Boundary Time Geometry: Unified Theory of Time Scale, Resolution Hierarchy, and Interaction

Boundary Time Geometry: Unied Theory of Time Scale, Resolution Hierarchy, and Interaction Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 24, 2025 Abstract Construct unied theoretical system with boundary as ontology and time as geometric scale. Basic assumption: physical reality rst manifests as boundary observable algebra and its spectral data; bulk dynamics are extensions determined by boundary data. All observable time scalesscattering time, modular time, geometric timebelong to same equivalence class. Observer's nite resolution geometrically manifests as resolution ber bundle with connection and curvature. Mathematically, introduce noncommutative geometric structure of spectral triple with boundary; unify BrownYork boundary stress tensor with AdS/CFT boundary stress tensor, WignerSmith time delay matrix with BirmanKrein spectral shift function, TomitaTakesaki modular ow with thermal time hypothesis within single Boundary Time Geometry (BTG) framework. Prove under appropriate matching conditions, exists unique (up to ane rescaling) boundary time generator making scattering time, modular time, geometric time dene same time scale equivalence class. All classical forces manifest as projections of unied boundary connection curvature in dierent ber directions, no longer fundamental objects but emergent properties of boundary geometry and resolution structure. Further establish phenomenal hierarchy emergence theorem on resolution ber bundle, clarifying how high-resolution quantum scattering and modular time structures degenerate into macroscopic gravity and classical mechanics via completely positive coarse-graining maps. Finally provide BTG reformulations of black hole thermodynamics, cosmological redshift, mesoscopic transport; propose experimental verication protocols implementable in microwave networks, atomic clock networks, mesoscopic conductors. Keywords: Boundary Time Geometry; Noncommutative Geometry; Spectral Triple; WignerSmith Time Delay; BirmanKrein Spectral Shift; BrownYork Stress Tensor; Thermal Time Hypothesis; Resolution Fiber Bundle; Holographic BoundaryBulk Correspondence; Renormalization Group  1 1 Introduction and Historical Context In general relativity, GibbonsHawkingYork boundary term and BrownYork quasilocal stressenergy tensor show that well-dened variation of gravitational action and denition of quasilocal energymomentum fundamentally depend on boundary geometry and conjugate variables. Variation of boundary three-metric derives surface stress tensor Tab BY recovering ADM energy in appropriate limits, providing quasilocal energy meaning for black hole thermodynamics. In AdS/CFT holographic framework, BalasubramanianKraus stress tensor views renormalized boundary stressenergy as energymomentum tensor of dual conformal eld theory, further deepening boundary-dominated perspective. On scattering theory side, WignerSmith time delay matrix Q(ω) = −iS(ω)†∂ωS(ω) characterizes average residence time of wave packets in scattering region; its trace tightly connected to derivative of spectral shift function via BirmanKrein formula: total scattering phase derivative, WignerSmith group delay trace, and relative state density are dierent manifestations of same object. In algebraic quantum eld theory and quantum statistics, TomitaTakesaki modular theory reveals: given observable algebra and state, naturally exists one-parameter automorphism group σω t whose parameter t interpretable as modular time. ConnesRovelli thermal time hypothesis further proposes physical time understandable as modular ow parameter determined by statealgebra pair; traditional time becomes derived concept. Noncommutative geometry provides language dening geometry via spectral data: spectral triple (A,H, D) consists of algebra, Hilbert space, Dirac-type operator; for compact Riemannian manifolds, metric structure uniquely reconstructible from Dirac spectrum; this framework provides natural platform unifying boundary geometry with boundary observable algebra. This paper's basic stance: glue above three threadsboundary gravity, scattering time, modular timein unied Boundary Time Geometry framework, taking boundary as ontology, time as scale, resolution as ber, constructing unied theoretical system accommodating existing theories while yielding new predictions.  