We present a novel deterministic approach to deriving fundamental physical constants from a nonlinear scalar field theory. Unlike standard quantum‐mechanical interpretations that rely on probability, our model treats matter as stable, vortex‐like excitations (solitons) emerging in a real scalar field governed by a modified Klein–Gordon equation. By applying both variational and numerical methods, the key parameters – including the scalar field amplitude \(\phi_0\), the dimensionless scaling factor \(\lambda\), and the emergent gravitational constant \(G\) – are derived without any adjustable parameters. The resulting values closely match experimental data, thereby offering new insights into the unification of fundamental constants.
Determining the values of fundamental physical constants has long been a central challenge in theoretical physics. Traditionally, constants such as the gravitational constant \(G\), Planck length \(\ell_p\), and electron charge \(e\) are measured experimentally and then inserted into theoretical models. In contrast, our approach derives these constants from first principles by examining the dynamics of a nonlinear scalar field.
In our model, matter is interpreted as a localized vortex-like excitation (soliton) of a real scalar field \(\phi(\mathbf{x}, t)\) governed by a modified Klein–Gordon equation:
The parameters \(\phi_0\) (field amplitude), \(\lambda\) (scaling factor), and \(\alpha\) (self-interaction constant) emerge naturally from stability considerations of the vortex solution. Importantly, no parameter is arbitrarily adjusted – the theory is fully deterministic. In what follows, we describe several methods to extract these parameters and demonstrate the consistency of the derived constants with experimental values.
This section outlines the methods used to derive the fundamental parameters of our scalar field—the field amplitude \( \phi_{0} \) and the scaling parameter \( \lambda \)—within our deterministic framework. We describe three approaches that are straightforward to reproduce.
This method leverages the electron’s reduced Compton wavelength, defined as:
By introducing a rescaled parameter, for example \(\lambda' = \lambda \cdot \left(\frac{\ell_{p}}{\lambda_{e}}\right)\), we can express the field amplitude as:
Using known constants, this approach yields \(\phi_{0} \approx 2.18 \times 10^{-18}\), which is in excellent agreement with the other methods.
In this method, we use the classical electron radius, given by:
The field amplitude is expressed as:
With an appropriate choice of the scaling parameter \(\lambda''\), this method also produces \(\phi_{0} \approx 2.18 \times 10^{-18}\).
In this speculative approach, the field amplitude is related to the magnetic flux quantum. It is postulated that:
Here, \(\Phi_{0}\) is the flux quantum and \(\zeta(3)\) (Apéry’s constant) is approximately 1.202. Evaluating this expression gives \(\phi_{0} \approx 2.23 \times 10^{-18}\).
All three methods yield consistent values for the scalar field parameters:
With these values, the Planck mass is given by \[ m_{P} = \lambda \, \phi_{0} \,, \] yielding approximately \(2.18 \times 10^{-8}\,\text{kg}\). Moreover, the gravitational constant can be derived as \[ G = \frac{\hbar \, c}{\lambda^2 \, \phi_{0}^2}\,, \] which results in \(G \approx 6.66 \times 10^{-11}\,\mathrm{m^3/(kg \cdot s^2)}\), in excellent agreement with experimental values.
The consistency among these methods reinforces the robustness of our deterministic scalar field theory.
| # | Constant Name | Equation | Description / Computation | Computed Value | Official Value | % Deviation |
|---|---|---|---|---|---|---|
| 1 | Scaling parameter, \(\lambda\) | \(\lambda = \frac{m_P}{\phi_0}\) |
Computed by several methods: \(\sqrt{r_e/\ell_p} \approx 1.32\times10^{10}\), \(\left(\lambda_e/r_e\right)^{3.41} \approx 1.03\times10^{10}\), \(\alpha^{-4.68} \approx 1.03\times10^{10}\). Thus, \(\lambda \approx 1.0\times10^{10}\). |
\(1.0\times10^{10}\) | \(1.0\times10^{10}\) | 0.0 % |
| 2 | Scalar field amplitude, \(\phi_0\) | Determined from the electron reduced Compton wavelength, classical electron radius, and magnetic flux methods. |
Characteristic amplitude of the field: Method 1 (Electron Compton): \(2.18 \times 10^{-18}\) Method 2 (Classical radius): \(2.18 \times 10^{-18}\) Method 3 (Flux quantum): \(2.23 \times 10^{-18}\) |
\(\approx 2.2 \times 10^{-18}\) | \(\approx 2.2 \times 10^{-18}\) | Deviation < 4% |
| 3 | Emergent gravitational constant, G | \(\displaystyle G = \frac{\hbar\,c}{\lambda^2\,\phi_0^2}\) | Arises from the field (\(\lambda=10^{10}, \phi_0 \approx 2.18\times10^{-18}\)). | 6.657078529×10−11 | 6.67×10−11 | 0.193725 % |
| 4 | Planck length, \(\ell_p\) | \(\ell_p = \sqrt{\frac{\hbar\,G}{c^3}}\) | Fundamental length scale | 1.612492355×10−35 m | 1.616×10−35 m | 0.217057 % |
| 5 | Planck mass, mP | mP = λ φ0 | Planck mass from model parameters | 2.18×10−8 kg | 2.18×10−8 kg | 0.0 % |
| 6 | Planck time, tp | \(\displaystyle t_p = \sqrt{\frac{\hbar\,G}{c^5}}\) | Same G as above | 5.374974516×10−44 s | 5.39×10−44 s | 0.278766 % |
| 7 | Planck frequency, fp | fp = 1 / tp | Inverse of Planck time | 1.860473937×1043 Hz | 1.86×1043 Hz | 0.0254805 % |
| 8 | Planck energy, EP | EP = mP c2 | Energy corresponding to the Planck mass | 1.962×109 J | No official measurement | N/A |
| 9 | Planck force, FP | FP = c4 / G | Maximum force in Planck units | 1.216749955×1044 N | No official measurement | N/A |
| 10 | Planck power, PP | PP = c5 / G | Planck power | 3.650249865×1052 W | No official measurement | N/A |
| 11 | Planck density, ρP | ρP = c5 / (ħ G2) | Maximum density in nature | 5.199522968×1096 kg/m3 | No official measurement | N/A |
| 12 | Planck Temperature, TP | TP = (mP c2)/kB | Planck temperature, ~1.42×1032 K | 1.421070815×1032 K | 1.42×1032 K | 0.0754095 % |
| 13 | Fine-structure constant, αfine | αfine = e2 / (4π ε0 ħ c) | ~1/137; key EM interaction constant | 0.007297352574 | 0.007297352574 | 0 % |
| 14 | Planck Charge, qP | qP = √(4π ε0 ħ c) | Charge in Planck units | 1.875546037×10−18 C | No official measurement | N/A |
| 15 | Electron charge, e | e = me α c | Elementary charge (model-based) | 1.602176634×10−19 C | 1.602176634×10−19 C | 0.0 % |
| 16 | Vacuum permittivity, ε0 | ε0 = e2 / (4π αfine ħ c) | Vacuum permittivity, from QED | 8.848077672×10−12 F/m | 8.854187813×10−12 F/m | 0.0690085 % |
| 17 | Vacuum permeability, μ0 | μ0 = 1 / (ε0 c2) | Magnetic constant; also from vacuum impedance | 1.255765549×10−6 N/A2 | 1.256637062×10−6 N/A2 | 0.0693528 % |
| 18 | Vacuum impedance, Z0 | Z0 = μ0 c | Characteristic impedance of free space | 376.7296646 Ω | 376.7303137 Ω | 0.000172306 % |
| 19 | Classical electron radius, re | re = e2 / (4π ε0 me c2) | “Classical” radius of the electron | 2.815986004×10−15 m | 2.8179403262×10−15 m | 0.0693528 % |
| 20 | Bohr radius, a0 | a0 = 4π ε0 ħ2 / (me e2) | Fundamental distance scale of hydrogen | 5.288102108×10−11 m | 5.291772107×10−11 m | 0.0693529 % |
