Operator Entropy and the Emergent Unification of Gravity and Gauge Interactions

! 1! Operator Entropy and the Emergent Unification of Gravity and Gauge Interactions Borros Arneth, Philipps University Marburg, Justus Liebig University Giessen, Germany, [email protected] Abstract Gravitation and gauge interactions are commonly treated as distinct manifestations of nature, yet mounting theoretical evidence suggests a shared informational origin. We present a framework in which spacetime, matter and gauge fields emerge from an entropy-maximizing projection of a complex operator 𝑍 "= 𝑀 %+ 𝑖𝑄 ) acting in a Diagram Hilbert Space (DHS) spanned by diagrammatic entities such as Feynman graphs, Wilson loops and knot invariants. The real and imaginary components of the projection reproduce Einstein’s and Maxwell’s equations, while higher-order correlations generate small, testable deviations. Within this formalism we derive the Bekenstein–Hawking entropy prefactor from diagram counting, reproduce the QCD one-loop β-function, obtain an operator-based renormalization-group flow, explain galactic rotation curves through entropic equilibrium of hidden eigenstates, and predict cross-sector couplings between the mass and charge operators. The approach provides a renormalizable, informationtheoretic route to the unification of gravity and gauge interactions. Introduction Thermodynamic reasoning has long hinted that gravitation may arise from statistical or informational principles. Bekenstein related horizon area to entropy [1]; Hawking showed that black holes radiate thermally [2]; Jacobson demonstrated that the Einstein equation can be interpreted as a local equation of state [3]; and Wald’s Noether-charge construction [4] and Verlinde’s entropic-force arguments [5] reinforced the view that spacetime dynamics can follow from entropy extremization. Gauge interactions, by contrast, are described in the language of local symmetry and field curvature. Wilson’s non-Abelian loops [6], Susskind’s holographic conjecture [7] and the AdS/CFT correspondence [8] revealed that gauge and geometric degrees of freedom can be dual aspects of one underlying structure. Topological quantum field theory [9, 10] and string-theoretic Wilson-loop studies [11] further emphasized the primacy of diagrammatic topology. ! 2! The Diagram Hilbert Space introduced here treats these diagrammatic objects as the fundamental basis of quantum reality. In this space a single complex operator 𝑍 "= 𝑀 %+ 𝑖𝑄 ) encodes both inertial-gravitational and gauge-charge sectors. Physical laws appear as macroscopic projections obtained by maximizing entropy subject to coarse constraints. The formalism naturally reproduces established results of QCD [12, 13] and renormalization theory [14–16], while linking them to gravitational thermodynamics [1– 5, 17–19]. Results Entropy-maximizing projection. A microscopic density matrix 𝜌 on the DHS maximizes 𝑆 = −Tr(𝜌ln2 𝜌) under fixed expectation values of coarse observables {𝒪!}, giving 𝜌⋆ ∝ exp2[− ∑𝜆!! 𝒪!]. Variations of 𝑆 yield field equations for the real and imaginary parts of 𝑍 ": Re{Π#$%[𝑍 "]} ⇒ 𝐺&' = 8𝜋𝐺 𝑇 &',2222222222222222222222222222Im{Π#$%[𝑍 "]} ⇒ ∇&𝐹&' = 0 thereby reproducing Einstein–Maxwell dynamics. Residual correlations proportional to 𝜖 = [𝑀 %, 𝑄 )]/𝑖𝐶 "produce higher-order couplings between curvature and field strength. Black-hole entropy from diagram counting. Enumerating distinct DHS diagrams that pierce a horizon yields a degeneracy Ω(𝐴) ∼ exp2(𝐴/4ℓ( )), reproducing the Bekenstein–Hawking prefactor [1–4, 17–19]. The area law follows from the proportionality of link crossings to horizon area, providing a microscopic origin for gravitational entropy. Gauge-sector consistency. Restricting 𝑄 ) to SU(3) color and coarse-graining over sub-diagrams produces the QCD one-loop β-function, 𝜇𝑑𝑔 𝑑𝜇 = − 𝑔* 16𝜋) (11 3𝐶+−4 3𝑇,𝑛-) recovering asymptotic freedom [12, 13]. The result confirms that the DHS operator algebra reduces to known gauge behavior in the perturbative limit. ! 3! Operator renormalization. A scale map ℛ. that integrates out sub-diagrams generates an intrinsic flow, 𝑑𝑍 " 𝑑ln2 𝜆 = 𝛽/ 0[𝑍 "] = 𝛽1[𝑀 %] + 𝑖𝛽2[𝑄 )] + 𝒪(𝜖), whose projection yields the standard renormalization-group equations [14, 15]. The gravitational coupling’s scale dependence follows from the entropy density of diagrams, consistent with asymptotic-safety expectations [16, 20]. Astrophysical and experimental predictions. Hidden eigenstates of 𝑀 % act as an entropic reservoir. Balancing visible and hidden sectors gives an additional acceleration 𝑔#$%(𝑟) ∝ (𝑐)/𝑟) ∂3ln2 Ω456(𝑟), reproducing the empirical baryon–acceleration relation in galaxies [21, 22] and echoing emergent-gravity scenarios [23, 24]. Weak non-commutativity between 𝑀 % and 𝑄 ) produces small curvature–field couplings 𝑅&'𝐹&.𝐹'., implying deviations from Einstein–Maxwell theory potentially measurable in strong-field or precision electrodynamics experiments. Discussion The results collectively indicate that geometry and gauge structure can be viewed as complementary projections of a single entropic operator. The five main achievements of the theory are: (1) derivation of the Bekenstein–Hawking entropy prefactor from diagram counting; (2) reproduction of the QCD one-loop β-function; (3) operator-based derivation of coupling constants and renormalization-group flow; (4) explanation of galactic rotation curves via entropic equilibrium of hidden eigenstates; and (5) prediction of testable Einstein–Maxwell deviations through mass–charge crossinteractions. These achievements anchor the framework in established physics while offering falsifiable extensions. The operator picture suggests that quantum gravity’s microstructure is not geometric but combinatorial; spacetime and gauge fields arise as statistical averages of diagrammatic configurations. The intrinsic RG flow unifies gravitational and gauge running couplings, hinting that coupling convergence may follow from information balance rather than symmetry enlargement. At cosmological scales, residual entropic curvature naturally provides an effective cosmological constant consistent with Planck-mission constraints [22]. ! 4! Conclusion The operator-entropic formulation supplies a unified, renormalizable foundation for gravity and gauge interactions. It bridges thermodynamic gravity, non-Abelian gauge theory and renormalization under a common informational principle. Future directions include explicit lattice-DHS implementations, quantitative fits to galactic dynamics, and investigation of neutrino-mass hierarchies and CP-violating phases as projections of complex-operator eigenstructures. If validated, the framework would recast the search for quantum gravity as a study of operator entropy in a topologically organized Hilbert space. Methods Diagram Hilbert Space. Basis vectors ∣ 𝐷⟩ correspond to labelled graphs and loop configurations; the inner product identifies topologically equivalent diagrams. Entropy projection. Coarse observables 𝒪! 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