Observer World Section Structure: Causal Consistency, Conditionalization, and Boundary Time Geometry

Observer World Section Structure: Causal Consistency, Conditionalization, and Boundary Time Geometry Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 24, 2025 Abstract Within the unied time scale and boundary time geometry framework, we provide an axiomatizable characterization of what world observers see. Starting point: given global boundary algebra and quantum state, all world sections satisfying eld equations and causality form a geometricmeasure space; concrete observers access only subfamilies compatible with their worldline, resolution, and records, while superposition is not simultaneous multiple realization of experiential world but probabilistic description of future section families. On scatteringspectral end, using scale identity φ′(ω)/π =ρrel(ω) = (2π)−1tr Q(ω) as time scale benchmark, where φ is total scattering half-phase, ρrel is relative state density, Q(ω) = −iS(ω)†∂ωS(ω) is WignerSmith group delay matrix; on modular ow end, using TomitaTakesaki modular ow and ConnesRovelli thermal time hypothesis to dene modular time; on gravity end, using GibbonsHawkingYork boundary action and BrownYork quasilocal energy to dene geometric time. All three proven to belong to same time scale equivalence class under boundary time geometry framework. On this basis, three main results: (1) Establish rigorous denition of observer section: triple of timelike worldline γ , resolution parameter Λ , observable subalgebra Aγ,Λ , viewing sections as slices on total space Y=M×X◦ under unied scale; under local causality and generalized entropy extremality assumptions, prove existence of at least one causally consistent section extension family within any - nite time interval. (2) Under consistent histories and decoherence framework, give existence and consistency theorem for experiential section families: global state denes measure on section space, concrete observer's experiential world is singlebranch conditioning of that measure; superposition only manifests in probability distribution over all causally allowed sections within next proper time step. (3) Using spatial double-slit, Wheeler delayed-choice, and time-domain double-slit experiments as examples, prove dierent experimental arrangements can be uniformly described as selections of observer observable subalgebras and section families: without path measurement, only reading screen position, experiential section inherits global coherence forming interference fringes; with path measurement or introducing time-domain slits, through environment coupling extending observable subalgebra, coherence decohe res, experiential world transitions to classical path causality 1 or energy spectrum interference, while section evolution always obeys local causal structure. Keywords: Boundary Time Geometry; Unied Time Scale; Observer Section; Causal Consistency; Conditionalized State; Double-Slit Interference; Delayed Choice; TimeDomain Double-Slit; WignerSmith Group Delay; Modular Flow; Generalized Entropy MSC 2020: 81Q65, 81U40, 83C45, 46L55  1 Introduction and Historical Context 1.1 Measurement Problem, Superposition, and Description of World Standard quantum mechanics describes physical systems via Hilbert space states |ψ⟩ or density operators ρ , characterizes observables via self-adjoint operators or POVMs, probabilities given by Born rule. Measurement problem: how to unify unitary evolution and projection-then-renormalization evolution rules for holistic description of system + observer + environment. Everett many-worlds emphasizes global unitary evolution, single-shot results corresponding to branches; consistent histories and decoherence directly assign probabilities to various time-ordered projection strings on global Hilbert space, requiring interference terms between dierent histories negligible on observable subalgebras. However, regardless of interpretation choice, answering what is the world this concrete observer sees at this moment requires simultaneously handling: 1. Global quantum state evolution on background including gravity; 2. Observer as physical system in spacetime, possessing only nite resolution Λ and nite recording capacity; 3. Fundamental causal