Relation of the Entropy-Originated Topological Framework (EOTF) to Conventional Quantum Pictures

1 Relation of the Entropy-Originated Topological Framework (EOTF) to Conventional Quantum Pictures Borros Arneth, Philipps University Marburg, Justus Liebig University Giessen, Germany, [email protected] Abstract The Entropy-Originated Topological Framework (EOTF) extends conventional quantum mechanics by embedding its algebraic, wave, and interaction representations within an entropic–topological manifold. While the Heisenberg, Schrödinger, and Dirac pictures traditionally provide unitarily equivalent formulations of quantum dynamics, the EOTF generalizes their equivalence to a deeper invariance of informational and geometric flow. Within this framework, operator evolution, wave function propagation, and interaction dynamics emerge as distinct projections of a unified entropic operator field defined on a diagram-Hilbert manifold. The correspondence preserves the predictive structure of quantum mechanics while revealing its geometric and thermodynamic origin—thereby linking quantum pictures to the emergent structure of spacetime, gauge interactions, and matter. 1 Introduction The mathematical structure of quantum mechanics admits several equivalent yet conceptually distinct formulations. Heisenberg’s matrix mechanics ¹, Schrödinger’s wave mechanics ², and Dirac’s interaction picture ³ each provide complementary perspectives on the same underlying Hilbert-space dynamics. Their equivalence—proved rigorously by von Neumann ⁴—became one of the defining insights of twentieth-century physics. Nevertheless, these representations differ in where they assign time dependence: Heisenberg attaches it to observables, Schrödinger to state vectors, and Dirac distributes it between both. The Entropy-Originated Topological Framework (EOTF) re-examines this trinity from a modern standpoint inspired by information geometry ⁵, topological field theory ⁶, and thermodynamic gravity ⁷. It regards the Hilbert space not as a purely linear vector 2 space but as a diagram-Hilbert manifold—a network of local projective spaces connected by entropic flux. Within this manifold, operators and states acquire geometric meaning: their evolution traces the flow of informational curvature rather than merely abstract unitary motion. In this way, the EOTF extends the unitary equivalence of conventional pictures to a topological equivalence under entropic projection. 2 Historical and Conceptual Context In 1925 Heisenberg’s matrix mechanics replaced unobservable trajectories with algebraic relations among measurable quantities ¹. Schrödinger’s 1926 wave mechanics ² offered an apparently different formalism—continuous wave equations in configuration space—but he soon showed the mathematical equivalence of his and Heisenberg’s approaches ⁸. Dirac’s 1927 papers ³ introduced the interaction picture and the now-standard operator calculus used in quantum electrodynamics. Feynman’s later path-integral formulation ⁹ recast Dirac’s ideas in spacetime language, while von Neumann ⁴ provided the rigorous Hilbert-space underpinning. Modern physics inherits these pictures as complementary lenses. In quantum field theory the Heisenberg picture dominates for relativistic invariance ¹⁰; in quantum chemistry and condensed matter the Schrödinger picture remains natural for bound systems; and in perturbative QFT the Dirac (interaction) picture is indispensable ¹¹. Yet all assume a flat, linear Hilbert geometry. The EOTF seeks to retain their predictive power while embedding them in a curved informational manifold, analogous to how general relativity embeds Newtonian mechanics. 3 The Heisenberg Picture and Operator Topology Heisenberg’s original insight was that measurable quantities—not unobservable trajectories—constitute the true objects of physics ¹. In modern notation an operator 𝐴𝐻(𝑡) evolves according to 𝑑𝐴𝐻 𝑑𝑡 =𝑖 ℏ[𝐻,𝐴𝐻]+∂𝐴𝐻 ∂𝑡 In the EOTF this equation remains valid but gains a geometric interpretation. The operators 𝐴𝐻 correspond to topological maps between local nodes of the diagram-Hilbert manifold. Their non-commutativity, [𝐴,𝐵]=𝑖ℏ𝐶, is not merely algebraic but reflects linking of informational flows, captured by invariants such as the Chern class or knot index 𝜒𝐴𝐵: [𝐴,𝐵] ⇒ 𝑖ℏ 𝜒𝐴𝐵 3 Hence the Heisenberg equation describes the transport of entropic curvature along informational trajectories. This operator-topological reading connects naturally with modern algebraic QFT ¹² and geometric quantization ¹³, where observables form fiber bundles over simplistic manifolds. In that sense the EOTF generalizes Heisenberg’s matrices into entropic connection forms, whose curvature corresponds to measurable dynamics. 