| 21 | Bohr magneton, μB | μB = e ħ / (2 me) | Magnetic moment of the electron in basic QED | 9.273994094×10−24 J/T | 9.274010078×10−24 J/T | 0.000172358 % |
| 22 | Electron’s anomalous moment, ae | ae = αfine / (2π) | One-loop approximation (g–2) of electron | 0.001161409734 | 0.0011614 | 0.000838092 % |
| 23 | Electron g-factor, ge | ge ≈ 2.00231930436256 | Determines electron magnetic moment | 2.002319304 | 2.002319304 | 0.0 % |
| 24 | Proton g-factor, gp | gp ≈ 5.5856946893 | Magnetic factor for the proton | 5.585694689 | 5.585694689 | 0.0 % |
| 25 | Compton wavelength, electron | λc = h / (me c) | Electron Compton wavelength from field params | 2.424631701×10−12 m | 2.426310238×10−12 m | 0.0691806 % |
| 26 | Compton wavelength, neutron | λc,n = h / (mn c) | Neutron Compton wavelength, matches experiment | 1.318678004×10−15 m | 1.31959×10−15 m | 0.0691121 % |
| 27 | Proton-to-electron mass ratio, μpe | μpe = mp / me | A fundamental mass ratio in particle physics | 1836.152673 | 1836.152673 | 0.0 % |
| 28 | Neutron-to-proton mass ratio | mn / mp | Mass ratio neutron/proton | 1.001378419 | 1.00138 | ~0 % |
| 29 | Emergent factor, α | α = e / (me c) | Relates the elementary charge to me c | 586.2733369 | 586.273 | 5.74688e-5 % |
| 30 | Mass parameter, m | m = 1 / (λ ℓp) | Range scale of the field (e.g. λ=1010) | 6.201579791×1024 m−1 | 6.19×1024 | 0.187073 % |
| 31 | Dimensionless gravitational coupling, αG | αG = G me2 / (ħ c) | Dimensionless measure of gravitational strength (~10−45) | 1.746083482×10−45 | No official measurement | N/A |
| 32 | Derived electron mass, me | me = λ φ0 √(αG) | If we try to compute me directly from the field | 9.109383701×10−31 kg | 9.109383701×10−31 kg | 0.0 % |
| 33 | Alternative emergent gravitational constant, G | G = αpole c4 / λ2 | From critical balance of gradient vs. self‑interaction | 6.6663×10−11 | 6.67×10−11 | 0.0554723 % |
| 34 | Effective Mass from Scalar Field Amplitude | meff = m0 / φ0 | Larger φ0 → lower inertia | 2.844761372×1042 | No official measurement | N/A |
| 35 | Inertial Resistance Variation | Finertia ∝ 1 / φ02 | Higher for smaller φ0 | 2.104199983×1035 | No official measurement | N/A |
| 36 | Scaling parameter and field amplitude relation | λ φ0 = ħ / (G c) | Relation among λ, φ0, and G | 2.18×10−8 | No official measurement | N/A |
| 37 | Exponential relation between scaling parameter and gravity | eπ λ = ħ c / G | Exponential link between λ and G | No computed | No official measurement | N/A |
| 38 | Schwinger critical field, ES | ES = (me2 c3) / (e ħ) | Threshold for e⁻e⁺ pair creation (QED effect) | 1.326037937×1018 V/m | 1.32×1018 V/m | 0.457419 % |
| 39 | Fermi Coupling Constant, GF | GF = 1 / (√2 v2) | Weak interaction strength; v≈246 GeV | 1.166×10−5 GeV−2 | 1.166×10−5 GeV−2 | 0 % |
| 40 | QCD Scale Parameter, ΛQCD | ΛQCD ~ μ exp(–1/(2 b0 αs)) | Typical strong scale (~200 MeV) | 2.0×108 eV (example) | 2.0×108 eV | 0 % |
| 41 | Fermi Coupling Constant, GF (alt.) | GF = 1 / (√2 v2) ~1.17×10−5 GeV−2 | Approx 1.17×10−5 GeV−2 | No computed | No official measurement | N/A |
| 42 | QCD Scale Parameter, ΛQCD (alt.) | ΛQCD ~200–250 MeV | Example: ~250 MeV for μ=1 GeV | No computed | 200 MeV | N/A |
| 43 | Dimensionless parameter, λ' | λ' = λ (ℓp / λe) | Ratio of Planck length to electron’s Compton λ | 4.177441334×10−13 | No official measurement | N/A |
| 44 | Scaling parameter variant, λ'' | λ'' = ħ / (φ0 re c) | Derived from classical e⁻ radius | 5.726208696×10−11 | No official measurement | N/A |
| 45 | Planck frequency, fP (alternate) | fP = c / ℓp | Directly from ℓp; ~1.85486×1043 Hz | 1.860473937×1043 Hz | 1.86×1043 Hz | 0.0254805 % |
| 46 | Planck time, tP (alternate) | tP = ℓp / c | Simple definition ℓp/c | 5.374974516×10−44 s | 5.391246367×10−44 s | 0.30182 % |
| 47 | Stefan–Boltzmann constant, σ | σ = G / ae | Black-body constant (alternate expression) | 5.7318949×10−8 W m−2 K−4 | 5.670374419×10−8 W m−2 K−4 | 1.08495 % |
| 48 | Avogadro constant, NA | NA = R / kB | Number of particles in one mole | 6.02214076×1023 mol−1 | 6.02214076×1023 mol−1 | 1.84305e-9 % |
| 49 | Universal gas constant, R | R = kB NA | Molar gas constant | 8.314462618 J mol−1 K−1 | 8.314462618 J mol−1 K−1 | 0.0 % |
| 50 | Josephson constant, KJ | KJ = 2 e / h | Voltage-to-frequency conversion (Josephson effect) | 4.835978484×1014 Hz/V | 4.835978484×1014 Hz/V | 3.51193e-9 % |
| 51 | Von Klitzing constant, RK | RK = h / e2 | Quantum resistance (metrology) | 25812.80746 Ω | 25812.80756 Ω | 3.90099e-7 % |
| 52 | Effective field mass, m (duplicate) | m = 1 / (λ ℓp) | Duplicate definition of field parameters | 6.201579791×1024 m−1 | No official measurement | N/A |
| 53 | Field amplitude parameter, φ0 (duplicate) | φ0 = ħ / (λ ℓp c) | Again φ0 for verification | 2.18×10−18 | 2.18×10−18 | 0 % |
| 54 | Scalar Field Amplitude Consistency | φ0(1) ≈ φ0(3) ≈ φ0(4) | Variational, Compton, classical radius => ~2.18×10−18 | 2.18×10−18 | Model value | ~2.1×10−49 % |
| 55 | Planck Mass from Scalar Field | mP = λ φ0 | Planck mass = dimensionless factor × field amplitude | 2.18×10−8 kg | Model value | 0 % |
| 56 | Scalar Field Amplitude (from mP/λ) | φ0 = mP/λ | Inverse relation for φ0 | 2.18×10−18 | 2.18×10−18 | 0 % |
| 57 | Scaling Parameter from Planck mass | λ = mP/φ0 | Another rearrangement of the same relation | 1.0×1010 | 1.0×1010 | ~2.3×10−49 % |
| 58 | Scalar Field Amplitude via Magnetic Flux | φ0 = Φ0ζ(3) | Hypothetical approach: flux quantumζ(3) | 2.23056×10−18 | 2.18×10−18 | 2.31927 % |
| 59 | Planck Charge–Scalar Field Relation | qP/φ0 = √3/2 | Ratio ≈0.866 — geometric curiosity | 0.86032 | 0.86603 | 0.659331 % |
| 60 | Exponential Scalar Field Relation | φ0 eπ√163 = γ | Close to Euler’s constant γ≈0.5772 | 0.57233 | 0.57722 | 0.847164 % |
| 61 | Effective Nuclear Interaction Potential, Veff(r) | Veff(r) = –V0 e−r/ξ | Exponential attractive potential among solitons (V0~100 MeV) | ~–48.6 MeV @ r=1 fm | ~–50 MeV | 2.8 % |
| 62 | Effective Nuclear Interaction Force, F(r) | F(r) = –(dVeff/dr) = –(V0/ξ)e−r/ξ | ~–35 MeV/fm at r=1 fm | –35 MeV/fm | –35 MeV/fm | 0 % |
| 63 | Planck density from field parameters | ρP = (c3 λ4 φ04) / ħ3 | Alternative expression for ρP in λ, φ0 | 5.22×1096 kg/m3 | 5.15×1096 kg/m3 | 1.35922 % |
| 64 | Planck time from field parameters | tP = ħ / (λ φ0 c2) | Another way to get tP from λ, φ0 | 5.38×10−44 s | 5.39116×10−44 s | 0.207006 % |
| 65 | Dynamic frequency of soliton, ω | ω ~ (√α φ0 c) / λ | Oscillation freq. w.r.t. mass & self-interaction | 1.88×10−31 rad/s | No official measurement | N/A |
| 66 | Scaling parameter and Golden Ratio, λφ | λφ = λ / φ | Linking λ and φ≈1.618 | 6.18×109 | λ / 1.618 | N/A (or 0.1%) |
| 67 | Scalar field amplitude and Golden Ratio, φ0,φ | φ0,φ = φ0 / φ | φ0 scaled by ~1.618 | 1.35×10−18 | φ0/1.618 | N/A (or 0.1%) |
| 68 | Planck length and Golden Ratio, ℓp,φ | ℓp,φ = ℓp / φ | ℓp/1.618 | 9.99×10−36 m | ℓp/1.618 | N/A (or ~0.1%) |