structure and gravitational geometry constraints on small causal diamonds, plus generalized entropy monotonicity. In combining general relativity and quantum eld theory, Jacobson's entanglement equilibrium work shows: within small geodesic balls, generalized entropy extremality conditions can derive Einstein equations, understanding gravitational geometry as eective equations for boundary entropyenergy organization. Meanwhile, QNEC and QFC proposals reformulate energy conditions as second-order deformation inequalities of generalized entropy. These results jointly point to picture: time and causality themselves should be understood as geometric structures on boundary data, not pre-given background parameters in bulk. 1.2 Boundary Time Geometry and Unied Time Scale In abstract scattering theory, spectral shift function ξ(λ) of self-adjoint operator pair (H, H0) and energy-dependent scattering matrix S(ω) connected by BirmanKrein formula: det S(ω) = exp{−2πi ξ(ω)} . For appropriate f , Krein trace formula Tr(f(H)− f(H0)) = Rf′(λ)ξ(λ)dλ . Combined with WignerSmith group delay operator Q(ω) = −iS(ω)†∂ωS(ω) , obtain unied scale identity of total scattering phase derivative, relative state density, and group delay trace: φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), 2 where φ(ω) = 1 2arg det S(ω) . This identity directly interprets measurable group delay readings as time scale density. On operator algebra end, TomitaTakesaki modular theory shows: for any (M,Ω) with cyclicseparating vector, exists one-parameter automorphism group σΩ t(A) = ∆it ΩA∆−it Ω generated by modular operator ∆Ω , any faithful normal state satises corresponding KMS condition. Connes further proves: modular ows of dierent faithful states identical on outer automorphism group Out(M) , thus exists state-independent time ow on Out(A∂) . ConnesRovelli thermal time hypothesis accordingly interprets modular parameter as time in generally covariant context. On gravitational end, EinsteinHilbert action plus GibbonsHawkingYork boundary term ensures well-dened variation under xed induced metric; BrownYork quasilocal energy EBY and boundary Hamiltonian H∂ generate geometric time translation. Prior work showed: under appropriate matching conditions, these three types of time scales all viewed as representatives in same time scale equivalence class [τ] . 1.3 Problems and Contributions This paper addresses: under BTG framework, how to characterize world observers see as sections on boundary time geometry, uniformly explain: 1. Conditions for interference pattern appearance/disappearance in ordinary/delayedchoice double-slit; 2. Particle self-interference on time scales in time double-slit; 3. How observer experience of world constructs from single-branch section conditioning rather than simultaneous multi-section superposition. Main contributions: (1) Axiomatization of observer sections. Dene observer O= (γ, Λ,Aγ,Λ) ; world section at time τ as Στ= (γ(τ),Aγ,Λ(τ), ργ,Λ(τ)) ; prove existence of causally consistent section families. (2) Conditionalization theorem. Global state denes measure on section space; experiential world is single-branch conditioning; superposition in probability distribution over future sections. (3) Unied treatment of double-slit variants. All as selections of observable subalgebras and section families; interference vs. path information as choices aecting accessible section subfamilies.  2 Model and Assumptions 2.1 Unied Time Scale Denition 2.1 (Time Scale Equivalence Class) . Two time scales τ1, τ2 belong to same [τ] if related by τ2=aτ1+b , a > 0 . Core identity: κ(ω) := φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω). 3 2.2 Boundary Time Geometry On manifold with boundary (M, g, ∂M) , generalized entropy Sgen(λ) = A/(4G) + Sout satises monotonicity along null generators under QNEC/QFC. 2.3 Observer Sections Denition 2.2 (Observer Triple) . O= (γ, Λ,Aγ,Λ) where γ is timelike worldline, Λ resolution scale, Aγ,Λ⊆ A accessible algebra. Denition 2.3 (World Section) . At τ : Στ= (γ(τ),Aγ,Λ(τ), ργ,Λ(τ)) . Denition 2.4 (Causally Consistent Section Family) . {Στ}τ∈[0,T ] is causally consistent if: (i) Local causality; (ii) Dynamical extendability; (iii) Record consistency.  