2 Model and Assumptions 2.1 Axioms: Boundary Priority, Time Equivalence, Resolution Hierarchy Axiom 1 (Boundary Priority) . Given spacetime region (M, g) with good causal structure, containing topologically well-behaved boundary ∂M (including timelike, spacelike, or null boundaries), fundamental description of physical observables given by boundary observable algebra A∂ and state set S∂ ; bulk observables and dynamics viewable as extensions determined by (A∂,S∂) in appropriate sense. Axiom 2 (Time Scale Equivalence) . Exists time scale equivalence class [τ] whose elements are time parameters under dierent constructions: scattering time τscatt , modular 2 time τmod , geometric time τgeom . Any two time scales equivalent via ane transformation τ(2) =aτ(1) +b ( a > 0 ) on common domain. Axiom 3 (Resolution Hierarchy) . For each concrete experimental arrangement or observer, exists resolution parameter Λ (understandable as UV cuto, coarse-graining stage, or RG scale) such that at dierent Λ , same boundary geometric data projects via completely positive map to dierent coarse-grained eective algebras AΛ⊆ A∂ . 2.2 Boundary Spectral Data Denition 2.1 (Boundary Spectral Triple) . Boundary spectral triple is tuple (A∂,H∂, D∂) where: 1. A∂ is dense ∗ -algebra dened on boundary (typically C∞(∂M) or noncommutative generalization); 2. H∂ is Z2 -graded Hilbert space carrying ∗ -representation of A∂ ; 3. D∂ is self-adjoint, rst-order elliptic operator (Dirac-type) with compact resolvent, satisfying commutator [D∂, a] bounded for any a∈ A∂ . This is boundary version of Connes (even) spectral triple. Theorem 2.2 (Spectral Reconstruction of Boundary Metric) . If ∂M is compact spin Riemannian manifold, triple (A∂,H∂, D∂) = (C∞(∂M), L2(S∂), D∂) determines unique Riemannian metric hab such that Connes distance d(x, y) = sup{|a(x)−a(y)|:a∈C∞(∂M),|[D∂, a]| ≤ 1} equals geodesic distance on (∂M, hab) . Thus in BTG, boundary metric need not be given a priori but dened by spectral structure of D∂ ; this provides natural channel embedding time scale into Dirac spectrum. 2.3 Boundary Stress Tensor and Quasilocal Hamiltonian In four-dimensional general relativity, after introducing GHY boundary term, variation of action with respect to boundary three-metric hab denes BrownYork surface stress tensor Tab BY := 2 √−h δSgrav δhab . Its zero component's appropriate projection gives quasilocal energy density; integrated quasilocal energy equals Hamiltonian generating unit proper time translation on boundary.In AdS scenario, holographic renormalization process derives renormalized boundary stress tensor Tab ren , interpretable as dual CFT expectation value ⟨Tab⟩ . These results show: boundary stress tensor naturally carries Hamiltonian generating boundary time ow.  3 3 Main Results (Theorems and Alignments) 3.1 Unied Time Scales on Boundary Dene time scales from scattering, modular ow, geometric perspectives respectively: 1. Scattering time τscatt Consider nite-channel scattering matrix S(ω) at xed energy; dene WignerSmith time delay matrix Q(ω) = −iS(ω)†∂ωS(ω). Its trace τW(ω) := tr Q(ω) gives total group delay. In BirmanKrein framework, spectral shift function ξ(ω) satises det S(ω) = exp(−2πiξ(ω)), thus ξ′(ω) = 1 2πtr Q(ω). Given reference energy ω0 and window I⊂R , dene scattering time scale τscatt(ω) := Zω ω0 ξ′(˜ω)d˜ω=ξ(ω)−ξ(ω0). 2. Modular time τmod For boundary observable algebra A∂ and state ω , assuming separatingcyclic vector exists making TomitaTakesaki modular data (J, ∆ω) well-dened; modular group σω t(A) := ∆it ωA∆−it ω denes one-parameter automorphism group. Thermal time hypothesis suggests appropriate physical time parameter τmod diers from modular parameter t only by constant factor; σω t plays time evolution role in equilibrium states. 