}\) |
| \(E_\alpha/E_0\) | \(\iota ^{-1}\) | \(\iota ^{-2}\) | \(\iota ^{-3}\) |
| \(p/p_0\) | \(\iota ^0\) | \(\iota ^{-1}\) | \(\iota ^{-2}\) |
After taking the logarithm, this table consists only of the numbers \(0\), \(+1\), \(+2\), \(-1\), and \(-2\). However, the physical information is not lost: it is carried over into the choice of reference quantities and the signs of the directions.
6. How operations are transformed
For consistent dimensionless numerical ratios, the following rules hold:
\[\tag{20} L(XY)=L(X)+L(Y), \] \[\tag{21} L\!\left(\frac{X}{Y}\right)=L(X)-L(Y), \] \[\tag{22} L(X^q)=qL(X). \] Therefore, any monomial formula
\[\tag{23} Y=C\prod_{k=1}^{n}X_k^{q_k} \] translates into a linear combination:
\[\tag{24} L_Y=L_C+\sum_{k=1}^{n}q_kL_{X_k}. \] The logarithm simplifies multiplication, division, and powers, but not ordinary addition of quantities: \(L(X+Y)\neq L(X)+L(Y)\). Therefore, the method directly linearizes power and multiplicative laws, but not arbitrary physical formulas.
7. Example 1: Deriving the Bohr radius
The Bohr radius is related to the classical radius of the electron in two steps:
\[\tag{25} a_0=\frac{r_e}{\alpha_{fs}^2}=r_e\iota ^2. \] In logarithmic form:
\[\tag{26} L_{a_0}=L_{r_e}+2L_{\iota }=-6.809169+2=-4.809169. \] The inverse transformation returns
\[\tag{27} a_0=(1\,{\rm m})\,\iota ^{-4{,}809169} \approx5{,}29177210\cdot10^{-11}\,{\rm m}. \] 8. Example 2: Energy at the Compton and Bohr Levels
From \(E=\hbar\omega\) it follows:
\[\tag{28} L_E=L_\hbar+L_\omega. \] For the Compton level:
\[\tag{29} L_{E_1}=-15{,}900586+9{,}776168=-6{,}124418, \] \[\tag{30} E_1\approx8{,}18710579\cdot10^{-14}\,{\rm J}=m_ec^2. \] The transition to the Bohr level decreases the geometric energy coordinate by exactly one:
\[\tag{31} L_{E_2}=L_{E_1}-1=-7{,}124418, \] \[\tag{32} E_{g,2}=\alpha_{fs}m_ec^2\approx3{,}72894\,{\rm keV}. \] An additional atomic projection onto the factor \(\alpha_{fs}\) means another negative step:
\[\tag{32a} L_{E_H}=L_{E_{g,2}}+L_{\alpha_{fs}} =-7{,}124418-1=-8{,}124418, \] \[\tag{32b} E_H=\alpha_{fs}E_{g,2} =\alpha_{fs}^2m_ec^2 \approx27{,}2114\,{\rm eV}. \] 9. Example 3: Calculating the Bohr Magneton
The Bohr magneton is defined as
\[\tag{33} \mu_B=\frac{e\hbar}{2m_e}. \] Using the reduced Compton length \(\bar\lambda_C=\hbar/(m_ec)\), the formula takes the geometric form:
\[\tag{34} \mu_B=\frac{ec\bar\lambda_C}{2}. \] Its logarithmic coordinate in consistent SI units is
\[\tag{35} L_{\mu_B}=L_e+L_c+L_{\bar\lambda_C}-L_2. \] | Magnitude | SI Value | Coordinate \(L\) |
|---|---|---|
| Charge \(e\) | \(1{,}602176634\cdot10^{-19}\,{\rm C}\) | \(-8{,}795856\) |
| Velocity \(c\) | \(2{,}99792458\cdot10^8\,{\rm m/s}\) | \(+3{,}966999\) |
| \(\bar\lambda_C\) | \(3{,}86159267\cdot10^{-13}\,{\rm m}\) | \(-5{,}809169\) |
| Number \(2\) | \(2\) | \(+0{,}140877\) |
All multiplications and divisions are replaced by one sum:
\[\tag{36} \begin{aligned} L_{\mu_B} &=-8{,}795856+3{,}966999-5{,}809169-0{,}140877\\ &=-10{,}778902. \end{aligned} \] Returning to the usual scale:
\[\tag{37} \mu_B=(1\,{\rm J/T})\,\iota ^{-10{,}778902} \approx9{,}27401006\cdot10^{-24}\,{\rm J/T}. \] Not only is the more compact calculation important here. The Compton level with coordinate \(-5{,}809169\), already present in \(M_2\), is directly included in the coordinate of the magnetic moment of the electron.It.
