The Ontological Structure of the Diagram Hilbert Space: Pre-Geometric Operators and the Emergence of Physical Reality

! 1! The Ontological Structure of the Diagram Hilbert Space: Pre-Geometric Operators and the Emergence of Physical Reality Borros Arneth, Philipps University Marburg, Justus Liebig University Giessen, [email protected] Abstract We explore the hypothesis that the physical world (space-time, matter, energy, entropy) is a projection of a deeper Hilbert space, which we call the Diagram Hilbert Space ℋ!. In this framework, time, energy and entropy already exist in non-geometric, relational guise, together with additional fundamental structures such as topological invariants, symmetries, information metrics and connectivity operators. We catalogue these operator-structures in ℋ!, show how they map via projection to familiar physical quantities, and discuss the ontological and physical implications of treating ℋ! and the emergent space-time layer as co-real. The result is a layered ontology in which physical conservation laws; gauge symmetries and geometric dynamics arise naturally from the relational architecture of ℋ!. 1 Introduction The paradigm of emergent space-time has become increasingly central in quantum gravity: rather than treating space-time as fundamental, many approaches propose that it arises from discrete, relational, non-geometric building blocks (see Oriti 2013) [1], Van Raamsdonk 2010) [2], Lee 2023) [3]. In this paper we go further: we posit a Hilbert space ℋ! of diagrammatic/topological states, whose elements are relations, connectivity graphs, or knot-like structures, lacking metric distance or temporal order. From this space we define operator structures that correspond (upon projection) to what we call time, energy, entropy, mass, charge, curvature and information flow. Treating both ℋ! and the emergent physical layer as equally real (“complementary realism”) provides a unified view: the usual physical quantities become mappings of deeper relational invariants. We aim to identify what else lives in ℋ! besides time, energy and entropy—in particular: topological operators, projective maps, symmetry algebras, information metrics, curvature operators and causal connectivity. Understanding this underlying ontological structure clarifies the origin of conservation laws, symmetries and dynamics in the emergent universe. ! 2! 2 Time, Energy and Entropy in ℋ! 2.1 Ordering Parameter (diagram-time) In ℋ! there is no external physical time coordinate 𝑡. Instead, we introduce an ordering parameter 𝜏, which labels algebraic transitions of diagrammatic states: 𝑖ℏ 𝑑 𝑑𝜏 ∣ Ψ!(𝜏)⟩ = 𝐻 .!∣ Ψ!(𝜏)⟩ Here 𝐻 .! is the generator of reconfiguration in ℋ!. That parameter 𝜏 plays the role of “diagram-time” but is distinct from the emergent physical time coordinate. 2.2 Topological Energy We define a topological operator 𝑇 0 on ℋ!, measuring invariants of a diagram (linking number, knot-degree, connectivity rank). Then define 𝐸 0!= 𝐻 .!+ 𝜆 𝑇 0 so that ⟨Ψ!∣ 𝐸 0!∣ Ψ!⟩ quantifies the “projective pressure” or binding of a diagrammatic configuration. Upon projection to physical space this quantity becomes the familiar energy 𝐸. Conservation of 𝐸 in the emergent layer follows from unitary evolution in ℋ!. 2.3 Projection Entropy We define a projection map Π : ℋ!→ ℋ"#$%. Then the projection entropy 𝑆&= −𝑘' Tr(Π ∣ Ψ!⟩⟨Ψ!∣ log@(Π ∣ Ψ!⟩⟨Ψ!∣)) measures the information-loss / alignment cost when projecting. Variations in 𝑆& correspond to entropic flows in the emergent universe (e.g., arrow of time phenomena). 3 Additional Operator-Structures in ℋ! Beyond the triad (time/energy/entropy) there exist further essential structures in ℋ!: ! 3! 3.1 Topological Operators 𝑇 0( These measure connectivity, knotting, linking, loop-structure of diagrams. They underpin gauge charges and topological sectors in the emergent layer. 3.2 Projective Operators Π( Different projectors Π)*, Π+, Π,, … map ℋ! into subspaces corresponding to spacetime, matter, gravity etc. Π(: ℋ!→ ℋ(⊂ ℋ!. 3.3 Metric Structure 𝑔F! An inner‐product or information‐metric on diagrams: 𝑑-(Γ., Γ-) = 1−∣ ⟨Γ.∣ Γ-⟩ ∣- which projects into the emergent spacetime metric 𝑔/0. 3.4 Symmetry Algebra 𝔄! Relational symmetries in ℋ! map to internal gauge symmetries and spacetime isometries after projection. 3.5 Information Metric / Fisher Tensor ℐ(1 A geometry on the space of states (Fisher information) that, under projection, gives rise to coupling‐constant flows and renormalization behavior. 3.6 Diagrammatic Curvature ℛ! Defined by noncommutativity of diagrammatic derivatives: ℛ!= [∇(, ∇1] ∣ Ψ!