The Electron as Unity: A Refoundation of Natural Units and the Identity of Particle Structure and Interaction

J. Rogers, SE Ohio

Abstract

We propose a system of natural units founded on the observable, stable properties of the electron. By setting the fundamental constants c, h (the Planck constant), and k_B to unity, and making the novel choice to set the electron mass m_e to unity, we establish the Electron Natural Unit (ENU) system. Within this framework, we demonstrate a profound physical identity: the classical electron radius r_e becomes numerically equal to the fundamental, linear electromagnetic coupling strength, amp_force_natural. This unifies the concept of a particle's geometric structure with its innate interaction strength. We further show that the Bohr radius a₀ is related to the classical electron radius r_e by the square of the fine-structure constant, revealing a geometric hierarchy of scales. Finally, we demonstrate that this framework naturally predicts the correct length scale for nucleons. This approach preserves the fundamental unity of physical dimensions (T=f=m=E=p=F=1/length) while providing a more powerful and physically grounded perspective on the structure of matter.

1. Introduction
Natural unit systems seek to simplify physical law by setting fundamental constants to unity. The conventional Planck system, by setting G=1, anchors itself in the abstract scale of quantum gravity. This paper proposes an alternative anchored in the most fundamental stable particle we can directly measure: the electron. By defining the electron's mass as the fundamental unit (m_e = 1), we construct a coordinate basis native to the world of quantum mechanics and electromagnetism, allowing fundamental truths about the structure of matter to emerge as simple identities.  This makes G the force between two electrons over a plank length. 

2. The Electron Natural Unit (ENU) System
We define the ENU system with the following four axioms:

1. The speed of light in vacuum, c = 1

2. The Planck constant, h = 1

3. The Boltzmann constant, k_B = 1

4. The electron rest mass, m_e = 1

From these axioms, the base units are uniquely determined. The fundamental unit of length is the Compton wavelength of the electron:
l_ENU = λ_c = h / (m_e c) ≈ 2.426 × 10⁻¹² m

The unit of energy is the rest energy of the electron:
E_ENU = m_e c² ≈ 0.511 MeV

The unit of time is t_ENU = l_ENU / c = λ_c / c ≈ 8.09 × 10⁻²¹ s.

3. The Unification of Structure and Interaction: 
The primary revelation of the ENU system emerges when we rigorously derive the dimensionless value for the classical electron radius (r_e).

The standard definition of r_e is r_e = e² / (4πε₀ m_e c²).
The fine-structure constant α is defined in terms of h as α = e² / (2ε₀ h c).

We can re-express r_e using α and h:
r_e = ( e² / (2ε₀ h c) ) * ( h / (2π m_e c) ) = α * ( h / (2π m_e c) )
r_e = (α / 2π) * (h / (m_e c))

To find the dimensionless value of r_e in the ENU system, we express it in terms of the ENU unit of length, λ_c = h / (m_e c):
r_e [ENU] = r_e [SI] / λ_c [SI] = [ (α / 2π) * (h / m_e c) ] / [ h / m_e c ]

The term (h / m_e c) cancels perfectly, leaving:
r_e [ENU] = α / 2π

We recognize α / 2π as the definition of the fundamental, linear electromagnetic coupling strength, amp_force_natural. Therefore, we have arrived at the profound identity:

r_e = amp_force_natural

This result is of colossal significance. In a coordinate system built from the electron's own properties, the numerical value of its classical radius is its fundamental electromagnetic coupling strength. The distinction between the electron's structure (r_e) and its interaction (amf) has vanished. They are two sides of the same coin.

4. The Geometric Hierarchy of Scales
We now extend this analysis to the Bohr radius (a₀), the fundamental length scale of chemistry. The Bohr radius can be expressed as a₀ = λ_c / (2πα).

Expressing this in ENU units, where λ_c = 1:
a₀ [ENU] = 1 / (2πα)

We can now find the ratio between the atomic scale (a₀) and the classical scale (r_e):
Ratio = a₀ / r_e = [ 1 / (2πα) ] / [ α / 2π ] = (1 / 2πα) * (2π / α) = 1/α²

This confirms the hierarchical structure. The fine-structure constant α is the geometric scaling factor connecting these different levels of reality. The Bohr radius and the classical electron radius are manifestations of the same underlying geometry, separated by a scale factor of α².

5. Extension to the Nucleon Scale
The proton-to-electron mass ratio is Γ_N = m_N / m_e ≈ 1836. In the ENU system, the nucleon mass is m_N [ENU] = 1836. The natural length scale of the nucleon is its Compton wavelength, λ_N = h / (m_N c).