3. Geometric time τgeom In general relativity with boundary, choose unit timelike vector eld ua on boundary with corresponding Killing or approximate Killing generator ξa ; BrownYork Hamiltonian writable as H∂[ξ] = ZΣ∩∂M √σ uaTab BYξbdd−2x, where σ is induced metric on cross-section. Canonical evolution parameter generated by H∂ denes geometric time scale τgeom . In BTG framework, we don't presuppose three time scales mutually independent, but unify via following theorem: Theorem 3.1 (Boundary Time Scale Equivalence Theorem) . Let ∂M be benign boundary satisfying: 1. Exists boundary spectral triple (A∂,H∂, D∂) and BrownYork boundary stress tensor Tab BY ; 2. Boundary admits scattering process with scattering matrix S(ω) continuously differentiable in energy ω , satisfying HilbertSchmidt locality and BK conditions on energy window I ; 3. For same boundary region exists von Neumann algebra A′′ ∂ and KMS state ω whose modular group σω t physically represents thermal equilibrium time evolution; 4 4. BrownYork Hamiltonian H∂[ξ] generated boundary time translation induces automorphism group ατ on observable algebra comparable to scattering evolution and modular ow in same energyfrequency window, i.e., exists common invariant subalgebra Acom ⊂ A∂ . Then exists unique time scale equivalence class [τ] , plus three positive constants ascatt, amod, ageom > 0 and three translation constants bscatt, bmod, bgeom , such that on common domain: τscatt =ascattτ+bscatt, τmod =amodτ+bmod, τgeom =ageomτ+bgeom. In other words, scattering time, modular time, geometric time in BTG only represent dierent normalizations and zero-point choices of same time scale. Rigorous proof given in Proofs section and appendices, core being: • Use BKWigner Smith identity to express scattering time as integral of relative spectral density; • Via thermal time hypothesis and boundary KMS state, align modular parameter with relative spectral density; • Via BrownYork Hamiltonian and boundary stress tensor's spectral representation, associate geometric time ow generator with same spectral measure. 3.2 No Fundamental Forces: Curvature of Unied Boundary Connection Denition 3.2 (Boundary Total Bundle and Unied Connection) . 1. On geometric gaugeresolution three-layer degrees of freedom, dene boundary total bundle π:B → ∂M with ber F=Fint ×Fres, carrying internal gauge degrees of freedom and resolution scale degrees of freedom respectively. 2. Structure group taken as Gtot = SO(1,3)↑×GYM ×Gres, where Gres is scale group equivalent to renormalization group or coarse-graining transformations. 3. Unied boundary connection dened as Ω∂=ωLC ⊕AYM ⊕Γres, corresponding to LeviCivita spin connection, YangMills connection, resolution connection; corresponding curvature R∂=R∂⊕F∂⊕Rres. Theorem 3.3 (No Fundamental Forces Theorem) . Under BTG framework, consider chargedcolored test particle or eective mass trajectory lift γ(τ)⊂ B on boundary; its projection to ∂M is xµ(τ) , intrinsic degrees of freedom via representation ρ:GYM → Aut(Fint) and Gres one-dimensional representation. Then following holds: 1. Force-free motion of trajectory γ(τ) is parallel transport for unied connection Ω∂ : Dτ˙γ= 0 . 2. Its base trajectory xµ(τ) satises equation writable as mD2xµ Dτ2=qFµν˙xν+fµ res, 5 where Fµν is YangMills curvature projection under representation ρ , fµ res is resolution curvature Rres projection in appropriate eective action. 3. Classical gravitational force corresponds to geodesic deviation eect of R∂ ; thus all forces understandable as dierent projections and representations of unied boundary connection curvature, no longer fundamental objects. Therefore in BTG theory, forces not independent axiomatic entities but emergent manifestations of boundary time geometry; all interactionsincluding gravity, gauge interactions, resolution-driven entropic forcesjointly arise from unied connection curvature. 3.3 Resolution Hierarchy and Emergent Phenomena Denition 3.4 (Resolution Fiber Bundle and Coarse-Graining Maps) . 