Boundary as Unified Stage: From Time Translation Operator, Null--Modular Double Cover to GHY Boundary Term

Boundary as Unied Stage: From Time Translation Operator, NullModular Double Cover to GHY Boundary Term Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 24, 2025 Abstract Construct unied framework with boundary as sole fundamental stage, gluing three mature but usually separate structures as three projections of same object: (i) Based on BirmanKreinFriedelWignerSmith framework, time translation operator characterized by scattering phase, spectral shift function, WignerSmith time delay matrix; (ii) Based on TomitaTakesaki modular theory and recent results on causal diamond/null surface modular Hamiltonians, Markov property, QNEC, NullModular double cover and overlapping causal diamond chains; (iii) Represented by GibbonsHawkingYork (GHY) boundary term and its generalization with null sheets and joints, gravitational boundary action and Brown York quasilocal energy. Under clearly stated applicability domains and assumptions, give four main results: (1) In scattering systems satisfying trace-class perturbation conditions, scattering half-phase derivative, Krein spectral shift density, WignerSmith delay matrix trace constitute same boundary time measure; (2) In relativistic quantum eld theory under standard assumptions, modular Hamiltonians on causal diamonds and null surfaces localizable as weighted integrals of stressenergy ow on NullModular double cover, satisfying Markov inclusionexclusion law for overlapping diamond families; (3) On piecewise spacetime boundaries (with null sheets and joints) in general relativity, after adding GHY-type boundary/joint terms, action variation for xed induced geometry well-dened; BrownYork boundary stress tensor generates boundary time translation as third type of boundary time; (4) With appropriate matching maps, these three boundary times uniable as dierent realizations of same one-parameter automorphism group, thus restating time, algebra, geometry simultaneously as dierent aspects of boundary data. Provide unied scale examples in black hole thermodynamics, AdS/CFT, scattering network experiments; propose several engineering schemes testable on mesoscale experimental platforms. 1 Keywords: Boundary Physics; Time Translation Operator; BirmanKrein Spectral Shift; WignerSmith Time Delay; TomitaTakesaki Modular Theory; NullModular Double Cover; Markov Property; GibbonsHawkingYork Boundary Term; BrownYork Quasilocal Energy; Holography  1 Introduction and Historical Context 1.1 Paradigm Shift from Bulk to Boundary Development of relativity, quantum eld theory, quantum gravity shows clear bulk-toboundary trend. Quantum scattering theory centered on S -matrix directly encodes dynamical information on spacetime asymptotic radiation boundary; BirmanKrein identity connects scattering determinant phase with spectral shift function, making bulk spectral changes readable from boundary scattering data. In algebraic quantum eld theory, TomitaTakesaki modular theory shows: given local algebra and faithful state, modular ow σω t is canonical one-parameter automorphism group of algebra; geometric realization given clear characterization by Bisognano Wichmann theorem, spherical region and null surface modular Hamiltonian local expressions. In general relativity, EinsteinHilbert bulk term alone cannot give well-dened variational principle for manifolds with boundary; must add GibbonsHawkingYork boundary term; when null sheets and joints present, must introduce corresponding null boundary and corner terms to ensure action dierentiability and HamiltonJacobi structure. These advances jointly suggest: truly computable physical objects often concentrated on boundary, while bulk more like reconstruction or evolution result of boundary data. 1.2 Three Seemingly Dierent Theoretical Paradigms Three specic paradigms this paper focuses on: (1) Scattering end: Time as boundary translation scale Let (H0, H0+V) be self-adjoint operator pair, V in appropriate trace class making wave operators and scattering matrix S(ω) well-dened satisfying BirmanKrein conditions. Then spectral shift function ξ(ω) exists satisfying det S(ω) = exp(−2πiξ(ω)), whose derivative ξ′(ω) has exact trace formula relation with scattering phase derivative and WignerSmith time delay matrix Q(ω) = −iS(ω)†∂ωS(ω) trace. This makes φ′(ω) , ξ′(ω) , tr Q(ω) jointly characterize boundary time measure, viewing time as translation parameter generated by boundary spectral data. (2) Algebraicgeometric end: NullModular double cover and overlapping causal diamond chains In Minkowski or AdS boundary CFT, for vacuum state restricted to wedge region, spherical diamond, or null surface region, modular Hamiltonian writable as local integral of stressenergy tensor on boundary, forming innite-dimensional Lie algebra and Markov property on null surfaces. 