In the ENU system, where h=1, c=1, and m_N=1836:
λ_N [ENU] = 1 / m_N = 1 / 1836

To verify this, we convert it back to SI units by multiplying by λ_c:
λ_N [SI] = (1 / 1836) * λ_c ≈ (1 / 1836) * (2.426 × 10⁻¹² m) ≈ 1.32 × 10⁻¹⁵ m
This value is the experimentally measured size of the proton and the range of the strong nuclear force. The framework, built only on the electron, naturally predicts the correct geometric scale for the nucleus.

6. The Invariant Unity Principle
This entire re-scaling is achieved without violating the fundamental unity T=f=m=E=p=F=1/length. The choice of m_e as our "1" for the mass axis is a choice of perspective. This perspective, however, proves to be uniquely powerful, causing historical paradoxes to resolve into simple statements of identity.

7.  E and N is not not equal to the rest in this unit chart
We now need constants to correct for E and N being offset in this unit chart.  But the same two constants against all the other values that do scale with each other. 

   1.0000000000e+00 J -> 1.2214328571e+13 J_Ph 8.18710578e-14

All the rest are equal to each other still because the unit scaling cancels out in the units, but E is an outlier that needs a constant in a unit chart where m_e =1 because it has units of measurement that are odd in power between the numerator and denominator.

Newton also has a slightly offset scaling, not as much as energy, and I am still trying to figure this out.  So newton also needs a constant in this unit system too.

    1.0000000000e+00 N ->  2.9635750469e+01 N_Ph  -> 3.37430294e-02

I think this is because the mass and length scaling in the numerator combine in the numerator but time scaling is squared in the denominator. 

   s          -> 8.09329979436845559909e-21 -> t_Ph      

   m          -> 2.42631023868461404300e-12 -> l_Ph      

   kg         -> 9.10938370149428874717e-31 -> m_Ph   

It is very interesting how these are interacting. 

We can see the same for momentum

   p          -> 2.73092453073611131183e-22 -> p_Ph

So in this unit chart T=m=f=1/length  = E/E_constant = N/N_constant = p/p_constant

This has reversed the need for constants between classical and quantum math.  But each constant in classical math just applies to the given scale, not to the unity of the measurements.

8. Conclusion
The adoption of Electron Natural Units, with the crucial axiom h=1, provides a profound new perspective on physical reality. It demonstrates that the classical radius of the electron is not a historical artifact but is the direct, numerical embodiment of the fundamental electromagnetic coupling strength (r_e = amp_force_natural). This unifies the concepts of a particle's structure and its interaction. The framework further reveals the geometric hierarchy connecting classical, quantum, and atomic scales, and correctly predicts the size of the nucleon. This suggests that the universe is a self-consistent geometric structure, where the laws of interaction emerge from the very nature of being, and the choice of a proper coordinate basis is the key to revealing this underlying simplicity.

Reality itself is the unified substrate. Our "understanding" of it is always a projection through a chosen coordinate system. Each system illuminates one set of relationships while casting shadows on others.

• The Planck system illuminates the Gravity-Relativity-Quantum relationship.

• The ENU system illuminates the EM-Structure-Quantum relationship.

The ultimate goal of physics, then, is not to find the one "perfect" system, but to become fluent in translating between them, to understand that they are all just different, valid perspectives on the same single, underlying, unified reality.

Appendix, second cut of the scaling in my PUCS framework

   Symbol   Constant Name                    Original Value   Rescaled Value   Units Applied                 Ratio

   -----------------------------------------------------------------------------------------------------------------------------

   --- Core Scaling Constants ---

   c        speed_of_light_c                 2.9979245800e+08 1.0000000000e+00 l_Ph t_Ph⁻¹                   2.99792458e+08

   h        planck_constant_h                6.6260701500e-34 1.0000000000e+00 J_Ph t_Ph                     6.62607015e-34

   Hz_kg    Hz_kg                            7.3724973238e-51 1.0000000000e+00 m_Ph Hz_Ph⁻¹                  7.37249732e-51

   k        boltzmann_constant_k             1.3806490000e-23 1.0000000000e+00 m_Ph l_Ph² t_Ph⁻² T_Ph⁻¹      1.38064900e-23

   K_Hz     K_Hz                             2.0836619123e+10 1.0000000000e+00 Hz_Ph T_Ph⁻¹                  2.08366191e+10

