Energy as a Topological Invariant in the Diagram Hilbert Space

! 1! Energy as a Topological Invariant in the Diagram Hilbert Space Borros Arneth, Philipps University Marburg, Justus Liebig University Giessen, [email protected] Abstract In this paper we propose that the quantity we call energy in the emergent physical world can be traced back to a topological invariant in a deeper state-space, which we call the Diagram Hilbert Space ℋ!. In this framework, time (in the usual sense) and geometry are emergent from ℋ! via projection operators, and energy corresponds to the projective pressure or stability measure of a diagrammatic state. We show how this reinterpretation resolves apparent paradoxes of quantum gravity, unifies matter and geometry, and recasts conservation laws as unitarity conditions in ℋ!. We outline formal definitions, operator relations, and potential empirical implications. 1 Introduction Energy is a central concept in physics: in classical mechanics it is the sum of kinetic and potential parts; in quantum mechanics it is the eigenvalue of the Hamiltonian; in relativity it is the source of gravitational curvature. Yet when we attempt to unify quantum mechanics and general relativity, the status of energy becomes problematic — particularly in pre-geometric approaches, emergent spacetime models, and in the “problem of time” in quantum gravity. In this work we propose that energy is fundamentally a topological invariant of a more primary state‐space, the Diagram Hilbert Space ℋ!. Within ℋ!, there is an internal ordering parameter (call it 𝜏) rather than physical time 𝑡, and geometric notions such as distance, curvature and metric only appear as projections. We argue that energy in the emergent spacetime corresponds to a measure of projective pressure or topological binding strength in ℋ!. Conservation of energy then follows from unitarity and information conservation in ℋ!. Our approach draws upon ideas of emergent spacetime (Hu 2009) [1], entanglement and geometry (Van Raamsdonk 2010) [2], emergent time (Oriti 2021) [3], entropic gravity (Verlinde 2016) [4], and relational quantum mechanics (Rovelli 1996) [5]. We build on ! 2! these by introducing a formal operator framework for the topological invariant that underlies energy. 2 The Diagram Hilbert Space ℋ! and Ordering Parameter We define ℋ! as a Hilbert space whose basis states ∣ Γ⟩ correspond to relational / diagrammatic structures (graphs, knot‐diagrams, network states) with no predefined metric, no spatial coordinates and no external time. The fundamental generator of change is an operator 𝐻 (! which induces evolution with respect to an internal parameter 𝜏: 𝑖ℏ 𝑑 𝑑𝜏 ∣ Ψ!(𝜏)⟩ = 𝐻 (! ∣ Ψ!(𝜏)⟩ Here 𝜏 is not the physical time coordinate; it is an ordering parameter for transitions in the diagrammatic state space. (See [3,6,7]). Within ℋ! we define a topological invariant operator 𝑇 1, which assigns to each diagram state a measure of its connectivity, loop‐oriented complexity, or linking number. We then define a topological “energy” operator 𝐸 1!= 𝐻 (!+ 𝜆 𝑇 1 where 𝜆 is a coupling parameter. A state ∣ Γ⟩ with large ⟨Γ ∣ 𝑇 1∣ Γ⟩ is more resistant to projection, more “topologically bound,” and thus corresponds to a larger emergent energy. 3 Projection to Physical Spacetime and Emergent Energy We introduce a projection operator Π"#: ℋ! → ℋ"# ! 3! that selects those diagrams which satisfy coherence conditions (e.g., adjacency, low entropic disorder) and maps them to emergent spacetime states. On ℋ"# one recovers usual coordinates 𝑥$, metrics 𝑔$%, Hamiltonians 𝐻 ("#, etc. We posit the correspondence 𝐸"# = ⟨Ψ!∣ Π"# & 𝐸 1! Π"# ∣ Ψ!⟩ Thus, the usual physical energy 𝐸"# of a matter or field excitation is the expectation of the Diagram Hilbert Space topological energy. Conservation of 𝐸"# follows from the unitarity of 𝐻 (! and the norm‐preserving property of Π"#. 4 Implications: Conservation and Gravity Because energy is now a topological invariant in ℋ!, conservation laws become direct consequences of unitarity: 𝑑 𝑑𝜏 ⟨Ψ!