2 For causal diamond D , introduce NullModular double cover composed of future/past null boundaries (E+, E−) ; modular Hamiltonian KD writable as KD= 2πX σ=±ZEσ gσ(λ, x⊥)Tσσ(λ, x⊥)dλ dd−2x, where Tσσ are null direction components. Modular Hamiltonians of overlapping diamond chains satisfy inclusion-exclusion and Markov stitching properties. (3) Gravity end: GHY boundary term, null boundaries, BrownYork quasilocal energy On spacetime manifold M with boundary, after adding GHY boundary term S=SEH +SGHY variation for xed induced metric well-dened; when boundary includes null sheets and joints, must supplement null boundary and corner terms. HamiltonJacobi analysis for regions with boundary yields BrownYork boundary stress tensor Tab BY =2 √−h δS δhab , whose time component gives quasilocal energy and Hamiltonian generating boundary time translation. These three routes respectively highlight spectralscattering, modularalgebraic, geometric gravitational aspects, but all depend on boundary: scattering dened by asymptotic boundary, modular ow localized on region boundary, gravitational action dierentiability determined by boundary terms. 1.3 Goals and Main Thread Goal: Provide mathematically self-consistent framework with clear applicability domain, viewing above three paradigms as three projections of same boundary structure. Core idea: 1. Take appropriate boundary data triple (∂M,A∂, µ∂) as fundamental object: ∂M geometric boundary, A∂ observable algebra or scattering algebra on it, µ∂ time scale measure from spectral shift or energy ow; 2. Restate time delay in scattering theory, modular ow in modular theory, Brown York boundary Hamiltonian in gravity all as dierent representations of one-parameter automorphism group on this boundary structure; 3. Under strictly limited conditions, prove these representations mutually equivalent, compressing timealgebrageometry three threads onto unied boundary stage.  2 Model and Assumptions Construct abstract model simultaneously accommodating scattering systems, algebraic QFT, gravitational boundaries; clarify applicability domains of all results. 3 2.1 Abstract Boundary Triple Denition 2.1 (Boundary Triple) . Boundary triple is data (∂M,A∂, ω∂) where: 1. ∂M is piecewise smooth three-dimensional manifold, decomposable into timelike, spacelike, null sheets and their joints C ; 2. A∂ is von Neumann algebra acting on Hilbert space H , containing boundary observables (scattering channels, boundary elds, quasilocal energy operators, etc.); 3. ω∂ is faithful normal state on A∂ ; GNS triple denoted (πω,Hω,Ωω) . Postulate 1 (Boundary Completeness) . Physical content of bulk region M completely reconstructible from some boundary triple (∂M,A∂, ω∂) (within given theory's applicability range); time evolution and response operators all determined by boundary one-parameter automorphism group and state evolution. This postulate has dierent concrete realizations in dierent contexts: wave operators and S -matrix in scattering theory, boundary CFT and bulk geometry in AdS/CFT, bulk solution reconstruction from boundary data in HamiltonJacobi perspective. 2.2 Scattering End Assumptions Assumption 1 (S.1: Trace-class perturbation and BK conditions) . On Hilbert space Hscatt , H0 and H=H0+V are self-adjoint operators, V in trace-class or stronger ideal making scattering matrix S(ω) exist for almost all energies ω with S(ω)−1∈S1 . Under these assumptions, Krein spectral shift function ξ(ω) and BirmanKrein identity det S(ω) = exp(−2πiξ(ω)) exist. Assumption 2 (S.2: Time delay matrix and trace formula) . WignerSmith time delay operator dened as Q(ω) = −iS(ω)†∂ωS(ω), Q(ω) is trace-class operator satisfying tr Q(ω) = 2πξ′(ω) in appropriate sense. Under these conditions, dene boundary time measure dµscatt ∂(ω) := 1 2πtr Q(ω)dω. 4 2.3 Modular Theory and NullModular Double Cover Assumptions Assumption 3 (M.1: Standard modular structure) . A∂⊂ B(H) is local algebra for causal region O , ω is vacuum or KMS state making Ωω cyclic and separating vector for A∂ , thus TomitaTakesaki modular operator ∆ and one-parameter modular ow σω t(A)=∆itA∆−it exist. Assumption 4 (M.2: Geometric modular ow) . For wedge region, spherical causal diamond, or null surface region O in vacuum state, modular ow's geometric action is corresponding region's Lorentz transformation or conformal Killing ow; modular Hamiltonian KO=−log ∆ writable as local integral of stressenergy tensor. Assumption 5 (M.3: Markov property and inclusion-exclusion) . For region families on null surface, modular Hamiltonians satisfy Markov property proved by CasiniTeste Torroba: for nested or overlapping regions along null line, conditional mutual information saturates strong subadditivity, corresponding to modular Hamiltonian inclusion-exclusion identities. Based on this, dene NullModular double cover: for causal diamond D , boundary null hypersurfaces decompose as E+∪E− ; dene weighted energy ow integral representation modular Hamiltonians on both sheets. 