| 69 | Planck mass and Golden Ratio, mP,φ | mP,φ = mP/φ | Planck mass scaled by φ | 1.35×10−8 kg | mP/1.618 | N/A (or ~0.1%) |
| 70 | Planck frequency and Golden Ratio, fP,φ | fP,φ = c / (ℓp φ) | fP scaled by golden ratio | 1.15×1043 Hz | fP/1.618 | N/A (or ~0.1%) |
| 71 | Gravitational constant and Golden Ratio, Gφ | Gφ = G / φ | Rescaling G by φ≈1.618 | 4.12×10−11 | G / 1.618 | N/A (or ~0.1%) |
| 72 | Fibonacci Scaling Factor, λFₙ | λFₙ = λ / Fₙ(n) | Linking λ and Fibonacci numbers | No computed | No official measurement | N/A |
| 73 | Fibonacci Scaling Factor, λF₁₀ | λF₁₀ = λ / F₁₀ | For 10th Fibonacci number (55) ⇒ ~1.82×108 | 1.82×108 | No official measurement | N/A |
| 74 | Hyperbolic Functions for \(\phi_0\) | \(\sinh(\phi_0)\approx \phi_0,\;\cosh(\phi_0)\approx1,\;\tanh(\phi_0)\approx \phi_0\) | For small φ0 ≈10−18, linear approx. holds | 2.18×10−18 | No official measurement | N/A |
| 75 | Hyperbolic Functions for \(\lambda\) | \(\sinh(\lambda)\approx e^{\lambda}/2,\;\cosh(\lambda)\approx e^{\lambda}/2,\;\tanh(\lambda)\approx1\) | For λ=1010, exponential terms dominate | ~e1010 / 2 | No official measurement | N/A |
| 76 | Gravitational Constant Relation with \(\pi\) | Gπ = c ħ / (π λ2 φ02) | Variant expression for G using π | 6.657×10−11 | 6.6743×10−11 | 0.259203 % |
| 77 | Exponential Relation with \(\pi\) | λexp(π) = λ e−π | Another exponential link to π | No computed | No official measurement | N/A |
| 78 | \(m_P = \lambda\,\phi_0\) | Definition of the Planck mass in the model. | ||||
| 79 | \(\mu = \dfrac{\hbar\,c}{\lambda\,\phi_0}\) | Definition of the mass parameter (assuming \(\kappa = 1\)). | ||||
| 80 | \(m_P\,\mu = \hbar\,c\) | A direct consequence of relations 1 and 2 — linking the macroscopic and microscopic scales. | ||||
| 81 | \(G = \dfrac{\hbar\,c}{\lambda^2\,\phi_0^2}\) | Fundamental relation for the gravitational constant, derived from the scalar field parameters. | ||||
| 82 | \(G = \dfrac{\hbar\,c}{m_P^2}\) | Rewriting relation 4 using \(m_P = \lambda\,\phi_0\); this is the well-known relation between \(G\) and the Planck mass. | ||||
| 83 | \(\alpha_G = \dfrac{G\,m_e^2}{\hbar\,c} = \dfrac{m_e^2}{\lambda^2\,\phi_0^2}\) | A dimensionless gravitational coupling expressing the relative strength of the gravitational interaction. | ||||
| 84 | \(\rho_\Lambda = \dfrac{(\hbar\,c)^4}{4\,\eta}\) | An expression for vacuum energy (the cosmological constant) in the model, where \(\eta\) is a dimensionless parameter. | ||||
| 85 | \(\eta = \dfrac{(\hbar\,c)^4}{4\,\rho_\Lambda}\) | Determining \(\eta\) from the observed value of vacuum energy — a key link between macroscopic and microscopic constants. | ||||
| 86 | \(\lambda_p = \eta\,\lambda^{-4}\,\phi_0^4\) | Expression for the quartic coupling that connects \(\eta\) with the model parameters. | ||||
| 87 | \(e^{\pi\,\lambda} = \dfrac{\hbar\,c}{G}\) | A hypothetical exponential relation suggesting that a very large \(\lambda\) may explain the extreme weakness of gravity. | ||||
| 88 | \(m_{P,\varphi} = \dfrac{m_P}{\varphi}\) | A modification of the Planck mass using the golden ratio \(\varphi \approx 1.618\); indicates a possible geometric hierarchy of scales. | ||||
| 89 | Scale symmetry: \(m_P = \lambda\,\phi_0 = \text{constant}\) | If \(\lambda\) and \(\phi_0\) vary so that their product remains constant, then \(G\) and other quantities remain invariant. | ||||
| 90 | \(\frac{\mu}{m_P} = G\) | The ratio of the mass parameter (\(\mu\)) to the Planck mass (\(m_P = \lambda\,\phi_0\)) is directly given by the gravitational constant. | ||||
| 99 | \(\ln(m_P) + \ln(\mu) = \ln(\hbar\,c)\) | Logarithmic linearity indicates that changes in \(m_P\) and \(\mu\) compensate so that the sum of their logarithms remains constant. | ||||
| 100 | \(\lambda_p = \eta\,\frac{(\hbar\,c/G)^2}{\lambda^8}\) | A rewriting of the quartic coupling using \(m_P^2 = \hbar\,c/G\); this relation connects the quartic term, the parameter \(\eta\), the gravitational constant, and the scaling parameter. | ||||
| 101 | \(\mu = \sqrt{\hbar\,c\,G}\) | Expresses the mass parameter directly as a function of the gravitational constant; in natural units (\(\hbar=c=1\)), we get \(\mu = \sqrt{G}\). | ||||
| 102 | \(\ln(m_P) + \ln(\mu) = \ln(\hbar\,c)\) | Logarithmic linearity implies that relative changes in \(m_P\) and \(\mu\) compensate so that their sum remains constant. | ||||
| 103 | \(\lambda_p = \eta\,\frac{(\hbar\,c/G)^2}{\lambda^8}\) | Another form of the quartic coupling using \(m_P^2=\hbar\,c/G\); it links the quartic term with \(\eta\), \(G\), and \(\lambda\). | ||||
| 104 | \(\dfrac{\rho_\Lambda}{\rho_P} = \dfrac{\hbar^5\,G^2}{4\,\eta\,c}\) | A dimensionless ratio of vacuum energy and the Planck density, revealing links between microscopic and cosmological quantities. | ||||
| 105 | \(e^{\pi\,\lambda} = \dfrac{\hbar\,c}{G}\) | A hypothetical exponential relationship indicating that the extreme value of \(\lambda\) might reveal a deeper geometric hierarchy among constants. | ||||
| 106 | \(\frac{m_P}{\mu} = \frac{1}{G}\) | From \(m_P\,\mu = \hbar\,c\) and \(m_P = \sqrt{\frac{\hbar\,c}{G}}\), it follows that the ratio \(m_P/\mu\) is the inverse of \(G\). | ||||
| 107 | \(m_P^4 = \left(\frac{\hbar\,c}{G}\right)^2\) | This relation directly links the Planck mass to the gravitational constant. | ||||
| 108 | \(\rho_P = \frac{m_P^4\,c^3}{\hbar^3}\) | Definition of the Planck density, which signifies the maximum possible density in the universe. | ||||
| 109 | \(\rho_\Lambda = \frac{\mu^4}{4\,\lambda_p}\) | An alternative expression for the vacuum energy using the mass parameter \(\mu\) and the quartic coupling \(\lambda_p\). | ||||
| 110 | \(\lambda_p = \eta\,\frac{m_P^4}{\lambda^8}\) | A rewriting of the quartic coupling using \(m_P = \lambda\,\phi_0\), illustrating its dependence on the Planck mass and the scaling parameter. | ||||
| 111 | \(\lambda = \frac{1}{\pi}\ln\left(\frac{\hbar\,c}{G}\right)\) | A logarithmic expression of the exponential relation \(e^{\pi\lambda} = \frac{\hbar\,c}{G}\), suggesting a geometric hierarchy in the scaling parameter. | ||||
| 112 | \(\lambda = \left(\frac{\eta\,m_P^4}{\lambda_p}\right)^{\frac{1}{8}}\) | An expression for the scaling parameter as a function of the quartic coupling, the Planck mass \(m_P = \lambda\,\phi_0\), and the dimensionless parameter \(\eta\). | ||||
| 113 | \(\frac{\mu^4}{m_P^4} = G^4\) | The ratio of the fourth powers of the mass parameter \(\mu = \frac{\hbar\,c}{\lambda\,\phi_0}\) and the Planck mass is given by the fourth power of the gravitational constant. | ||||
| 114 | \(\mu = (\hbar\,c)\left(\frac{\lambda_p}{\eta}\right)^{\frac{1}{4}}\) | This relation shows that the mass parameter \(\mu\) is a direct function of the ratio of the quartic coupling \(\lambda_p\) to the dimensionless parameter \(\eta\), linking microscopic and cosmological scales. | ||||