(𝜏) ∣ 𝐸 1!∣ Ψ!(𝜏)⟩ = 0<<<<<< ⇒ <<<<<<<< 𝑑 𝑑𝑡 𝐸"# = 0 Furthermore, gravitation — the curvature of spacetime in response to energy–momentum — emerges as the adjustment of the projection mapping in response to changes in topology. One may write: ∇'()𝐸! = 𝛿ℰ![Γ] 𝛿Γ <<<<< ⟹ <<<<<<<ℛ*+ ∝ ∇'()𝐸*+ mirroring the projective form of Einstein’s equation (as in Verlinde 2016, Hu 2009). 5 Quanta in ℋ!: Energy Content and Stability In this framework, a quantum excitation in emergent spacetime corresponds to a sub‐ diagram state in ℋ! which is both topologically stable (minimises ℰ! subject to 𝒯 constraints) and coherently projectable. Its “mass” or “energy” in spacetime arises from its diagrammatic binding: 𝑚𝑐,= ⟨Γ ∣ 𝐸 1!∣ Γ⟩ ! 4! Hence, quanta “have energy” because their underlying diagrammatic states carry topological invariants. Their dynamics in spacetime reflect transitions or deformations of these states in ℋ!. 6 Testable Predictions and Discussion 1. Because energy corresponds to a topological invariant, one expects discrete spectra for certain high-topology states, analogous to quantized solitons in field theory. 2. If spacetime is emergent from ℋ!, then at very high energies (close to the binding threshold) one may observe deviations from standard dispersion relations, possibly testable in high-energy cosmic rays or early‐universe cosmology. 3. Conservation of energy becomes tied to information conservation; any process violating standard energy conservation in spacetime would correspond to non‐ unitary evolution in ℋ!, hence strongly suppressed or forbidden. 7 Conclusion We have proposed that energy in the emergent physical world is the image of a topological binding invariant in a deeper Diagram Hilbert Space. Time and geometry themselves are emergent. This view harmonizes conservation laws, mass/energy generation, and gravitational coupling in a unified topology‐projection framework. Future work will focus on explicit models of ℋ!, numerical simulations of diagrammatic binding spectra, and links to experimental signatures. References 1. B. L. Hu, Emergent/Quantum Gravity: Macro/Micro Structures of Spacetime, arXiv:0903.0878 (2009). 2. M. Van Raamsdonk, Building up spacetime with quantum entanglement, arXiv:1005.3035 (2010). 3. D. Oriti, The complex timeless emergence of time in quantum gravity, arXiv:2110.08641 (2021). 4. E. P. Verlinde, Emergent Gravity and the Dark Universe, arXiv:1611.02269 (2016). 5. C. Rovelli, Relational Quantum Mechanics, Int. J. Theor. Phys. 35, 1637 (1996). 6. S. Sung-Sik Lee, Massless graviton in a model of quantum gravity with emergent spacetime, Phys. Rev. D 108, 024054 (2023). ! 5! 7. A. Connes & C. Rovelli, Von Neumann algebra automorphisms and the time– thermodynamics relation in generally covariant quantum theories, Class. Quant. Grav. 11, 2899 (1994). 8. G. E. Volovik, Emergent physics: Fermi point scenario, arXiv:0801.0724 (2008). 9. J. Erlich, Stochastic Emergent Quantum Gravity, arXiv:1807.08105 (2018). 10. E. Gesteau, Emergent spacetime and the ergodic hierarchy, Phys. Rev. D 110, 106005 (2024). 11. L. Freidel, Group field theory: an overview, Int. J. Theor. Phys. 44, 1769 (2005). 12. F. Markopoulou, Quantum causal histories, Class. Quantum Grav. 17, 2059 (2000). 13. J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998). 14. H. Nicolai & K. Peeters, Living Rev. Relativ. 9, 5 (2006). 15. S. Weinberg, The Quantum Theory of Fields, Vol. II, Cambridge Univ. Press (1996). 16. T. Jacobson, Thermodynamics of spacetime: the Einstein equation of state, Phys. Rev. Lett. 75, 1260 (1995). 17. Z. K. Zou, Time and Gravity in a Novel Quantum Thermodynamic Spacetime Framework, Cambridge Open Engage (2025). 18. M. Schneider, Spacetime Emergence: An (In)effective Story, Phil. Phys. 2 (2024) 18. 19. J. Ambjørn, J. Jurkiewicz & R. Loll, Causal dynamical triangulations and the quest for quantum gravity, Phys. Rev. D 72, 064014 (2005). 20. A. Kauffman, Knots and Physics, World Scientific (1991).