4 The Schrödinger Picture and Entropic-Wave Projection Schrödinger introduced his wave equation in 1926 ² as 𝑖ℏ ∂ ∂𝑡∣Ψ𝑆(𝑡)⟩=𝐻 ∣Ψ𝑆(𝑡)⟩ with ∣Ψ𝑆∣2 interpreted by Born as a probability density ¹⁴. Within the EOTF the formal equation is unchanged, but ∣Ψ𝑆⟩ represents a distributed entropic field across the diagram-Hilbert manifold. Its modulus squared denotes a local entropic density—the probability of finding curvature, or equivalently localized mass–energy, in informational space. The Hamiltonian becomes 𝐻ent =𝐻QM +Φtop(𝑆,∇𝑆) where 𝐻QM is the conventional quantum Hamiltonian and Φtop encodes curvatureinduced terms. The evolution is thus projectively unitary: linear unitarity is recovered when the entropic curvature 𝑆 is constant. This viewpoint resonates with information-geometric formulations of quantum theory ⁵, thermodynamic derivations of gravity ⁷, and quantum entropy dynamics in open systems ¹⁵. It also parallels the geometric phase interpretation of wavefunctions in curved parameter spaces ¹⁶. 4 5 The Dirac Picture and Interaction Dynamics Dirac’s 1927 formulation ³ introduced the interaction picture, pivotal to modern perturbation theory and quantum field interactions ¹¹. The state evolves as ∣Ψ𝐼(𝑡)⟩=𝑒𝑖𝐻0𝑡/ℏ ∣Ψ𝑆(𝑡)⟩ and the evolution operator 𝑈𝐼(𝑡)=𝒯exp⁡ ⁡[−𝑖 ℏ∫ 𝐻𝐼 𝑡 0(𝑡′) 𝑑𝑡′] In the EOTF this structure remains but its components acquire entropic meaning: 𝐻0 describes a flat informational background, while 𝐻𝐼 represents curvature coupling between submanifolds of the diagram space. The full propagator becomes a path integral over topological configurations ⁹, 𝑈ent =∫𝒟[Σ] 𝑒−𝑆ent[Σ]/ℏ where 𝑆ent is an entropic action functional. In the low-curvature limit, this reduces exactly to Dirac–Dyson perturbation theory ¹¹. Thus the interaction picture finds its natural geometric generalization within EOTF as the description of entropic–gauge coupling. 6 Comparative Synthesis The three canonical pictures can be compared as different projections of a single underlying entropic operator field. Conventional Picture Time Dependence EOTF Interpretation Heisenberg Operators evolve; states fixed Local entropic curvature flow between diagram nodes Schrödinger States evolve; operators fixed Global propagation of entropic density Dirac Both evolve (split by 𝐻0+𝐻𝐼) Coupling of free entropic background and topological curvature 5 Their unitary equivalence in standard quantum theory becomes a topological equivalence under entropic projection. The traditional conservation of probability corresponds to conservation of total entropic flux. In this sense, Heisenberg, Schrödinger, and Dirac formulations are embedded within EOTF rather than replaced by it. 7 Topological Equivalence of Quantum Pictures In the conventional formulation of quantum mechanics, the Heisenberg, Schrödinger and Dirac pictures are related by unitary transformations that leave all measurable quantities invariant. In the EOTF this invariance acquires a topological interpretation: the transformation 𝑈(𝑡) acts not on a flat Hilbert space but on a curved, diagram-Hilbert manifold in which entropy defines the local curvature. A unitary rotation in standard quantum mechanics corresponds here to a projection along a topological fiber of informational flow. Formally, the transformation 𝐴𝐻(𝑡)=𝑈†(𝑡)𝐴𝑆𝑈(𝑡) becomes in the EOTF Πobs =𝒫Σ[𝑈(𝑡)] Πent 𝒫Σ[𝑈(𝑡)]† where the projection operator 𝒫Σ selects the observable submanifold Σ of the informational manifold. When the entropic curvature 𝑆 is constant, these topological projections reduce to the familiar linear unitaries; when 𝑆 varies, they encode the geometric deformation of probability amplitudes in an informationally curved space. This notion echoes geometric-phase theory ¹⁶ and the fibre-bundle interpretation of gauge invariance ¹³. Probability conservation in standard mechanics, ⟨Ψ∣Ψ⟩=const, becomes the conservation of total entropic flux, aligning with information-conservation theorems discussed in statistical thermodynamics ¹⁷ and quantum information geometry ⁵. 8 Embedding within Conventional Quantum Theory To ensure physical continuity, the EOTF must recover the entire structure of quantum mechanics in the flat-entropy limit. This embedding can be made explicit: 6 1. Hilbert-space continuity: When the entropic curvature 𝑆 ⁡→ ⁡const, the diagramHilbert manifold becomes flat, the projectors 𝒫Σ commute globally, and the framework collapses to the standard Hilbert space of von Neumann ⁴. 2. Operator algebra: The Heisenberg commutation relation [𝑥,𝑝]=𝑖ℏ emerges from the local linking number 𝜒𝑥𝑝 of conjugate entropic flows, reproducing canonical quantization. The operator topology thus generalizes, not alters, Heisenberg’s algebra ¹ . 