[Paper Review] Theoretical calculation of the fine-structure constant and the permittivity of the vacuum

G. B. Mainland, B. Mulligan|arXiv (Cornell University)|May 26, 2017

Electrical and Electromagnetic Research2 references3 citations

TL;DR

This paper proposes a theoretical derivation of the fine-structure constant α and the vacuum permittivity ε₀ by modeling the quantum vacuum as a dielectric medium dominated by virtual, bound lepton-antilepton pairs—primarily positronium-like states. Using classical dielectric theory adapted to quantum vacuum fluctuations, it derives α ≈ 1/139, close to the experimental 1/137.036, and ε₀ ≈ 8.98×10⁻¹² C²/(N·m²), near the experimental 8.85×10⁻¹² C²/(N·m²).

ABSTRACT

Light traveling through the vacuum interacts with vacuum fluctuations similarly to the way that light traveling through a dielectric interacts with ordinary matter. And just as the permittivity of a dielectric can be calculated, the permittivity $ε_0$ of the vacuum can be calculated, yielding an equation for the fine-structure constant $α$. The most important contributions to the value of $α$ arise from the interaction of photons with charged lepton-antilepton vacuum fluctuations that appear in the vacuum as on-shell, bound states. Considering these contributions only to first order in alpha, the fully screened $α\cong 1/(8^2\sqrt{3π/2}) \cong 1/139$.

Motivation & Objective

• To derive the fine-structure constant α and vacuum permittivity ε₀ from first principles using quantum vacuum polarization effects.
• To model the vacuum as a dielectric medium influenced by virtual, bound states of charged lepton-antilepton pairs.
• To explain the value of α through the collective response of virtual positronium-like states to electromagnetic fields.
• To assess the contribution of virtual quark-antiquark pairs and other virtual states to α and ε₀, showing their negligible impact at this approximation level.
• To explore the implications of vacuum structure on the speed of light in the early universe, particularly through changes in ε₀.

Proposed method

• Adapts classical dielectric permittivity theory (based on oscillating dipoles) to the quantum vacuum by replacing real oscillators with virtual, bound lepton-antilepton pairs.
• Uses the Heisenberg uncertainty principle to estimate the energy and time scales of virtual particle fluctuations.
• Applies the classical permittivity formula to virtual positronium states, treating them as harmonic oscillators with resonant frequency ω₀ and dipole moment p_j.
• Derives the vacuum permittivity ε₀ as a sum over contributions from virtual positronium, muon-antimuon, and tau-antitau pairs, each contributing equally.
• Converts the expression for ε₀ into a formula for α using the standard definition α = e²/(4πε₀ħc), leading to α ≈ 1/139.
• Evaluates the consistency of the result by expressing ε₀ in terms of h, e, c, and μ₀, yielding values close to experimental measurements.

Experimental results

Research questions

• RQ1 Can the fine-structure constant α be derived from the vacuum polarization due to virtual lepton-antilepton bound states?
• RQ2 What is the contribution of virtual positronium-like states to the vacuum permittivity ε₀?
• RQ3 How do virtual quark-antiquark pairs affect the value of α, and why are their contributions negligible?
• RQ4 What implications does this model have for the speed of light in the early universe, given changes in ε₀?
• RQ5 Could higher-order vacuum effects, such as virtual diparapositronium formation, reduce the effective α and improve agreement with experiment?

Key findings

• The theoretical value of 1/α is derived as 8²√(3π/2) ≈ 138.93, which is within 1.4% of the experimental value 137.036.
• The calculated vacuum permittivity is ε₀ ≈ 8.98×10⁻¹² C²/(N·m²), in close agreement with the experimental value of 8.85×10⁻¹² C²/(N·m²).
• Virtual positronium, muon-antimuon, and tau-antitau pairs each contribute equally to ε₀, with the total contribution being three times that of a single positronium state.
• The derivation shows that the electron mass cancels out, implying that the contribution to ε₀ is independent of the lepton's mass, a consequence of the scaling in the energy and frequency relations.
• Contributions from virtual quark-antiquark pairs (e.g., π⁰, η, η′ mesons) are suppressed due to high resonance frequencies and large decay widths, making them negligible at this level of approximation.
• The model suggests that in the early universe, when virtual lepton pairs could not bind due to high temperature, ε₀ would have been significantly smaller, implying a potentially larger speed of light at that time.