2.4 Gravitational Boundary Assumptions Assumption 6 (G.1: Action with boundaries/null boundaries) . On four-dimensional Lorentzian manifold M , gravitational action takes standard form S=1 16πG ZM √−g(R−2Λ) d4x+S∂, where S∂=Stl/sp GHY +Snull N+Scorner C are GHY boundary terms on timelike/spacelike sheets, improved terms on null sheets, corner terms at joints. Assumption 7 (G.2: Variation well-denedness and BrownYork stress tensor) . On variation family xing boundary induced metric (and appropriate equivalent data for null boundaries), δS contains only bulk terms, yielding Einstein equations. For given timelike boundary three-sheet 3B , action variation with respect to boundary metric de- nes BrownYork boundary stress tensor Tab BY =2 √−h δS δhab , whose contraction with boundary Killing vector ξa gives quasilocal energy and Hamiltonian generating boundary time translation. Under these assumptions, view BrownYork Hamiltonian as third type of boundary time generator.  5 3 Main Results (Theorems and Alignments) State four main results establishing correspondences among three theoretical threads. Theorem 3.1 (A: BKWignerSmithTime Scale Identity) . Under Assumptions S.1 S.2, dene: • Total scattering phase Φ(ω) = arg det S(ω) , half-phase φ(ω) = 1 2Φ(ω) ; • Krein spectral shift function ξ(ω) satisfying det S(ω) = exp(−2πiξ(ω)) ; • WignerSmith time delay operator Q(ω) = −iS(ω)†∂ωS(ω) . Then almost everywhere nite derivatives ξ′(ω) and φ′(ω) exist such that φ′(ω) π=ξ′(ω) = 1 2πtr Q(ω) holds almost everywhere. Thus measure dµscatt ∂(ω) := 1 2πtr Q(ω)dω equivalent to spectral shift measure dξ(ω) and scattering phase scale π−1dφ(ω) , viewable as unied boundary time scale. Denition 3.2 (NullModular Double Cover) . In d -dimensional Minkowski spacetime, consider causal diamond with vertices p, q : D(p, q) = J+(p)∩J−(q). Boundary composed of future null hypersurface N+ and past null hypersurface N− . Dene NullModular double cover e ED:= E+⊔E−, where E± are two smooth leaves of N± after removing joint points; introduce ane parameter λ and transverse coordinates x⊥ on each leaf. Theorem 3.3 (B: Modular Hamiltonian Null Measure Localization) . Under Assumptions M.1M.2, for Minkowski vacuum state restricted to causal diamond D local algebra A(D) , modular Hamiltonian writable as KD= 2πX σ=±ZEσ gσ(λ, x⊥)Tσσ(λ, x⊥)dλ dd−2x , where: 1. T++ =Tvv , T−− =Tuu are stressenergy tensor components along two null directions; 2. Weight functions gσ(λ, x⊥) determined solely by causal diamond geometric data, linearly degenerating at endpoints. Moreover, for overlapping causal diamond family {Dj} on same null surface, modular Hamiltonians satisfy inclusion-exclusion identity K∪jDj=X k≥1 (−1)k−1X j1<···<jk KDj1∩···∩Djk, and Minkowski vacuum state restricted to these regions satises Markov property: for appropriately nested A, B, C , conditional mutual information I(A:C|B)=0 equivalent to above inclusion-exclusion identity. 6 Theorem B shows : Modular Hamiltonian completely localizable on null measure boundary; NullModular double cover provides purely boundary geometric realization of modular ow. Theorem 3.4 (C: GHYBrownYork Boundary Hamiltonian) . Under Assumptions G.1 G.2, consider spacetime region M with timelike/spacelike/null sheets and joints; total action S=1 16πG ZM √−g(R−2Λ) d4x+Stl/sp GHY +Snull N+Scorner C satises: 1. For all metric variations δgµν xing induced geometry (and equivalent data on null boundaries) on boundary, total variation δS =1 16πG ZM √−g Gµν δgµν d4x, yielding Einstein equations; 2. For given timelike boundary 3B with timelike Killing vector ξa , BrownYork boundary stress tensor Tab BY =2 √−h δS δhab denes boundary Hamiltonian Hgrav ∂[ξ] = ZB √σ uaTab BY ξbd2x generating quasilocal time translation along 3B time direction ξa ; when 3B extends to innity, this Hamiltonian converges to ADM or Bondi mass. Thus Hgrav ∂ viewable as third type of boundary time generator, at same level as Null Modular and scattering end generators. Denition 3.5 (Unied Boundary Time Generator) . Given boundary triple (∂M,A∂, ω∂) , call self-adjoint operator H∂ unied boundary time generator if: 1. In scattering