3. State evolution: The Schrödinger equation retains its form but gains an entropic potential Φtop. When this term vanishes, the system obeys the ordinary Schrödinger dynamics ² , ¹⁴. 4. Interaction picture: The decomposition 𝐻=𝐻0+𝐻𝐼 corresponds to the separation of background and curvature terms. In the limit of negligible curvature, 𝐻𝐼 ⁡→ ⁡0, the EOTF evolution operator reduces to the standard Dyson series ¹¹, ¹⁸. Hence all conventional pictures are contained as special projections of the EOTF. The framework is thus conservative in empirical content while being generative in conceptual scope. 9 Relations to Quantum Field Theory and Modern Developments The entropic–topological reinterpretation finds its natural arena in quantum field theory (QFT) and gravitational unification. In local QFT ¹⁰, observables form operator algebras attached to spacetime regions. Within EOTF these algebras acquire geometric meaning: the commutator of fields corresponds to the entropic linking of informational fibres, consistent with Haag’s algebraic formulation ¹². In gauge theory, topological invariants such as Chern classes ⁶ and Wilson loops describe the global structure of gauge fields. EOTF identifies these as manifestations of entropic curvature. The non-Abelian gauge symmetry of the Standard Model can thus be viewed as the symmetry of entropic flux conservation across interconnected submanifolds, connecting naturally with geometric unification attempts ¹⁹ and entropic gravity concepts ⁷, ²⁰. At the quantum-gravitational level, Ashtekar’s reformulation of general relativity ²¹ and Rovelli’s loop-quantum-gravity programme ²² interpret geometry in terms of connection variables and spin networks—mathematically similar to the diagram-Hilbert representation of EOTF. Witten’s topological quantum field theory ⁶ and Maldacena’s AdS/CFT correspondence ²³ suggest that geometry and information are dual; the EOTF realises this duality operationally within the language of quantum mechanics. The entropic manifold’s curvature can be interpreted as the information metric studied by Fisher, Amari, and later applications in quantum geometry ⁵. This bridges the EOTF with 7 the work of Padmanabhan ⁷ and Verlinde ²⁰ on thermodynamic gravity, where spacetime curvature arises from informational degrees of freedom. 10 Physical Implications and Predictive Coherence Because the EOTF reproduces standard quantum mechanics in its flat limit, it is automatically consistent with all known quantum experiments. Its novelty lies in predicting subtle corrections when entropic curvature is non-zero—typically in regimes of extreme density, temperature, or gravitational field. Several implications follow:  Entropic curvature and mass generation: Local gradients of 𝑆 contribute to effective mass terms, reminiscent of Higgs-like mechanisms ²⁴ but derived from informational geometry rather than scalar fields.  Gauge coupling unification: Renormalisation-group flow in entropic space leads to convergence of effective couplings, analogous to SU(5)/SO(10) models ²⁵ but encoded as topological invariants within the diagram-Hilbert manifold.  Quantum thermodynamics: The conservation of entropic flux implies a generalised second law for quantum processes, compatible with quantuminformation theorems ¹⁷ and black-hole entropy relations ²⁶.  Emergent spacetime: The informational manifold can be mapped onto a coarsegrained spacetime geometry where curvature corresponds to entropic density, aligning with spacetime-from-entanglement ideas ²⁷ and relational interpretations ²⁸,29,30. The framework therefore offers not only conceptual unification but also quantitative pathways to connect microscopic information flow with macroscopic gravitational dynamics. 11 Conclusion The Entropy-Originated Topological Framework (EOTF) positions the three canonical quantum pictures within a broader informational geometry. The Heisenberg picture becomes the algebra of topological operator flow; the Schrödinger picture emerges as entropic wave propagation; and the Dirac picture describes interaction between informational submanifolds. Their traditional unitary equivalence is elevated to a topological invariance of entropic flux. Crucially, this generalization leaves intact every successful prediction of quantum mechanics and quantum field theory. In the limit of constant entropy the EOTF reduces precisely to the established formalisms, while in regimes of strong entropic curvature it provides new geometric corrections linking quantum information, gauge symmetry and gravitation. 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