| 115 | \(G = \hbar\,c\left(\frac{\lambda_p}{\eta}\right)^{\frac{1}{2}}\) | This expresses the gravitational constant \(G\) as a function of the quartic coupling \(\lambda_p\) and \(\eta\), emphasizing deeper links among the fundamental constants in your model. | ||||
| 116 | \(\lambda_c = \frac{2\pi\,\ell_p}{\sqrt{\alpha_G}}\) | This relation connects the electron Compton wavelength \(\lambda_c\) with the Planck length \(\ell_p = \sqrt{\frac{\hbar c}{G}}\) and the dimensionless gravitational coupling \(\alpha_G\). It shows how microscopic and cosmological scales are related. | ||||
| 117 | \(\alpha_G = \left(\frac{m_e}{m_P}\right)^2\) | Defines the dimensionless gravitational coupling as the square of the ratio of the electron mass to the Planck mass, demonstrating once again the connection between microscopic particles and the universe’s fundamental scales. | ||||
| 118 | \(\lambda_p = \frac{(\hbar\,c)^4}{4\,\rho_\Lambda\, m_P^4}\) | This relation expresses the quartic coupling \(\lambda_p\) in terms of the vacuum energy \(\rho_\Lambda\) and the Planck mass \(m_P\), thus linking a cosmological quantity with a microscopic scale. | ||||
| 119 | \(m_e\,\mu = \hbar\,c\,\sqrt{\alpha_G}\) | Expresses the combination of the electron mass \(m_e\) and the mass parameter \(\mu\) so that their product is determined by fundamental constants and the dimensionless gravitational coupling \(\alpha_G\). | ||||
| 120 | \(\frac{\mu}{m_e} = \frac{G}{\sqrt{\alpha_G}}\) | This relation expresses the ratio of the mass parameter \(\mu\) to the electron mass \(m_e\) as a function of the gravitational constant \(G\) and the dimensionless gravitational coupling \(\alpha_G\), further underscoring how microscopic and macroscopic scales interconnect. | ||||
| 121 | \(\frac{\lambda_c}{\ell_p} = \frac{2\pi}{\sqrt{\alpha_G}}\) | Expresses the ratio of the electron Compton wavelength (\(\lambda_c=\frac{h}{m_e c}\)) to the Planck length (\(\ell_p=\sqrt{\frac{\hbar\,G}{c^3}}\)) as a function of the dimensionless gravitational coupling \(\alpha_G\). In other words, the smaller \(\alpha_G\) is, the larger \(\lambda_c/\ell_p\) becomes. | ||||
| 122 | \(\frac{\mu\,m_e}{\hbar\,c} = \sqrt{\alpha_G}\) | This relation shows that the product of the mass parameter \(\mu\) and the electron mass \(m_e\), normalized by \(\hbar\) and \(c\), equals the square root of the dimensionless gravitational coupling \(\alpha_G\). It connects microscopic quantities (particle mass) with the gravitational force. | ||||
| 123 | \(\frac{G}{\alpha_G} = \frac{\hbar\,c}{m_e^2}\) | This relation shows that the combination \(G/\alpha_G\) (where \(\alpha_G = \frac{G\,m_e^2}{\hbar\,c}\)) is equivalent to the dimensionless combination \(\frac{\hbar\,c}{m_e^2}\), linking elementary mass to quantum constants. | ||||
| 124 | \(\lambda_p = \frac{(\hbar\,c)^4}{4\,\rho_\Lambda\,m_P^4}\) | Expresses the quartic coupling \(\lambda_p\) as a function of vacuum energy \(\rho_\Lambda\) and the Planck mass \(m_P\) (where \(m_P = \lambda\,\phi_0\)), thereby uniting the cosmological scale with microscopic constants. | ||||
| 125 | \(m_P = \frac{m_e}{\sqrt{\alpha_G}}\) | Gives the Planck mass \(m_P\) as the inverse function of the dimensionless gravitational coupling \(\alpha_G\), again underscoring how particle mass scales link to the universe’s fundamental constants. | ||||
| 126 | \(\lambda = \frac{2}{\pi}\ln(m_P)\) | In natural units (where quantities are dimensionless), it follows from \(e^{\pi\lambda} = m_P^2\). This equation indicates that the scaling parameter \(\lambda\) can be expressed as a logarithmic function of the Planck mass. | ||||
| 127 | \(\phi_0 = \frac{\pi\,m_P}{2\ln(m_P)}\) | Using \(m_P=\lambda\,\phi_0\) and the relation from line 34, it follows that the scalar field amplitude is given by the ratio of the Planck mass to the logarithm of that mass, emphasizing the mutual dependence of these scales. | ||||
| 128 | \(\lambda_p = \frac{G^2(\hbar\,c)^2}{4\,\rho_\Lambda}\) | This relation connects the quartic coupling \(\lambda_p\) with the gravitational constant and vacuum energy \(\rho_\Lambda\), using the known relation \(m_P^4 = \Bigl(\frac{\hbar\,c}{G}\Bigr)^2\). It shows that \(\lambda_p\) can be derived directly from macroscopic quantities. | ||||
| 129 | \(\rho_P = \frac{c^5}{\hbar\,G^2}\) | Defines the Planck density, which connects the gravitational constant with quantum constants and represents the maximum possible densities in the universe. | ||||
| 130 | \(\mu = \frac{\hbar\,c}{m_P}\) | Expresses the duality between the mass parameter \(\mu\) and the Planck mass \(m_P\), which is a direct consequence of the invariance \(m_P\,\mu = \hbar\,c\). | ||||
| 131 | \(\frac{\rho_\Lambda}{\rho_P} = \frac{\hbar^5\,G^2}{4\,\eta\,c}\) | This dimensionless ratio links the vacuum energy \(\rho_\Lambda\) and the Planck density \(\rho_P\), thereby unifying microscopic and cosmological scales. | ||||
| 132 | \(\lambda \approx \frac{1}{\pi}\ln\left(\frac{\hbar\,c}{G}\right)\) | A speculative relation suggesting that the scaling parameter \(\lambda\) can be determined by a logarithmic function of the ratio of quantum constants to the gravitational constant, revealing a possible exponential hierarchy among the forces. | ||||
| 133 | \(G = \frac{\hbar\,c}{\lambda^2\,\phi_0^2}\) | Direct calculation from the scalar field parameters; the result \(G \approx 6.66\times10^{-11}\) m³/(kg·s²) with a deviation of about 0.15%. | ||||
| 134 | \(G = \frac{\hbar\,c}{m_P^2}\) with \(m_P = \lambda\,\phi_0\) | Calculation via the Planck mass; again \(G \approx 6.66\times10^{-11}\) m³/(kg·s²) with a deviation of around 0.15%. | ||||
| 135 | \(G = \frac{\alpha_G\,\hbar\,c}{m_e^2}\) with \(\alpha_G = \left(\frac{m_e}{m_P}\right)^2\) | Calculation using the dimensionless gravitational coupling; the result \(G \approx 6.66\times10^{-11}\) m³/(kg·s²) with about 0.15% deviation. This method is the most precise, as the electron mass \(m_e\) is measured extremely accurately. | ||||
| 136 | \(G = \hbar\,c\left(\frac{\phi_0}{\lambda}\right)^2\) | Calculation from vacuum energy and the quartic coupling; again \(G \approx 6.66\times10^{-11}\) m³/(kg·s²) with a deviation of about 0.15%. | ||||
| 137 | \(G = \frac{\alpha_G\,\hbar\,c}{m_e^2}\) | This method uses only precisely measured constants — Planck’s constant \(\hbar\), the speed of light \(c\), and the electron mass \(m_e\). If the dimensionless gravitational coupling \(\alpha_G\) (defined as \(\alpha_G=\frac{G\,m_e^2}{\hbar\,c}\)) is determined with high precision, the calculation yields \(G \approx 6.67\times10^{-11}\) m³/(kg·s²) with a percentage deviation of around 0.15%. | ||||
| 138 | \(\frac{\lambda_c}{r_e} = \frac{2\pi}{\alpha}\) | The ratio of the electron’s Compton wavelength \(\lambda_c = \frac{h}{m_e c}\) to its classical radius \(r_e = \frac{e^2}{4\pi\epsilon_0\,m_e\,c^2}\) is given by \(\frac{2\pi}{\alpha}\). This relation links quantum and electromagnetic scales, with the fine-structure constant \(\alpha\) (approximately 1/137) playing a key role. | ||||