representation, H∂ 's spectral decomposition measure for energy variable ω equivalent to dµscatt ∂(ω) ; 2. In algebraic representation, modular Hamiltonian satises K∂= 2πβ−1H∂ for some positive β , producing TomitaTakesaki modular ow; 3. In geometric representation, BrownYork Hamiltonian writable as Hgrav ∂[ξ] = ⟨H∂, J[ξ]⟩ where J[ξ] is boundary charge functional associated with Killing vector ξ . Theorem 3.6 (D: Boundary Trinity Principle) . Assume following matching structure exists: 1. Scattering system's incoming/outgoing channels embeddable into separable subalgebra of some QFT's boundary algebra A∂ , making scattering phase φ(ω) consistent with modular Hamiltonian's spectral phase in asymptotic regions; 2. QFT's stressenergy tensor expectation value connected to BrownYork boundary stress tensor via holographic dictionary or semiclassical Einstein equations; 3. Boundary Killing time and scattering energy normalization constants satisfy thermal time hypothesis normalization condition: modular ow parameter diers from physical time by constant scale. 7 Under these conditions, exists unique (up to global ane transformation) unied boundary time generator H∂ making Scattering time delay ⇐⇒ Modular ow parameter ⇐⇒ BrownYork boundary time equivalent in common domain. More specically, positive constants c1, c2 exist making φ′(ω) π=ξ′(ω) = 1 2πtr Q(ω), KD= 2πZTσσgσ, Hgrav ∂[ξ] = Z√σ uaTab BYξb satisfy H∂=Zω dµscatt ∂(ω) = c1KD+c−1 2Hgrav ∂ giving three realizations of same one-parameter group e−itH∂ in dierent representations of same boundary Hilbert space. Theorem D doesn't claim unied generator automatically constructible in arbitrary theories, but points out under above matching conditions, three threads naturally compress onto same boundary time object.  4 Proofs Provide proof skeletons of main theorems; ner technical details and special cases in appendices. 4.1 Proof Skeleton of Theorem A BirmanKrein identity shows under Assumption S.1, spectral shift function ξ(ω) exists such that det S(ω) = exp(−2πiξ(ω)), thus Φ(ω) = arg det S(ω) = −2πξ(ω). Half-phase φ(ω) = −πξ(ω) . Dierentiating at smooth points: φ′(ω) = −πξ′(ω). On other hand, EisenbudWignerSmith time delay theory shows: in potential scattering systems satisfying moderate decay and regularity conditions, WignerSmith time delay operator denable as Q(ω) = −iS(ω)†∂ωS(ω), proving its trace satises tr Q(ω) = 2π ξ′(ω). 8 Combining two equations gives φ′(ω) π=ξ′(ω) = 1 2πtr Q(ω), completing Theorem A proof. Rigorous treatment requires considering exceptional zeromeasure sets on ω ; see Appendix A. [Proofs of Theorems B, C, D condensed for space...]  5 Model Applications Show unied boundary framework applicability in three typical physical scenarios: black hole thermodynamics, AdS/CFT, nite-scale scattering networks. 5.1 Black Hole Thermodynamics For static black hole, event horizon is null boundary; surface gravity κ and Hawking temperature TH=κ/(2π) jointly characterize thermal properties. QFT vacuum state outside horizon in wedge region's modular ow equivalent to Euclidean time translation around horizon, period being βH= 1/TH . In unied boundary framework: 1. Scattering end : Consider xed-energy scattering outside black hole; group delay tr Q(ω) energy dependence encodes eective potential barrier and quasinormal mode structure near horizon; 2. NullModular end : Horizon itself viewable as part of NullModular double cover; modular Hamiltonian localized on null generator, proportional to Tvv integral; 3. Gravity end : For boundary two-sphere surrounding black hole, BrownYork quasilocal energy approaches ADM mass at innity, forms Legendre structure with Bekenstein Hawking entropy and temperature product near horizon. Unied generator H∂ gives compatible time translation and heatgeometry relations at three endpoints, restating black hole thermodynamics as purely boundary phenomenon. 5.2 AdS/CFT and Holographic Time Reconstruction In AdS/CFT duality, boundary CFT time evolution generated by Hamiltonian HCFT ; modular Hamiltonian and generalized entropy extremal surfaces jointly determine bulk geometric response. Unied boundary framework provides natural language: • Scattering end : Bulk high-energy process scattering delay corresponds to boundary CFT correlation function phase structure; • NullModular end : Boundary CFT spherical region modular ow corresponds via JLMS to bulk Killing ow and minimal surface; • Gravity end : These Killing ows generated by BrownYork boundary Hamiltonian, action readable from GHY term HamiltonJacobi variation. Thus holographic time reconstruction understandable as unied generator H∂ 's two representations on CFT and AdS sides. 9