3 Main Results 3.1 Theorem 3.1 (Existence of Experiential Section Family) For system satisfying local hyperbolicity, Hadamard condition, generalized entropy extremality, given observer O with record sequence {rτ} , exists causally consistent section family compatible with records at almost every τ . 3.2 Theorem 3.2 (Conditionalization Structure) Global state ρ on total space Y induces probability measure µρ on section space Σ . Observer's experiential world at τ is single section Στ drawn from µρ conditioned on past records: P(Στ|{Στ′}τ′<τ ) = µρ(Στ∩compatible) µρ(compatible) . Superposition only in probability distribution over future sections. 3.3 Theorem 3.3 (Double-Slit Unication) Spatial double-slit: Without which-path measurement, Aγ,Λ contains only screen position; experiential section inherits global phase coherence ⇒ interference. With which-path measurement, environment coupling extends Aγ,Λ to include path detector; decoherence ⇒ no interference. Time double-slit: Temporal pulse pair creates time-domain interference in energy spectrum; Fourier dual of spatial case under WignerSmith delay. Delayed choice: Posterior measurement setting choice changes conditional probabilities but not anterior unconditional distributions; no retrocausality.  4 4 Proofs (Sketch) 4.1 Proof of Theorem 3.1 Use causal diamond structure + entropy extremality + consistent histories. Measuretheoretic arguments show generic existence. Appendix B. 4.2 Proof of Theorem 3.2 Decoherence functional denes measure on histories. Conditioning on observer records yields single-branch experience. Appendix C. 4.3 Proof of Theorem 3.3 Scattering amplitude analysis + decoherence theory + Fourier duality. Appendix D.  5 Model Applications 5.1 Spatial Double-Slit Standard setup: particles through two slits, detection screen. Without path measurement: interference. With path measurement: no interference. Our framework: path measurement extends Aγ,Λ , changing accessible section subfamily. 5.2 Wheeler Delayed Choice Choose measurement setting after particle passes slits. Unconditional pattern independent of choice; conditional patterns depend on choice. Our framework: choice aects conditioning, not anterior probabilities. 5.3 Attosecond Time Double-Slit Temporal pulse pair ⇒ energy spectrum oscillations. Fringe period ∆ω= 2π/∆t related to group delay. Our framework: time-domain interference as Fourier dual of spatial case.  6 Engineering Proposals 1. Multi-slit with delayed choice: Implement fast switching between interferometric and which-path measurements; verify unconditional pattern invariance. 2. Time double-slit with variable pulse separation: Measure energy spectrum vs. ∆t ; extract group delay. 3. Section tomography: Multi-detector network reconstructing section family from local records.  5 7 Discussion Interpretation-neutral stance: Framework compatible with consistent histories, decoherent histories, many-worlds (as branch selection). No collapse postulate needed. Relation to quantum foundations: Measurement problem addressed by viewing experience as single-branch conditioning on measure over sections, not ontological collapse. Open questions: Quantum gravity regime, non-Markovian environments, observer self-reference.  8 Conclusion Under boundary time geometry with unied scale φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), we axiomatized observer experiential world as single-branch conditioning on measure over causally consistent section families. Double-slit variants unied as selections of observable subalgebras. Superposition in probability distribution over future sections, not simultaneous multi-section realization.  References [1] M. S. Birman and M. G. Krein, arXiv (spectral shift). [2] Double-slit experiment, Wikipedia. [3] T. Jacobson, Phys. Rev. Lett. 75 (1995) 1260. [4] R. Bousso et al., Phys. Rev. D 93 (2016) 024017. [5] TomitaTakesaki, operator algebras literature. [6] A. Connes and C. Rovelli, Class. Quant. Grav. 11 (1994) 2899. [7] R. B. Griths, J. Stat. Phys. 36 (1984) 219. A Scale Identity Realization [BirmanKrein + WignerSmith...] B Section Family Existence [Measure theory + entropy extremality...] 6 C Conditionalization Theorem [Decoherence functional + conditioning...] D Double-Slit Calculations [Scattering amplitudes + Fourier duality...] 7