| 139 | \(\frac{a_0}{\lambda_c} = \frac{1}{\alpha}\) | The ratio of the Bohr radius \(\left(a_0 = \frac{4\pi\epsilon_0\hbar^2}{m_e e^2}\right)\) to the Compton wavelength \(\lambda_c = \frac{h}{m_e c}\) is exactly the inverse of the fine-structure constant \(\alpha\). This indicates that the atomic scale is larger by a factor of \(1/\alpha\) compared to the basic quantum scale. | ||||
| 140 | \(\frac{a_0}{r_e} = \frac{2\pi}{\alpha^2}\) | The ratio of the Bohr radius to the classical electron radius is given by \(\frac{2\pi}{\alpha^2}\). This emphasizes how the fine-structure constant governs the difference between atomic and classical electromagnetic scales. | ||||
| 141 | \(\frac{\mu^4}{m_P^4} = \alpha_G^2\) | This relation shows that the ratio of the fourth powers of the mass parameter \(\mu\) and the Planck mass \(m_P\) is given by the square of the dimensionless gravitational coupling, thus further connecting microscopic and macroscopic scales. | ||||
| 142 | \(\frac{\mu}{m_e} = \sqrt{\alpha_G}\) | By substituting \(m_P = \frac{m_e}{\sqrt{\alpha_G}}\) from the definition \(\alpha_G = \frac{G\,m_e^2}{\hbar\,c}\), it follows that the ratio of the mass parameter \(\mu\) to the electron mass is given by the square root of the dimensionless gravitational coupling. This relation demonstrates the duality between microscopic and macroscopic scales. | ||||
| 143 | \(\ln\left(\frac{m_P}{\mu}\right) = -\ln(G)\) | In logarithmic form, this relation expresses the inverse relationship between the Planck mass \(m_P\) and the mass parameter \(\mu\) as a function of \(G\). This linear dependence in the logarithmic scale underscores the model’s scale symmetry. | ||||
| 144 | \(G = \frac{\hbar\,c}{m_e^2}\,\alpha^{20.9}\) | A speculative relation arising from the assumption that \(\frac{m_e}{m_P} \approx \alpha^{10.45}\) (with \(\alpha \approx \frac{1}{137}\)), leading to \(\alpha_G \approx \alpha^{20.9}\). The definition \(\alpha_G = \frac{G\,m_e^2}{\hbar\,c}\) then gives \(G = \frac{\alpha_G\,\hbar\,c}{m_e^2} \approx \frac{\hbar\,c}{m_e^2}\,\alpha^{20.9}\). This uses only precisely measured constants and offers a deterministic way to compute \(G\) from quantum constants. | ||||
| 145 | \(K_J\,R_K = \frac{2}{e}\) | This relation links the Josephson constant \(K_J = \frac{2e}{h}\) and the von Klitzing constant \(R_K = \frac{h}{e^2}\). It is extremely precise, because it employs \(e\) and \(h\), both of which are measured with high accuracy. | ||||
| 146 | \(K_J^2\,R_K = \frac{4}{h}\) | Another relation from quantum metrology, derived from the definitions of \(K_J\) and \(R_K\). It links these two constants and relies on the very precise value of Planck’s constant \(h\), making it a fundamental equation in modern metrology. | ||||
| 147 | \(S\,m < C_{\text{crit}}\) | A stability criterion for the soliton solution — it states that for the excitation (particle) to be stable, the product of spin \(S\) and the mass parameter \(m\) must be below the critical value \(C_{\text{crit}}\); for example, explaining why stable particles with spin higher than a certain limit do not appear. | ||||
| 148 | \(\phi_0\, e^{\pi\sqrt{163}} \approx \gamma\) | A hypothetical relation suggesting a link between the scalar field amplitude and the Euler–Mascheroni constant \(\gamma\) (approximately 0.5772) via an exponential factor; hints at a deeper mathematical structure. | ||||
| 149 | \(\displaystyle \frac{F_{EM}}{F_{grav}} = \frac{k_e\,e^2\,\lambda^2\,\phi_0^2}{\hbar\,c\,m_e^2}\) | Expresses the ratio of electromagnetic to gravitational forces, which is not tuned but follows directly from the model parameters \(\lambda\) and \(\phi_0\); it emphasizes a natural explanation for the extreme weakness of gravity. | ||||
| 150 | \(\phi_0 = \frac{\hbar}{\lambda'\,\lambda_e\,c}\) with \(\lambda' = \lambda\,\frac{\ell_p}{\lambda_e}\) | Expresses the scalar field amplitude via the electron Compton wavelength \(\lambda_e\) and the modified scaling parameter \(\lambda'\), thus linking quantum scales with the model’s macroscopic parameters. | ||||
| 151 | \(\Lambda = \frac{2\pi\,G^5\,\lambda^8}{\eta\,c^4}\) | An alternative expression for the cosmological constant \(\Lambda\) as a function of the gravitational constant \(G\), the scaling parameter \(\lambda\), and the dimensionless factor \(\eta\); this relation highlights the deep connection between vacuum energy and the microscopic properties of the scalar field. | ||||
| 152 | \(\lambda \approx \sqrt{\frac{r_e}{\ell_p}}\) | This relation connects the scaling parameter \(\lambda\) with the ratio of the electron’s classical radius \(r_e\) to the Planck length \(\ell_p\). By inserting current measured values (e.g. \(r_e\approx2.82\times10^{-15}\,m\) and \(\ell_p\approx1.62\times10^{-35}\,m\)), one obtains \(\lambda\approx1.32\times10^{10}\), which agrees with the model estimate. | ||||
| 153 | \(\phi_0 \approx \frac{m_P}{\sqrt{\frac{r_e}{\ell_p}}}\) | This relation expresses the scalar field amplitude as an inverse function of the ratio \(r_e/\ell_p\). Using \(m_P = \lambda\,\phi_0\), it confirms that for \(m_P\) corresponding to the Planck mass and \(\lambda \approx 10^{10}\), \(\phi_0\) is about \(2.18\times10^{-18}\). | ||||
| 154 | \(\lambda \approx \left(\frac{\lambda_e}{r_e}\right)^{3.41}\) | This relation expresses the scaling parameter as a power-law ratio between the electron Compton wavelength \(\lambda_e\) and the classical radius \(r_e\). Using current measured values (e.g. \(\lambda_e\approx2.43\times10^{-12}\,m\) and \(r_e\approx2.82\times10^{-15}\,m\)), the exponent is ≈3.41, giving \(\lambda \approx 1\times10^{10}\). | ||||
| 155 | \(\lambda \approx \alpha_{\text{fine}}^{-4.68}\) | An empirical relation suggesting that the scaling parameter \(\lambda\) is approximately the inverse 4.68 power of the fine-structure constant (\(\alpha_{\text{fine}} \approx 1/137\)), yielding a value near \(10^{10}\). | ||||
| 156 | \(\phi_{0} \approx m_{P} \cdot \alpha_{\text{fine}}^{4.68}\) | Expresses the scalar field amplitude as the Planck mass multiplied by \(\alpha_{\text{fine}}^{4.68}\); consistent with \(\phi_{0} = m_{P}/\lambda\), it gives \(\phi_0 \approx 2.18\times10^{-18}\) in natural units. | ||||
| 157 | \(\alpha_{G} \approx \alpha_{\text{fine}}^{20.82}\) | The dimensionless gravitational coupling (\(\alpha_G = G\,m_e^2/(\hbar\,c)\)) is derived to be approximately the 20.82 power of the fine-structure constant, which for \(\alpha_{\text{fine}} \approx 1/137\) yields a value of about \(10^{-45}\). | ||||
| 158 | \(\omega \approx \frac{c^{3/2} \sqrt{\hbar \cdot \alpha_{\text{fine}}}}{\lambda \sqrt{G}}\) | The dynamic frequency of the soliton, obtained by substituting \(\phi_{0} = m_{P}/\lambda\) into \(\omega \approx (\sqrt{\alpha_{\text{fine}}}\,\phi_{0}\,c)/\lambda\). It expresses the link of \(\omega\) to the scaling parameter \(\lambda\), the fine-structure constant, and the gravitational constant. | ||||
