What question did this study set out to answer?

To derive a unified Lagrangian where electromagnetism and gravity are connected through a 5-dimensional framework.

May 3, 2026Open Access

Lagrangian of the Fine-Structure Constant: Analytical Derivation of Quantum Gravity (PEPS 5D Formulation)

Key Points

To derive a unified Lagrangian where electromagnetism and gravity are connected through a 5-dimensional framework.
Developed a unified Lagrangian consisting of three terms: volume, electromagnetic, and hinge terms.
Defined fundamental cell parameters like Planck sphere radius and informational volume from holography.
Analyzed derived relationships between fine-structure constant and gravitational coupling.
Unified Lagrangian expressed as L_unified = L_volume + L_EM + L_hinge.
Derived bare gravitational coupling without free parameters, indicating structural coherence.
Establishes fine-structure constant and bare gravitational coupling as eigenvalues of a topological operator.

Abstract

We present a unified Lagrangian formulation in which electromagnetism and gravity emerge from complementary projections of a single 5-dimensional Projected Entangled Pair State (PEPS). The fundamental cell is a Planck sphere of radius lₚ, and its informational volume is axiomatically defined as Iₛ = π/6 from spinorial holography. The complete unified Lagrangian is the sum of three terms: Lᵤnified = Lᵥolume + LEM + Lₕinge where: - Lᵥolume = ½∂_μΦ∂^μΦ - μ²/2 (Φ² - (π⁴+1) ) ² (tesseract volume potential, gravitational sector) - LEM = √ (-g (t) ) (ḋ² - 1/64 (∇d) ²) (Polyakov string with tension T=8, electromagnetic sector from Alpha Series, concept DOI: https: //doi. org/10. 5281/zenodo. 19802606) - Lₕinge = Φ² |e^-iπ/6 Ψd - (4/ (3π) ) Ψ_Φ|² (hinge term that locks the string state Ψd to the gravitational volume state Ψ_Φ at the minimal spinorial resolution angle χ = π/6) The hinge term Lₕinge is the mathematical suture that unifies the two sectors. It does not stand alone; it couples Lᵥolume and LEM into a single coherent structure. Solving the combined action on-shell yields the bare gravitational coupling: ln (Gbare mₑ² / ħ c) = -π/3 (π⁴ + 1) No free parameters are introduced. The fine-structure constant α (derived in the Alpha Series) and the bare gravitational coupling αG (derived here) are two eigenvalues of the same topological operator, distinguished only by the dimensionality of their projection: 1D for electromagnetism (circular MPS, bond dimension D=45), 5D for gravity (PEPS, Planck sphere cell). The construction is closed, falsifiable, and requires no external parameters beyond π, the string tension T=8, and the overlap angle π/6. Keywords: Quantum gravity, fine-structure constant, PEPS, MPS, topological invariants, Lagrangian unification, Alpha Series Alpha Series: https: //doi. org/10. 5281/zenodo. 19802606

Lagrangian of the Fine-Structure Constant: Analytical Derivation of Quantum Gravity (PEPS 5D Formulation)

Key Points

Abstract

Cite This Study

Also Consider

Also Consider