Diagram-Hilbert Space as a Generative Substrate for Matter, Gauge Fields and Gravity

! 1! Diagram-Hilbert Space as a Generative Substrate for Matter, Gauge Fields and Gravity Borros Arneth, Philipps University Marburg, Justus Liebig University Giessen, Germany, [email protected] Abstract Quantum field theory and general relativity describe matter and geometry in formally distinct languages. Recent work on the combinatorial resummation of Feynman diagrams demonstrates that diagrammatic ensembles possess internal algebraic structure rather than being mere bookkeeping devices. Here this observation is developed into a universal formalism: a diagram-Hilbert space in which diagrams form an orthonormal basis and physical observables emerge from entropic projection. Within this space, the Standard Model and gravity appear as complementary limits of one generative substrate. Mass spectra arise from entropy-driven projection, reproducing QCD hadronic hierarchies while preserving quark-level inputs. Coarse-graining of diagram ensembles yields an effective information metric whose entropy gradients obey Einstein-like relations. The resulting framework unifies matter, gauge fields and curvature without introducing additional degrees of freedom and provides falsifiable predictions for hadronic spectra, cosmological corrections and synthetic gauge analogues. 1 Introduction Quantum chromodynamics (QCD) explains confinement and asymptotic freedom [1–3], yet the origin of hadronic mass patterns remains conceptually opaque [4]. The Higgs mechanism sets quark masses but not their emergent hierarchies; gravitational mass appears separately in general relativity [5]. Diagrammatic perturbation theory captures strong-interaction phenomena but scales factorially with order. Kozik [6] showed that Feynman diagrams can be reorganized combinatorially, implying a hidden algebraic structure. Interpreting this structure as a Hilbert manifold suggests a route to unify QFT and gravitation through information-theoretic and entropic principles [7–10]. This paper formalizes that idea. ! 2! 2 Diagram-Hilbert Space Framework Define the Hilbert space ℋ𝒟=span{∣𝐷⟩},--------------------------⟨𝐷"∣𝐷⟩=𝛿[$!],[$] 𝑤(𝐷) (2.1) where each basis vector represents a topologically distinct diagram with weight 𝑤(𝐷). Observables act as 𝑂 4= 5𝑂$!$ $,$!∣𝐷"⟩⟨𝐷∣,------------------⟨𝑂⟩=Tr(𝜌𝑂 4) Tr𝜌----------------- (2.2) Gauge invariance corresponds to invariance of a subspace projector 𝑃 4'.). (2.3). Combinatorial compression maps factorial diagram sums to polynomial classes 𝔠 as shown in [6], indicating that ℋ𝒟possesses a tractable topology analogous to tensornetwork manifolds [11, 12]. 3 Projective Operators and Entropic State Selection Physical states are selected by maximizing entropy 𝑆=−𝑘*Tr(𝜌ln-𝜌) (3.1) under constraints Tr(𝜌𝐶 @+)=𝑐+. The variational solution 𝜌=𝑍,-exp-(−∑𝜆++ 𝐶@+) defines equilibrium ensembles (3.1). A projector 𝑃 4. satisfying 𝑃 4.𝜌𝑃 4.=𝜌 (3.2) isolates stable subspaces. Expectation values computed within these subspaces, ⟨𝑂⟩.=Tr(𝑃 4.𝜌𝑂 4) Tr(𝑃 4.𝜌) (3.3) yield effective observables such as mass and coupling. ! 3! 4 Mass Generation via Projection Effective mass arises from projected time evolution 𝑀 H/00 =ℏ 𝑐1 𝑃 4. i∂2 𝑃 4.(4.1) so that 𝑚/00(𝐷)=M𝐷 ∣ ∣ 𝑀 H/00 ∣ ∣ 𝐷N/⟨𝐷∣𝐷⟩ Spectral eigenvalues in the projected subspace satisfy 𝐻 H/00 ∣𝜓3⟩=𝐸3∣𝜓3⟩,---------------------𝑚3𝑐1=∂(+ 5")𝐸3(4.2) Color confinement enforces ----------------------------------------------------------------------𝑃 4color-singlet𝑃 4.=𝑃 4. (4.3) so only singlet combinations acquire non-zero 𝑚/00. Comparing with lattice QCD data [13] reproduces meson–baryon mass ratios. Topological tension operators 𝒯 4topo link heavy-flavor splittings to diagram connectivity (4.4). Inertial and gravitational mass expectations coincide (Eq. 5.3), satisfying the equivalence principle. 5 Spacetime and Gravity as Entropic Projections Ensemble statistics define an information metric [14]: 𝑔78 =∂7∂8[−ln-𝑍(𝜃)]≃5𝑝9 9 ∂7ln-𝑝9 ∂8ln-𝑝9(5.1) Entropy flux across a local horizon obeys 𝛿𝑄=𝑇𝛿𝑆, leading to 𝐺9: +Λ𝑔9: =2𝜋 𝜂 𝑇9: (5.2) as in Jacobson’s thermodynamic derivation [7]. Expectation equality ⟨𝑀 H';<=⟩.=⟨𝑀 H)>/;?⟩.=ℏ 𝑐1Tr(𝑃 4.𝜌 i∂2) Tr(𝑃 4.𝜌) (5.3) ! 4! identifies gravitational and inertial mass. This connects entropic gravity [8, 9] with quantum-informational geometry [14–16]. 6 QCD as Worked Example Sector Correlation functions Π@(𝑞1)=𝑖 e dA𝑥 𝑒)B ⋅ DM0 ∣ ∣ 𝑇 𝐽(𝑥)𝐽E(0) ∣ ∣ 0N= 5 𝒜(𝐷) $∈ℭ# (6.1) map directly onto ℋ𝒟. Projected spectral densities (6.2) yield discrete hadronic states consistent with lattice averages [13]. Heavy-flavor mass splittings correlate with topological tension averages (4.4). The same projection formalism, applied to electroweak diagrams, reproduces Higgssector mass generation as a special case [17]. 7 Phenomenology and Computation Projection-renormalization flow in diagram space obeys 𝑑𝜌(ℓ) dℓ =𝒞 @𝜌−𝜌𝒞 @E,---------------------------------𝜌(ℓ)=𝑃 4.(ℓ)𝜌(ℓ)𝑃 4.(ℓ)(7.1) linking coarse-graining to coupling running [18]. Diagram compression reduces complexity from factorial to polynomial (7.2), consistent with Kozik’s results [6] and modern machine-learning approaches [19]. Predictions include: (i) small but measurable deviations in heavy-flavor spectra; (ii) entropic curvature corrections resembling ΛCDM dark-energy terms [20]; (iii) analog dynamics in synthetic gauge condensates. 8 Discussion The diagram-Hilbert formalism connects perturbative QFT, lattice phenomenology and gravitational thermodynamics within a single entropic topology. It preserves all verified ! 5! Standard-Model limits while providing structural unification. 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