10. What does logarithmic representation provide?
The resulting system allows us to divide a physical quantity into two parts: its dimensional standard and its numerical coordinate on the \(\iota \) scale. As a result:
1. The vast range of values is replaced by a compact range of decimal numbers.
2. Multiplicative physical laws become linear relationships.
3. Relationships equal to powers of \(\alpha_{fs}\) turn into integer distances.
4. Invariants manifest themselves as constant sums or differences of table rows.
5. Several physical quantities can be described by a single state coordinate \(N\).
However, the logarithmic notation itself is not yet a new physical law. It becomes an element of the model only when the choice of base, the direction of steps, and the observed connections between levels can be deduced from geometric postulates.
11. A Constant Base as a First Approximation
At the first stage, it is convenient to assume the base to be constant:
\[\tag{38} g_0=\iota =\frac{1}{\alpha_{fs}(0)}. \] Then the scale is uniform:
\[\tag{39} r_N=r_0g_0^N, \qquad \omega_N=\omega_0g_0^{-N}, \qquad E_N=E_0g_0^{-N}. \] In this form of the table, \(M_1\) and \(M_2\) are completely determined by the initial state and the integer or continuous parameter \(N\).
12. Transition to Effective \(\alpha_{fs}(Q)\)
More generally, the effective electromagnetic coupling depends on the characteristic energy scale or transferred momentum \(Q\). Therefore, the constant base can be expanded to
\[\tag{40} g(Q)=\frac{1}{\alpha_{fs}(Q)}. \] If the change in \(g(Q)\) is small over the region under consideration, a local step can be used:
\[\tag{41} \Delta N\approx \frac{\Delta\ln r}{\ln g(Q)} =-\frac{\Delta\ln\omega}{\ln g(Q)}. \] When the base changes significantly, the simple logarithm with one base is replaced by the integral coordinate:
\[\tag{42} dN=\frac{d\ln r}{\ln g[Q(r)]} =-\frac{d\ln\omega}{\ln g[Q(\omega)]}, \] \[\tag{43} N(r)=\int_{r_0}^{r} \frac{d\ln\rho}{\ln g[Q(\rho)]}. \] When \(g(Q)=g_0={\rm const}\), the general formula automatically returns the original scale:
\[\tag{44} N(r)=\frac{\ln(r/r_0)}{\ln g_0} =\log_{g_0}\!\left(\frac{r}{r_0}\right). \] The formula \(g(Q)=1/\alpha_{fs}(Q)\) is still an intended generalization, not a proven consequence of the geometric model. To prove it, it is necessary to independently derive the law of change of \(\alpha_{fs}(Q)\) from the postulates of the model. Substituting the well-known \(\alpha_{fs}\) run from quantum electrodynamics would provide an interpretation, but not an independent conclusion.
Conclusion
The extended table shows that the sequence \(r_e\rightarrow\bar\lambda_C\rightarrow a_0\) generates not only three radii. Along with the radius, the angular frequency, cyclic frequency, period, energy, and momentum change in concert. In ordinary units, these quantities appear disparate, but in coordinates with the base \(\iota =\alpha_{fs}^{-1}(0)\), their transitions are reduced to the steps \(+1\) and \(-1\).
Table \(M_2\) thus represents the system of physical relationships as linear geometry: the laws \(\omega r=c\), \(E=\hbar\omega\), \(E=pc\), \(Tf=1\) are expressed as constant sums and differences of coordinates, and the calculation of composite quantities, such as the Bohr magneton, is reduced to the addition of several numbers. The constant base provides a working first model; the transition to \(\alpha_{fs}(Q)\) leaves the possibility of describing a non-uniform, energy-dependent scale in the future.