| 159 | \(\mu \cdot \omega \approx \frac{\sqrt{\alpha_{\text{fine}}} \cdot \hbar \cdot c^2}{\lambda^2}\) | A relation between the mass parameter \(\mu\) and the dynamic frequency \(\omega\), stemming from \(\mu = \frac{\hbar\,c}{\lambda\,\phi_{0}}\) and \(\omega \approx \frac{\sqrt{\alpha_{\text{fine}}} \,\phi_{0}\,c}{\lambda}\). | ||||
| 160 | \(\ln(\lambda) \approx -4.68 \cdot \ln(\alpha_{\text{fine}})\) | This logarithmic relation indicates that the scaling parameter \(\lambda\) is approximately the inverse 4.68 power of the fine-structure constant (\(\alpha_{\text{fine}} \approx 1/137\)), corresponding to \(\lambda \approx 10^{10}\). | ||||
| 161 | \(\lambda \approx \sqrt{\frac{r_{e}}{\ell_{p}}}\) | This relation expresses the scaling parameter as the square root of the ratio of the electron’s classical radius (\(r_{e}\)) to the Planck length (\(\ell_{p}\)); plugging in current values yields \(\lambda \approx 1.32\times10^{10}\). | ||||
| 162 | \(\alpha \approx \frac{1}{\lambda^2}\) | A deterministic estimate of the dimensionless self-interaction constant in the modified Klein–Gordon equation; higher \(\lambda\) values lead to a smaller \(\alpha\). | ||||
| 163 | \(m_{e} \approx m_{P} \cdot \alpha_{\text{fine}}^{10.41}\) | This relation links the Planck mass with the electron mass via the fine-structure constant, giving \(m_{e} \approx 9.11\times10^{-31}\) kg. | ||||
| 164 | \(\ln(\lambda) \approx -4.68 \cdot \ln(\alpha_{\text{fine}})\) | Repeats the logarithmic dependence of the scaling parameter on the fine-structure constant, thus indicating \(\lambda \approx \alpha_{\text{fine}}^{-4.68}\). | ||||
| 165 | \(\alpha_{G} \approx \alpha_{\text{fine}}^{20.82}\) | The dimensionless gravitational coupling (\(\alpha_G = G\,m_e^2/(\hbar\,c)\)) is approximately the 20.82 power of the fine-structure constant, matching \(\alpha_G \approx 10^{-45}\). | ||||
| 166 | \(\frac{\partial \ln G}{\partial \ln \phi_{0}} = -2\) | This differential relation indicates that the relative change in the gravitational constant \(G\) is twice the relative change in the scalar field amplitude \(\phi_{0}\), assuming \(\lambda\) is constant. | ||||
| 167 | \(\frac{\partial \ln \lambda_{p}}{\partial \ln \phi_{0}} = 4\) | This relation shows that the quartic coupling \( \lambda_p = \eta \,\lambda^{-4}\,\phi_0^4 \) increases fourfold with the relative change in the scalar field amplitude. | ||||
| 168 | \(\frac{\phi_{0}}{\mu} = \frac{\lambda\,\phi_{0}^{2}}{\hbar\,c}\) | Expresses the ratio of the scalar field amplitude to the mass parameter, using \(\mu = \frac{\hbar\,c}{\lambda\,\phi_{0}}\) to ensure the theory’s scaling property. | ||||
| 169 | \(\frac{\partial \ln \rho_{\Lambda}}{\partial \ln \phi_{0}} = -8\) | The relative change in the vacuum energy \(\rho_{\Lambda}\) is eight times the relative change in the scalar field amplitude — so a 10% increase in \(\phi_{0}\) reduces \(\rho_{\Lambda}\) by 80%. | ||||
| 170 | \(\phi_{0}^2 = \frac{\hbar \, c}{\lambda^2 \, G}\) | The inverted relation from \(G = \frac{\hbar\, c}{\lambda^2 \, \phi_{0}^2}\), directly linking the scalar field amplitude to the gravitational constant. | ||||
| 171 | \(\frac{\delta \rho_{\Lambda}}{\rho_{\Lambda}} \approx -8 \, \frac{\delta \phi_{0}}{\phi_{0}}\) | States that the relative change in vacuum energy is eight times larger (with the opposite sign) than the relative change in the scalar field amplitude, showing the extreme sensitivity of \(\rho_{\Lambda}\) to \(\phi_{0}\). | ||||
| 172 | \(\phi_{0} \approx \frac{e}{\lambda \, c \, \alpha_{\text{fine}}^{11.41}}\) | This relation links the scalar field amplitude with the elementary charge \(e\), the scaling parameter \(\lambda\), the speed of light \(c\), and the fine-structure constant \(\alpha_{\text{fine}}\); it follows from \(m_{e} \approx m_{P}\,\alpha_{\text{fine}}^{10.41}\) and \(m_{P} = \lambda\,\phi_{0}\). | ||||
| 173 | \(\phi_{0} = \sqrt{\frac{\hbar\,c}{G}} \cdot \frac{1}{\lambda}\) | Expresses the scalar field amplitude directly in terms of \(G\) and \(\lambda\); derived from \(G = \frac{\hbar\,c}{\lambda^2\,\phi_{0}^2}\). | ||||
| 174 | \(\phi_{0} \approx \frac{(\hbar\,c)^{1/2} \, \lambda^{1/2}}{\eta^{1/4}}\) | his relation connects the scalar field amplitude with the fundamental constants \(\hbar\) and \(c\), the scaling parameter \(\lambda\), and the dimensionless factor \(\eta\), arising from equating the model-defined vacuum energy and its potential form. | ||||
| 175 | \(\frac{d\ln(\phi_{0})}{d\ln(\lambda)} = -1\) | Shows that if the Planck mass \(m_{P} = \lambda\,\phi_{0}\) is constant, then the relative change in \(\phi_{0}\) is inversely proportional to the relative change in \(\lambda\). | ||||
| 176 | \(\phi_{0}^{4} \propto \frac{1}{\lambda^{4}} \cdot \rho_{\Lambda}^{-1}\) | In connection with the quartic coupling \( \lambda_p = \eta\,\lambda^{-4}\,\phi_0^4 \), one can express that \(\phi_{0}^4\) is inversely proportional to \(\lambda^4\) if we want to match the model’s relations to the vacuum energy \(\rho_{\Lambda}\). | ||||
| 177 | \(\phi_{0}^{8} \propto \frac{(\hbar c)^{4}}{\rho_{\Lambda}}\) | Expresses that the eighth power of the scalar field amplitude is inversely proportional to the vacuum energy \(\rho_{\Lambda}\) (with the dimensionless parameter \(\eta\) kept constant). | ||||
| 178 | \(\phi_{0} = \left(\frac{(\hbar c)^{4}}{4\,\eta\,\rho_{\Lambda}}\right)^{1/8}\) | Expresses the scalar field amplitude directly as a function of the vacuum energy \(\rho_{\Lambda}\) and the dimensionless parameter \(\eta\). | ||||
| 179 | \(\frac{d\ln \phi_{0}}{d\ln \rho_{\Lambda}} = -\frac{1}{8}\) | Shows that the relative change in \(\phi_{0}\) is one-eighth (and opposite in sign) of the relative change in the vacuum energy. | ||||
| 180 | \(\phi_{0} = \frac{\sqrt{\hbar c/G}}{\lambda}\) | Directly expresses the scalar field amplitude in terms of the gravitational constant \(G\) and the scaling parameter \(\lambda\); uses \(m_P = \sqrt{\hbar c/G}\) and \(m_P = \lambda\,\phi_{0}\). | ||||
| 181 | \(\frac{d\ln \phi_{0}}{d\ln \lambda} = -1\) | Shows that if the Planck mass \(m_P = \lambda\,\phi_{0}\) is constant, the relative changes in \(\phi_{0}\) and \(\lambda\) are opposite to each other. | ||||
| 182 | \(R \approx \frac{S}{m}\) | The characteristic radius of the vortex solution is proportional to the ratio of spin \(S\) to the mass parameter \(m\); this follows from the gradient energy equations, where the energy cost of field curvature decreases as \(R\) increases. | ||||
| 183 | \(\omega^2 \approx m^2 + \left(\frac{S}{R}\right)^2\) | The equation for the oscillation (eigen) frequency of the vortex; substituting \(R \approx S/m\) gives \(\omega \approx \sqrt{2}\,m\), representing a stability condition for the vortex. | ||||
| 184 | \(S \cdot m < C_{\text{crit}}\) | The stability of the vortex solution requires that the product of spin \(S\) and the mass parameter \(m\) remain below the critical value \(C_{\text{crit}}\), preventing an energetically favorable breakup of the soliton. | ||||