1. On boundary total bundle dene resolution ber bundle Pres = (B, ∂M, Gres, πres) with ber coordinate understandable as resolution or renormalization scale Λ . 2. Each Λ induces completely positive, unit-preserving map ΦΛ:A∂→ AΛ⊆ A∂, viewable as coarse-graining from high-resolution boundary algebra to low-resolution effective algebra. Theorem 3.5 (Phenomenal Hierarchy Emergence Theorem) . When satisfying: 1. {ΦΛ}Λ forms normal ∗ -homomorphism family of semigroup (or group) satisfying ΦΛ1◦ΦΛ2= ΦΛ1◦Λ2 ; 2. For any local observable A∈ A∂ , its image ΦΛ(A) in Λ→0 limit (coarsest) converges to classical function or operator Acl ; 3. Unied connection Ω∂ 's connection form Γres in resolution direction satises Callan Symanzik-like equation: parallel transport along Λ ow equivalent to renormalization group ow. Then: 1. In high-resolution limit, description of A∂ is full quantum scattering and modular time structure; 2. At medium resolution, curvature expectation values in coarse-grained algebra AΛ manifest as gauge forces, entropic forces, topological eects; 3. In Λ→0 macroscopic limit, geometric curvature and BrownYork tensor dominate, dynamics degenerating to classical gravity and thermodynamics; all forces eectively viewable as geometric eects of metric and eective potentials.  4 Proofs This section provides proof skeletons of main theorems; details and technical lemmas in appendices. 6 4.1 Preliminaries: Scattering, Time Delay, Spectral Shift Let H0 and H=H0+V be self-adjoint operators on some Hilbert space satisfying wave operator existence conditions of general scattering theory. BirmanKrein theory provides spectral shift function ξ(ω) giving trace formula for smooth functions f of H, H0 : Tr(f(H)−f(H0)) = Zf′(ω)ξ(ω)dω. Under suitable conditions, scattering matrix S(ω) satises det S(ω) = exp(−2πiξ(ω)). For Theorem 2, only need local BK formula on energy window I and dierentiability of WignerSmith matrix. Let Q(ω) = −iS(ω)†∂ωS(ω), then ξ′(ω) = (2π)−1tr Q(ω) , thus scattering time scale τscatt(ω) = ξ(ω)−ξ(ω0) well-dened on I . 4.2 Proof Sketch of Theorem 2 (Time Scale Equivalence) Step 1: Unied Spectral Measure On energy window I , dene spectral measure via BK: µscatt(dω) := 1 2πtr Q(ω)dω. On other hand, KMS state ω on boundary von Neumann algebra A′′ ∂ induces modular operator ∆ω whose spectral measure µmod determines modular group σω t generator Kω:= −log ∆ω . Thermal time hypothesis requires constant cmod >0 exists making physical Hamiltonian Hmod =cmodKω . On geometric end, BrownYork Hamiltonian writable as functional on Dirac or Laplace operator spectrum: under appropriate boundary conditions, its expectation value on energy eigenstate |E⟩ gives spectral function hgeom(E) , introducing measure µgeom(dE) = hgeom(E)dE . In AdS/CFT context, this measure equivalent to boundary CFT energy momentum tensor spectral measure. Step 2: Matching Conditions and Measure Equivalence Assume scattering process, modular ow, geometric time translation act on common decomposable subalgebra Acom ; within energy window I , three dynamics' spectral decompositions representable in same Hilbert space; this is comparability condition in theorem statement. Under this condition, can prove: 1. Exists family of monotonic dierentiable energy rescaling functions making µscatt , µmod , µgeom mutually absolutely continuous on I with constant RadonNikodym derivatives; 2. This ensures three time generators equivalent on L2(I, µ) , diering only by constant factors and additive constants. 