⟩ Upon projection, this becomes spacetime curvature and gauge field strengths 𝐹 /0. 3.7 Connectivity / Causal Tensor 𝐶(1 Encodes how diagram nodes are entangled or connected. Its projection defines the emergent causal structure (light cones, causal ordering). ! 4! 4 Mapping to the Emergent Physical Layer The operator‐structures in ℋ! map to familiar physical entities under projection: ℋ! Operator Emergent Physical Correlate 𝜏 Physical time coordinate 𝑡 𝐸 0! Energy, Hamiltonian 𝐻 𝑆& Thermodynamic / information entropy 𝑇 0( Gauge charges, topological sectors 𝑔F! Metric 𝑔/0 ℛ! Curvature 𝑅/023, field strength ℐ(1 Coupling constants running 𝐶(1 Causality, light‐cone structure This mapping explains the origin of conservation laws (from unitarity in ℋ!), gauge symmetries (from topological invariants), and geometry (from connectivity patterns). 5 Ontological Implications 5.1 Dual‐Real Ontology We treat ℋ! and the emergent physical layer as equally real: Reality consists of two co‐real layers – relational/topological and geometric/phenomenal – linked by projective maps. This eliminates the hierarchy of “fundamental vs emergent” and gives a symmetric ontology. 5.2 Conservation and Symmetry from Unitarity Unitarity in ℋ! ensures conservation of 𝐸 0!, which projects to conservation of energy in the physical layer. Gauge symmetries arise from invariants under transformations in ℋ!. 5.3 Emergence of Dynamics, Geometry and Matter Dynamics, geometry and matter are all modes of projection of ℋ!. The “laws of physics” in the emergent layer are shadows of relational algebra in the deeper space. ! 5! 6 Testable Consequences and Discussion If this framework is physically meaningful, one expects: • Discrete or quantized spectra of topological invariants in ℋ! → discrete mass/charge levels in emergent physics. • Deviations from classical dispersion at high energies where diagrammatic effects are large. • Links between information‐theoretic quantities (Fisher metrics) and physical coupling running or black‐hole entropy. These offer paths to empirical tests, perhaps in early universe cosmology or high‐energy particle experiments. 7 Conclusion We have sketched the ontological map of the Diagram Hilbert Space ℋ!, identifying a rich set of operator‐structures beyond time/energy/entropy. Their projections yield the familiar physical world. Treating both layers as equally real transforms the origin of conservation laws, symmetries and geometry: they are all relational and projective. Further work will focus on explicit models of ℋ!, quantitative mapping of projection operators, and testable predictions. References 1. D. Oriti, Disappearance and emergence of space and time in quantum gravity, arXiv:1302.2849 (2013). 2. M. Van Raamsdonk, Building up spacetime with quantum entanglement, arXiv:1005.3035 (2010). 3. S.-S. Lee, Massless graviton in a model of quantum gravity with emergent spacetime, Phys. Rev. D 108, 024054 (2023). 4. T. Padmanabhan, Structural aspects of gravitational dynamics and the emergent perspective of gravity, arXiv:1208.1375 (2012). 5. J. B. Hartle, Generalizing quantum mechanics for quantum spacetime, arXiv:grqc/0602013 (2006). 6. G. E. Volovik, Emergent Physics: Fermi point scenario, arXiv:0801.0724 (2008). 7. V. Balasubramanian, D. Berenstein, A. Lewkowycz, et al., Emergent classical spacetime from microstates of an incipient black hole, JHEP 2019:197 (2019). 8. J. Queiruga, Non-metric construction of spacetime defects, Class. Quantum Grav. 37, 185009 (2020). 9. A. Kauffman, Knots and Physics, World Scient. (1991). ! 6! 10. L. Hardy, Found. Phys. 33, 379 (2003). 11. H. Nicolai & K. Peeters, Living Rev. Relativ. 9, 5 (2006). 12. J. Ambjørn, J. Jurkiewicz & R. Loll, Causal dynamical triangulations and the quest for quantum gravity, Phys. Rev. D 72, 064014 (2005). 13. “From complexity geometry to holographic spacetime”, Phys. Rev. D 108, 106020 (2023). 14. P. Chiarelli, Quantum Spacetime Geometrization: QED at High Curvature…, Quantum Rep. 6(1), 14-28 (2024). 15. “Group field theories: decoupling spacetime emergence from the ontology of nonspatiotemporal entities”, Eur. J. Phil. Sci. 14, 22 (2024). 16. E. P. Verlinde, Emergent Gravity and the Dark Universe, arXiv:1611.02269 (2016). 17. “Spacetime-emergent ring toward tabletop quantum gravity experiments”, Phys. Rev. Research 5, 023168 (2023). 18. L. Freidel, Group Field Theory: an overview, Int. J. Theor. Phys. 44, 1769 (2005). 19. Y. Manton & P. Sutcliffe, Topological Solitons, Cambridge Univ. Press (2004). 20. B. Swingle, Spacetime from entanglement, Annu. Rev. Condens. Matter Phys. 9, 345 (2018).