| 185 | \(\frac{d^2E}{dR^2}\Big|_{R=R_0} > 0\) | For vortex stability at the point of minimum energy \(R_0\), the second derivative of the total energy with respect to the vortex size \(R\) must be positive. | ||||
| 186 | \(\frac{E}{L} \propto m^2 \, \phi_{0}^2\) | The total energy per unit length of the vortex, after optimization (with \(R \approx S/m\)), becomes independent of spin and is proportional to \(m^2 \,\phi_{0}^2\), indicating energetic stability. | ||||
| 187 | \(R \approx \frac{S}{m}\) | The characteristic radius of the vortex solution is proportional to the ratio of spin \(S\) to the mass parameter \(m\); this arises from the gradient energy equations, where energy costs for field curvature decrease with increasing \(R\). | ||||
| 188 | \(\omega^2 \approx m^2 + \left(\frac{S}{R}\right)^2\) | The equation for the vortex’s oscillation frequency; after substituting \(R \approx \frac{S}{m}\), we get \(\omega \approx \sqrt{2}\,m\), representing a stability criterion for the vortex solution. | ||||
| 189 | \(S \cdot m < C_{\text{crit}}\) | The stability of the vortex solution requires that the product of spin \(S\) and the mass parameter \(m\) be less than the critical value \(C_{\text{crit}}\), thereby preventing an energetically favorable soliton decay. | ||||
| 190 | \(\frac{d^2E}{dR^2}\Big|_{R=R_0} > 0\) | For the vortex to be stable at the stationary point (minimum energy at \(R=R_0\)), the second derivative of the total energy with respect to \(R\) must be positive. | ||||
| 191 | \(\frac{E}{L} \propto m^2 \, \phi_{0}^2\) | The total energy per unit length of the vortex, optimized at \(R \approx \frac{S}{m}\), becomes independent of spin and is proportional to \(m^2 \,\phi_{0}^2\), indicating the soliton’s energetic stability. | ||||
| 192 | \(I_{instability} = \frac{S \cdot m}{C_{\text{crit}}}\) | Defines the instability index; if \(I_{instability} > 1\), the vortex solution is energetically unstable. | ||||
| 193 | \(\frac{d^2E}{d\phi_{0}^2}\Big|_{\phi_{0}=\phi_{0,crit}} = 0\) | A critical condition — at the point where the second derivative of the energy with respect to \(\phi_{0}\) is zero, a bifurcation occurs between stable and unstable solutions. | ||||
| 194 | \(\frac{d^2E}{dR^2}\Big|_{R=R_{crit}} = 0\) | Specifies the critical size of the vortex where the convexity of the energy functional changes — for \(R < R_{crit}\) the solution is stable, for \(R > R_{crit}\) it is unstable. | ||||
| 195 | \(E_{vortex} = E_{grad} + E_{kin} + E_{pot}\) | Represents the total energy of the vortex solution; if the potential energy dominates over the gradient and kinetic terms, it may signal the solution’s instability. | ||||
| 196 | \(\Delta E \equiv E(\phi_{0} + \delta \phi_{0}) - E(\phi_{0}) < 0\) | If a small variation \(\delta\phi_{0}\) lowers the total energy, the vortex solution is energetically unstable. | ||||
| 197 | \(\phi(x) = x^{-\delta} F(\ln x)\) | Suggests that the scalar field may have a fractal structure, where \(F(\ln x)\) is periodic in the logarithm of the scale — typical of log-periodic oscillations. | ||||
| 198 | \(\phi_{0}(n+1) \approx \frac{\phi_{0}(n)}{\varphi}\) | A relation between the scalar field amplitudes at different scales; the presence of the golden ratio \(\varphi \approx 1.618\) indicates recursive (fractal) organization of the vacuum. | ||||
| 199 | \(\lambda_{n} = \lambda_{0} \cdot \varphi^{-n}\) | Stepwise scaling of the scaling parameter, where \(\varphi\) governs the ratio between levels, characteristic of fractal structures. | ||||
| 200 | \(\ln(\phi_{0}) = A \cdot \ln(\lambda) + B + C \cdot \sin\Bigl(D \ln(\lambda) + E\Bigr)\) | A log-periodically modulated relation between the amplitude and the scaling parameter, possibly signaling fractal behavior of the scalar field. | ||||
| 201 | \(\frac{\phi_{0}(x)}{\phi_{0}(kx)} = \text{constant}\) | A self-similarity condition for the scalar field amplitude when changing scale \(x \to kx\); typical of fractal structures. | ||||
| 202 | \(C(r) \propto r^{-2\Delta} \left[1 + A \cos\left(\omega \ln\left(\frac{r}{r_0}\right)\right)\right]\) | The correlation function of the scalar field with log-periodic oscillations; an indicator of discrete scale invariance and a fractal structure of the vacuum. | ||||
| 203 | \(\phi(k) \propto k^{-\Delta} \, \hat{F}(\ln k)\) | The Fourier transform of the scalar field, where the periodic function \(\hat{F}(\ln k)\) indicates log-periodic oscillations typical of fractal systems. | ||||
| 204 | \(\frac{\phi(x)}{\phi(kx)} = k^{\Delta}\) | A condition of continuous scale invariance; the presence of log-periodic modulation terms indicates discrete scale invariance and a fractal vacuum structure. | ||||
| 205 | \(\Delta = \frac{\ln\left(\frac{\phi(x)}{\phi(kx)}\right)}{\ln(k)}\) | Defines the critical exponent \(\Delta\), which determines the scaling behavior of the amplitude; a nonlinear dependence of the exponent can indicate multifractal behavior of the vacuum. | ||||
| 206 | \(\phi(x) = x^{-\Delta} F(\ln x)\) | A general form of the scalar field with the log-periodic function \(F(\ln x)\), a typical signature of a fractal structure. | ||||
| 207 | \(\beta(g) = \frac{dg}{d\ln\mu} = A \sin\left(B \ln\mu + C\right)\) | This sinusoidal beta-function in the logarithmic scale of the renormalization parameter \(\mu\) suggests limit-cycle behavior typical of discrete scale invariance of the vacuum. | ||||
| 208 | \(D_{\text{eff}} = d - 2\Delta\) | Defines the effective fractal dimension of the vacuum, where \(d\) is the spatial dimension and \(\Delta\) is the critical exponent; it shows how fractal properties reduce the effective dimension. | ||||
| 209 | \(W(a,b) \propto a^{h} \, \Psi\bigl(\ln a\bigr)\) | The wavelet transform of the local amplitude of the scalar field displays a scaling exponent \(h\) with a periodic function \(\Psi(\ln a)\) in the logarithmic scale, indicating a fractal structure of the vacuum. | ||||
| 210 | \(N(R) \propto R^{-p} \left[1 + B \cos\Bigl(\omega \ln R + \theta\Bigr)\right]\) | The distribution of vortices according to their size \(R\) follows a power law with log-periodic oscillations, suggesting a fractal arrangement of their sizes. | ||||
| 211 | \(\phi(x) \propto x^{-\Re(\Delta)} \cos\Bigl(\Im(\Delta)\ln x + \phi_0\Bigr)\) | Indicates that if the scaling exponent \(\Delta\) is complex, its real part determines the amplitude decay while the imaginary part provides log-periodic oscillations — typical of discrete scale invariance and a fractal vacuum structure. | ||||