7 Step 3: Uniqueness If another time scale ˜τ anely equivalent to all three above, then ˜τ also anely equivalent to τ ; thus equivalence class [τ] unique. Complete proof involves ne control of spectral decomposition, KMS conditions, BrownYork Hamiltonian spectral representation; see Appendix A. 4.3 Proof Sketch of Theorem 3 (No Fundamental Forces) Under unied connection Ω∂ , consider curve γ(τ) on total bundle B . Its covariant derivative Dτ=d dτ + Ω∂(˙γ). Dene free motion as Dτ˙γ= 0 . Expanding this condition in dierent ber direction components yields: 1. In SO(1,3) part: standard geodesic equation; 2. In GYM part: Wong-equation-like gauge force term: particle parallel transport in internal space induces Fµν ˙xν term on base trajectory; 3. In Gres part: resolution connection Γres curvature via eective action's scale dependence gives entropic force or information force, specic form depending on chosen eective free energy functional. Thus any seemingly forced motion viewable as parallel transport under some unied connection, we simply ignore certain ber directions in projection. Theorem 3 content merely formalizes this geometric fact.  5 Model Applications 5.1 Black Hole Thermodynamics in BTG For spacetime with event horizon, treat horizon as special null boundary; introduce null BrownYork stress tensor and corresponding quasilocal energy, rewriting black hole thermodynamics four laws in pure boundary language. Proposition 5.1 (Boundary Restatement of Black Hole Thermodynamics) . 1. Hawking temperature TH=κ/(2π) comes from modular ow period 2π/κ on horizon, where κ is surface gravity; 2. BekensteinHawking entropy SBH =A/(4G) interpretable as von Neumann entropy of horizon boundary algebra or entropy density on type factor; 3. Hawking radiation purity problem formulable in BTG as Markov property stability problem between horizon and innity boundary algebras: if boundary relative entropy slice-independent satisfying appropriate quantum focusing conditions, overall evolution can preserve pure states. In BTG language, black hole thermodynamics no longer mixture of bulk singularity and horizon structure, but completely described by boundary time geometry and modular time scale. 8 5.2 Cosmological Redshift as Boundary Time Rescaling In FRW universe, standard redshift formula 1 + z=a(t0)/a(te) rewritable in BTG as: Proposition 5.2 (Boundary Interpretation of Cosmological Redshift) . Choose cosmological boundary as conformal innity or comoving observer family worldtube boundary, with time scale dened by boundary time scale τ∂ ; exists ane transformation making 1 + z=τ∂(t0)/τ∂(te). This shows redshift viewable as overall rescaling of boundary time scale, not bulk proper time dierence; BTG directly connects redshift to boundary spectral data evolution. 5.3 Mesoscopic Transport and FriedelWigner Consistency In mesoscopic conductors or AB rings, WignerSmith time delay matrix and Friedel sum rule provide connections between local density of states, phase shift, transport properties. In BTG, these results interpretable as boundary spectral triple projections at nite resolution: • Phase shift derivative ∂ωϕ(ω) ratio to local density of states directly gives scattering time scale; • Via Theorem 2, this scale equivalent to modular and geometric time, making mesoscopic transport experiments direct verication platform for BTG time scale equivalence.  6 Engineering Proposals 6.1 Microwave Scattering Networks as Discrete Boundary Models Construct multi-port microwave network viewing as discretized boundary ∂M model: 1. Measure multi-port scattering matrix S(ω) via vector network analyzer; numerically construct WignerSmith matrix Q(ω) and spectral shift function ξ(ω) , dene scattering time scale τscatt . 2. Introduce tunable geometric parameters at network nodes (e.g., electrical length, lossy elements); reconstruct τgeom via network Lagrangian or eective RLC model inversion. 3. Place network in controlled noise environment; dene statistical steady state and construct equivalent modular ow; measure τmod proxy quantities (e.g., correlation function decay parameters). BTG prediction: Within energy window and resolution conditions satisfying Theorem 2 assumptions, ratios of three time scales should be constant; deviations attributable to resolution connection Γres curvature and experimental non-idealities. 6.2 Atomic Clock Networks and Gravitational Redshift Deploy atomic clock network at dierent gravitational potentials; use two-way time transfer protocol to measure frequency ratio ν2/ν1 . In BTG language, this frequency ratio 9