   R        molar_gas_constant_R             8.3144626180e+00 5.4857990886e-07 m_Ph l_Ph² t_Ph⁻² mol_Ph⁻¹ T_Ph⁻¹ 1.51563382e+07

   R_K      klitzing_constant_Rk             2.5812807450e+04 9.9999999964e-01 m_Ph l_Ph² t_Ph⁻¹ C_Ph⁻²      2.58128075e+04

   Φ₀       magnetic_flux_quantum_Phi0       2.0678338480e-15 4.9999999989e-01 m_Ph l_Ph² t_Ph⁻¹ C_Ph⁻¹      4.13566770e-15

   F        faraday_constant_F               9.6485332120e+04 5.4857990885e-07 mol_Ph⁻¹ C_Ph                 1.75882001e+11

   K_J      josephson_constant_KJ            4.8359784840e+14 1.9999999999e+00 m_Ph⁻¹ l_Ph⁻² t_Ph C_Ph       2.41798924e+14

   G₀       conductance_quantum_G0           7.7480917290e-05 1.9999999998e+00 m_Ph⁻¹ l_Ph⁻² t_Ph C_Ph²      3.87404586e-05

   c_1      first radiation constant         3.7417718520e-16 6.2831853069e+00 m_Ph l_Ph⁴ t_Ph⁻³ pi          5.95521486e-17

   c_1L     first radiation constant sr      1.1910429720e-16 1.9999999993e+00 m_Ph l_Ph⁴ t_Ph⁻³             5.95521486e-17

   c_2      second_radiation_constant_c2     1.4387768776e-02 1.0000000001e+00 l_Ph T_Ph                     1.43877688e-02

   --- Units of Measure ---

   1 kg     1 kg                             1.0000000000e+00 1.0977691058e+30 m_Ph                          9.10938370e-31

   1 m      1 meter                          1.0000000000e+00 4.1214844831e+11 l_Ph                          2.42631024e-12

   1 s      1 second                         1.0000000000e+00 1.2355899638e+20 t_Ph                          8.09329979e-21

   Hz       1 Hz                             1.0000000000e+00 8.0932997944e-21 t_Ph⁻¹                        1.23558996e+20

   A        Amp force                        1.0000000000e-07 1.1614097323e-03 m_Ph l_Ph t_Ph⁻² A_Ph⁻²       8.61022576e-05

   A        Amp force scaled in alpha        1.0000000000e-07 2.9635750469e-06 m_Ph l_Ph t_Ph⁻² A_old⁻²      3.37430294e-02

   N        1 N                              1.0000000000e+00 2.9635750469e+01 N_Ph                          3.37430294e-02

   J        1 J                              1.0000000000e+00 1.2214328571e+13 J_Ph                          8.18710578e-14

   e        elementary_charge_e              1.6021766340e-19 1.0000000000e+00 C_Ph                          1.60217663e-19

   Na       avogadro_constant_Na             6.0221407600e+23 5.4857990887e-07 mol_Ph⁻¹                      1.09776911e+30

   --- Masses ---

   me       electron_mass_me                 9.1093837015e-31 1.0000000000e+00 m_Ph                          9.10938370e-31

   m_{π^±}  pion_charged_mass_pi_pm          2.4880679000e-28 2.7313240736e+02 m_Ph                          9.10938370e-31

   m_{π^0}  pion_neutral_mass_pi0            2.4061802000e-28 2.6414302864e+02 m_Ph                          9.10938370e-31

   m_μ      muon_mass_mu                     1.8835316270e-28 2.0676828298e+02 m_Ph                          9.10938370e-31

   m_τ      tau_mass_tau                     3.1674700000e-27 3.4771507094e+03 m_Ph                          9.10938370e-31

   m_{Λ^0}  lambda_baryon_mass_Lambda0       1.9883800000e-27 2.1827821345e+03 m_Ph                          9.10938370e-31

   mp       proton_mass_mp                   1.6726219237e-27 1.8361526734e+03 m_Ph                          9.10938370e-31

   mn       neutron_mass_mn                  1.6749274980e-27 1.8386836617e+03 m_Ph                          9.10938370e-31

   u        atomic_mass_unit_u               1.6605390666e-27 1.8228884862e+03 m_Ph                          9.10938370e-31

   m_H      Higgs_boson_mass_H               2.2320400000e-25 2.4502645548e+05 m_Ph                          9.10938370e-31

   m_Z      Z_boson_mass_Z                   1.6255700000e-25 1.7845005252e+05 m_Ph                          9.10938370e-31

   m_W      W_boson_mass_W                   1.4326900000e-25 1.5727628201e+05 m_Ph                          9.10938370e-31