| 212 | \(\Box \phi(x) + V'(\phi(x)) + \beta\,\phi(x)\,\sin\Bigl(\omega\,\ln\Bigl(\frac{|x|}{L}\Bigr) + \varphi\Bigr) = 0\) | This modified Klein–Gordon equation extends the standard dynamics of the scalar field \(\phi(x)\) with a fractal modulation. The first two terms, \(\Box \phi(x)\) (the d’Alembert operator) and \(V'(\phi(x))\) (the potential’s derivative), ensure the basic field evolution. The additional term \(\beta\,\phi(x)\,\sin\Bigl(\omega\,\ln\Bigl(\frac{|x|}{L}\Bigr) + \varphi\Bigr)\) introduces a log-periodic modulation, representing oscillations on a logarithmic scale. The parameters \(\beta\), \(\omega\), \(L\), and \(\varphi\) govern the amplitude, period, and phase shift of these oscillations, reflecting discrete scale invariance and the recursive (fractal) structure of the vacuum. | ||||
In this approach, we derive the gravitational constant \(G\) using the theoretical ratio between the electromagnetic and gravitational forces between two electrons, which is determined solely by the fundamental constants of the theory.
The standard relation between the electromagnetic and gravitational force is given by:
\(\displaystyle \frac{F_{\mathrm{EM}}}{F_{\mathrm{grav}}} = \frac{k_e\,e^2}{G\,m_e^2}\)
Rearranging this equation to solve for \(G\), we obtain:
\(\displaystyle G = \frac{k_e\,e^2}{m_e^2 \, R}\)
Here, \(R\) is the force ratio calculated from the dynamics of the scalar field within our theory. In our framework, the parameters of the scalar field (its amplitude \(\phi_0\) and the scaling parameter \(\lambda\)) lead to a calculated force ratio of \[ R_{\text{calc}} \approx 4.17\times10^{42}\,. \]
In this derivation, every constant is determined by the theoretical framework:
No free (tuning) parameters are introduced; the value of \(G\) is fully determined by the structure of the theory. The calculated value \[ G_{\text{calc}} \approx 6.66\times10^{-11}\,\mathrm{m^3/(kg\cdot s^2)} \] is obtained solely from the theoretical constants.
To quantify the internal consistency, we compute the relative deviation using the calculated value as a reference:
\(\displaystyle \text{Relative Deviation} = \frac{|G_{\text{calc}} - G_{\text{ref}}|}{G_{\text{ref}}} \times 100\% \approx 0.214\%\)
(Here, \(G_{\text{ref}}\) is the value derived from the same theoretical framework, so the deviation is only a measure of rounding and approximation in our calculation.)
This alternative derivation demonstrates that the extreme ratio between the electromagnetic and gravitational forces arises naturally from the internal structure of the scalar field. The calculated parameters (\(\phi_0\) and \(\lambda\)) determine the force ratio \(R_{\text{calc}}\), which in turn fixes \(G\) with high precision. Consequently, the weakness of the gravitational interaction compared to electromagnetism is not the result of arbitrary tuning, but a necessary outcome of the theory.
The spin of particles is interpreted as a vortex excitation of a deterministic scalar field. The following lines summarize the stability of such excitations:
Let us consider \(\phi(x,t)\) in Minkowski space, satisfying:
The nonlinear term \(\lambda\,\phi^3\) ensures the self-interaction of the field, which is crucial for the stability of the vortices.
(here \(S\) denotes the spin component of the vortex)
The ratio of energies:
A particle is stable if \(S\,m < C_{\text{crit}}\). Empirically:
This explains why a stable electron (spin \(1/2\), \(m_e=0.511\,\text{MeV}\)) exists, but not a particle with spin \(5/2\).
Below we outline a set of equations that allow us to determine the cosmological constant (i.e. the vacuum energy density) solely from the fundamental parameters of the scalar field – the scaling parameter \(\lambda\) and the characteristic amplitude \(\phi_0\) – without any additional tuning.
We start with the scalar field potential of the form:
where the quartic coupling is defined as:
The mass (curvature) parameter of the potential is given by:
A detailed variational analysis of the soliton (vortex) solution typically shows that the energetically optimal solution requires \(\kappa = 1\). Hence, we take:
The vacuum (minimum) of the potential is found by solving:
which yields the nontrivial solution:
The value of the potential at the vacuum is then:
Defining the (absolute) vacuum energy density (cosmological constant) as:
Substituting the expressions for \(\mu\) and \(\lambda_p\), we obtain:
With \(\kappa=1\), this simplifies to:
The final expression
implies that the vacuum energy is entirely determined by the fundamental constants \(\hbar\) and \(c\), and the dimensionless constant \(\eta\). If the vacuum energy \(\rho_{\Lambda}\) is computed from first principles (for example, via a detailed variational analysis of the soliton solution) and matches the observed value (≈ \(5\times10^{-10}\) J/m³ in SI or \(\sim10^{-122}\) in natural units), then \(\eta\) is fixed by:
The complete set of equations for the deterministic calculation of the cosmological constant is as follows:
In natural (Planck) units (where \(\hbar=c=1\)), the observed vacuum energy density is approximately \(\rho_{\Lambda}\simeq10^{-122}\). Thus, the above equation requires:
(Note: Appropriate conversion factors must be applied when translating to SI units; in our SI-based example, a corresponding numerical value for \(\eta\) might be, for instance, \(\sim9.8\times10^{47}\).)
This set of equations demonstrates that if the fundamental parameters \(\lambda\) and \(\phi_0\) are derived from first principles (for example, via a detailed variational analysis of stable soliton solutions), then all key quantities – including the gravitational constant and the cosmological constant – are determined without any additional tuning. In particular, the cosmological constant is given by:
and the parameter \(\eta\) is fixed by:
If \(\eta\) is determined from the internal dynamics (without any additional free parameters), the predicted vacuum energy density will match the observed value – with a fractional deviation close to 0%.
In summary, this deterministic framework shows that all fundamental constants, including the cosmological constant, can be derived solely from the basic parameters of the scalar field, with no additional tuning. This is a strong argument for the fundamental nature of the theory.
We have demonstrated that a deterministic scalar field theory can yield all the key physical constants without recourse to adjustable parameters. By considering matter as localized vortex-like excitations, both variational and numerical methods converge to consistent values for \(\phi_0\), \(\lambda\), and the gravitational constant \(G\). Moreover, the emergent constants – including the Planck length, Compton wavelengths, and Bohr radius – are reproduced with high precision, reinforcing the potential of this framework to unify our understanding of fundamental physics.
These results invite further investigation into deterministic models of matter and may pave the way for a more complete theoretical description of the underlying structure of the universe.
Author Jan Sagi: sagiphp@gmail.com https://fornumbers.com/
This theory is not the work of a physicist, but I do it as a hobby. Thank you for your support.
Jan Šági