   --- Planck Units ---

   l_P      planck_length_lP                 1.6162550000e-35 6.6613699033e-24 l_Ph                          2.42631024e-12

   t_P      planck_time                      5.3912470000e-44 6.6613706856e-24 t_Ph                          8.09329979e-21

   m_P      planck_mass                      2.1764340000e-08 2.3892220059e+22 m_Ph                          9.10938370e-31

   T_P      planck_temperature               1.4167840000e+32 2.3892221087e+22 T_Ph                          5.92989658e+09

   --- Stoney Units ---

   l_S      stoney_length_lS                 1.3800000000e-33 5.6876485867e-22 l_Ph                          2.42631024e-12

   t_S      stoney_time_tS                   4.6000000000e-43 5.6837138335e-23 t_Ph                          8.09329979e-21

   m_S      stoney_mass_mS                   1.8500000000e-09 2.0308728457e+21 m_Ph                          9.10938370e-31

   T_S      stoney_temperature_TS            1.4400000000e+32 2.4283728758e+22 T_Ph                          5.92989658e+09

   --- Other Natural Unit Systems ---

   eV       electronvolts_to_energy_EeV      1.6021766340e-19 1.0000000000e+00 A_Ph t_Ph                     1.60217663e-19

   l_n      natural_length_ln                2.8179403262e-15 1.1614097329e-03 l_Ph                          2.42631024e-12

   t_n      natural_time_tn                  9.3132257462e-24 1.1507328263e-03 t_Ph                          8.09329979e-21

   --- Electromagnetic & Atomic ---

   G_newton gravitational_G_Newtons          6.6743000000e-11 1.9779788936e-09 l_Ph m_Ph t_Ph⁻²              3.37430294e-02

   G        gravitational_constant_G         6.6743000000e-11 2.7880912441e-46 l_Ph³ m_Ph⁻¹ t_Ph⁻²           2.39385996e+35

   k_e      coulombs_constant_k_e            8.9875517870e+09 1.1614097322e-03 m_Ph l_Ph³ t_Ph⁻² C_Ph⁻²      7.73848500e+12

   ε₀       vacuum_permittivity_epsilon0     8.8541878128e-12 6.8517999542e+01 C_Ph² m_Ph⁻¹ l_Ph⁻³ t_Ph²     1.29224260e-13

   μ₀       vacuum_permeability_mu0          1.2566370621e-06 9.5597510142e-43 m_Ph l_Ph t_Ph⁻² C_Ph⁻²       1.31450815e+36

   Z_0      character impedance vacuum       3.7673031341e+02 1.4594705129e-02 m_Ph l_Ph² t_Ph⁻¹ C_Ph⁻²      2.58128075e+04

   [ε₀]_au  atomic unit of permittivity      1.1126500560e-10 1.5944462436e-193 l_Ph⁻³ m_Ph⁻¹ t_Ph⁻⁴ C_Ph⁻²   6.97828516e+182

   E_h      hartree_energy_Eh                4.3597447221e-18 5.3251354519e-05 m_Ph l_Ph² t_Ph⁻²             8.18710578e-14

   μ_B      bohr_magneton_muB                9.2740100783e-24 7.9577471545e-02 l_Ph² t_Ph⁻¹ C_Ph             1.16540648e-22

   μ_N      nuclear_magneton_muN             5.0507837461e-27 4.3339245531e-05 l_Ph² t_Ph⁻¹ C_Ph             1.16540648e-22

   --- Wavelengths & Diameters ---

   λ_e_C    electron_compton_wavelength      2.4263102387e-12 9.9999999999e-01 l_Ph                          2.42631024e-12

   λ_n_C    neutron Compton wavelength       1.3195909058e-15 5.4386734424e-04 l_Ph                          2.42631024e-12

   b        Wien wl d law                    2.8977719550e-03 2.0140523526e-01 l_Ph T_Ph                     1.43877688e-02

   R_inf    rydberg infinite                 1.0973731568e+07 2.6625677260e-05 l_Ph⁻¹                        4.12148448e+11

   r_e      electron_radius_re               2.8179403262e-15 1.1614097329e-03 l_Ph                          2.42631024e-12

   a₀       bohr_radius_a0                   5.2917721090e-11 2.1809956636e+01 l_Ph                          2.42631024e-12

   Å_star   Angstrom star                    1.0000149500e-10 4.1215460993e+01 l_Ph                          2.42631024e-12

   --- Thermodynamic & Other ---

   σ        stefan_boltzmann_constant_sigma  5.6703744190e-08 4.0802624637e+01 m_Ph t_Ph⁻³ T_Ph⁻⁴            1.38970825e-09

   g₀       standard_gravity_g0              9.8066500000e+00 2.6474367954e-28 l_Ph t_Ph⁻²                   3.70420552e+28

   κ        quantum of circulation           3.6369475467e-04 4.9999999932e-01 l_Ph² t_Ph⁻¹                  7.27389510e-04

   Λ        cosmological_constant_Lambda     1.0980000000e-52 6.4639055490e-76 l_Ph⁻²                        1.69866343e+23

   --- Material Properties ---

   E        youngs_modulus_steel             2.0000000000e+11 3.4893022206e-11 m_Ph l_Ph⁻¹ t_Ph⁻²            5.73180502e+21

   k        thermal_conductivity_copper      4.0100000000e+02 5.7033897236e-07 m_Ph l_Ph t_Ph⁻³ T_Ph⁻¹       7.03090652e+08

   ρ        resistivity_silicon              2.3000000000e+03 3.6723685814e+10 m_Ph l_Ph³ t_Ph⁻³ A_Ph⁻²      6.26298790e-08

   K_IC     fracture_toughness_alumina       3.5000000000e+00 3.9201570854e-16 m_Ph l_Ph⁻⁰⋅⁵ t_Ph⁻²          8.92821365e+15

   T_C      curie_temperature_iron           1.0430000000e+03 1.7588839649e-07 T_Ph                          5.92989658e+09

   γ        surface_tension_water            7.2800000000e-02 5.2347222051e-12 m_Ph t_Ph⁻²                   1.39071372e+10

   μ_n      mobility_silicon                 1.5000000000e-01 1.6087663625e+48 m_Ph⁻¹ l_Ph⁰ t_Ph⁴ A_Ph       9.32391449e-50

   β        compressibility_diamond          2.0000000000e-12 1.1463610049e+10 m_Ph⁻¹ l_Ph t_Ph²             1.74465111e-22

   R        reflectivity_aluminum            9.2000000000e-01 9.2000000000e-01                               1.00000000e+00

   d_11     piezoelectric_coefficient_quartz 2.3000000000e-12 4.8439707596e+05 m_Ph⁻¹ l_Ph⁻¹ t_Ph³ A_Ph      4.74817069e-18

   α        thermal_expansion_invar          1.2000000000e-06 7.1158758904e+03 T_Ph⁻¹                        1.68637005e-10

   --- Scaling Factors Used ---

   s          -> 8.09329979436845559909e-21 -> t_Ph      

   m          -> 2.42631023868461404300e-12 -> l_Ph      

   kg         -> 9.10938370149428874717e-31 -> m_Ph      

   K          -> 5.92989657531983184814e+09 -> T_Ph      

   A          -> 1.97963336921590276063e+01 -> A_Ph      

   mol        -> 9.10938370149428874717e-31 -> mol_Ph    

   pi         -> 1.00000000000000000000e+00 -> pi        

   amu        -> 1.00000000000000000000e+00 -> amu       

   Hz         -> 1.23558996380663906304e+20 -> Hz_Ph     

   N          -> 3.37430294209089473578e-02 -> N_Ph      

   J          -> 8.18710577681875380216e-14 -> J_Ph      

   W          -> 1.01159057304606083781e+07 -> W_Ph      

   Pa         -> 5.73180502455025244570e+21 -> Pa_Ph     

   Hz         -> 1.23558996380663906304e+20 -> Hz_Ph     

   C          -> 1.60217663399999989376e-19 -> C_Ph      

   V          -> 5.10998949995843868237e+05 -> V_Ph      

   Ω          -> 2.58128074593045130314e+04 -> Ω_Ph      

   S          -> 3.87404586493182367881e-05 -> S_Ph      

   F          -> 3.13538146020266874311e-25 -> F_Ph      

   H          -> 2.08910789302461781931e-16 -> H_Ph      

   Wb         -> 4.13566769692386059435e-15 -> Wb_Ph     

   T          -> 7.02510749387830495834e+08 -> T_Ph      

   Bq         -> 1.23558996380663906304e+20 -> Bq_Ph     

   Gy         -> 8.98755178736817760000e+16 -> Gy_Ph     

   Sv         -> 8.98755178736817760000e+16 -> Sv_Ph     

   kat        -> 1.12554630780301171377e-10 -> kat_Ph