Boundary Language as a Unified Physical Framework:\\ From Scattering Phase, GHY Boundary Term to Modular Flow and Generalized Entropy as a Single Structure

Boundary Language as a Unied Physical Framework: From Scattering Phase, GHY Boundary Term to Modular Flow and Generalized Entropy as a Single Structure Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 20, 2025 Abstract This paper proposes and systematizes the concept of boundary language, rewriting physical theories as algebraicgeometric structures about what is allowed to be exchanged on causal cut surfaces, with bulk eld theory being merely one realization of this structure. Taking the boundary observable algebra and boundary state as fundamental objects, we unify three seemingly independent theoretical paradigms into the same framework: (1) in scattering theory, the spectral shift function, total scattering phase, and WignerSmith group delay; (2) in general relativity, the GibbonsHawkingYork (GHY) boundary term and BrownYork quasilocal energy; (3) in operator algebras, the TomitaTakesaki modular ow and relative entropy monotonicity. The core idea is: time is not a parameter of ow within the bulk that is given a priori, but rather a unied translation parameter generated by what is allowed in terms of ux balance and information monotonicity in the boundary language; all observable delays, energies, and generalized entropy variations are dierent projections of the same boundary structure. Mathematically, we formalize this framework as three boundary language axioms: (A1) Conservation and Flux Axiom, viewing the boundary as a balancing interface for energy, charge, and information ux; (A2) Time Generation Axiom, viewing the one-parameter ∗ -automorphism group dened on the boundary and its generator as the source of time scale; (A3) Monotonicity and Consistency Axiom, represented by relative entropy monotonicity and its geometric forms (quantum focussing, quantum null energy condition, etc.), excluding supercausality and negative entropy transport. In the scattering realization, we prove: the boundary language satisfying A1A3 necessarily induces the scale identity on a well-posed short-range scattering system κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), where φ(ω) is the total scattering half-phase, ρrel(ω) is the relative state density, and Q(ω) is the WignerSmith group delay operator. This identity unies the phase gradient, spectral shift density, and group delay trace as a single boundary object called time scale. 1 In the gravity realization, we show: the GHY boundary term and the Brown York quasilocal energy positivity are necessary conditions for the boundary language A1A2 on the geometric side; thus ADM time, proper time, and Killing time are restated as translations generated by the boundary Hamiltonian. In the operator algebra realization, we provide the canonical model of boundary language through TomitaTakesaki modular theory and implement A3 through relative entropy monotonicity, thereby characterizing the time arrow as the monotonic evolution of relative entropy with modular time under a natural class of conditions. Finally, through three model classesone-dimensional potential scattering, static black hole exterior regions, and Rindler wedgeswe demonstrate how the boundary language produces experimentally observable time delays, quasilocal energy, and Unruh temperature, and give several testable spectralgeometricinformation theoretical predictions. Detailed appendices provide proofs of key propositions such as the scatteringspectral shiftgroup delay scale identity, the variational completeness of the GHY term, and relative entropy monotonicity, and introduce the error geometry framework of nite-order EulerMaclaurin and Poisson discipline to ensure a controllable mapping from boundary readings to experimental data. Keywords: Boundary Language; Scattering Phase; WignerSmith Group Delay; GHY Boundary Term; BrownYork Quasilocal Energy; TomitaTakesaki Modular Flow; Relative Entropy; Time Scale  1 Introduction 1.1 Research Motivation and Overall Structure Traditional eld theory and gravitational theory often start from the bulk: given a manifold (M, g) with a metric, a bulk action S[Φ, g] and its variational equations, then supplemented by boundary conditions. However, in three classes of seemingly unrelated theories, the boundary has long played the role of truly observable: 1. In scattering and spectral theory, the total scattering phase φ(ω) , the BirmanKren spectral shift function ξ(ω) , and the WignerSmith group delay operator Q(ω) = −iS†∂ωS are completely dened by the in/out asymptotic innite boundaries. 2. In general relativity, the variation of the EinsteinHilbert bulk action requires introducing the GibbonsHawkingYork (GHY) boundary term to achieve variational completeness, while the BrownYork quasilocal energy and quasilocal stress tensor are strictly boundary data. 3. In operator algebras and algebraic quantum eld theory, the TomitaTakesaki modular ow σω t is completely determined by the algebrastate pair (M, ω) , with its parameter t viewed as intrinsic time; the monotonicity of relative entropy and generalized entropy inequalities depend only on the inclusion relationships between boundary accessible algebras. These facts suggest: the boundary itself, rather than the bulk, is the natural stage for unied physical structure . Based on this, this paper proposes the concept 2 of boundary language, dening physical theory as a triple structure on a certain causal cut surface Σ⊂M LΣ= (A∂, ω, F), where A∂ is the boundary observable algebra, ω is the boundary state, and F is a family of ux functionals used to characterize the exchange of energy, charge, and information across Σ . Bulk eld theory, geometry, and scattering constructions are merely dierent ways of realizing this triple. 1.2 Core Point: Time as Boundary Translation The core claim of this paper can be summarized as:  Given a causal cut surface Σ , physical theory rst gives allowed exchanges across Σ this is the boundary language;  Time is not a continuous parameter given to the bulk a priori, but is derived from a one-parameter ∗ -automorphism group {αt}t∈R internal to the boundary language and its generator;  When conservation and monotonicity conditions are satised, this time belongs to the same time scale equivalence class as the frequency derivative of the scattering phase, the gravitational boundary Hamiltonian, and the modular ow parameter. More specically, we will prove the scale identity in the scattering realization κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), and show in gravity and modular ow realizations: ADM/proper time and modular time can be mapped into the same equivalence class through this scale. 1.3 Article Structure The structure of the full text is as follows: Section 2 gives the strict denition of boundary language and the three axioms. Section 3 realizes the boundary language in short-range scattering theory and proves the scale identity. Section 4 shows how the GHY term and BrownYork quasilocal energy implement A1A2 in gravitational theory with boundaries. Section 5 discusses modular ow and relative entropy monotonicity, demonstrating the implementation of A3 on the algebraic side. Section 6 integrates the three-terminal structure, introducing the unied time scale equivalence class. Section 7 gives several typical models and testable predictions. Appendices AD provide detailed proofs of key propositions and the error geometry framework.  2 Boundary Language and Three Axioms 2.1 Causal Cut Surface and Boundary Algebra Let (M, g) be a Lorentzian manifold with good causal structure, and Σ⊂M be a codimension-one submanifold dividing spacetime into inside Min and outside Mout . We call Σ a causal cut surface. 3 Denition 2.1 (Boundary Observable Algebra) . The boundary observable algebra associated with the causal cut surface Σ is a C∗ -algebra A∂ , whose elements correspond to observables that can be determined solely through readings on Σ , such as:  In/out eld operators and functions of the S-matrix on scattering channels;  Boundary-induced metric, extrinsic curvature, and geometric quantities composed of them;  Local algebras on wedge regions or Cauchy surface boundaries in algebraic quantum eld theory. Denition 2.2 (Boundary State) . A boundary state is a positive normalized linear functional ω:A∂→C on A∂ , representing the expectation value of boundary observables under given physical congurations (bulk elds, metric, external sources, etc.). 2.2 Boundary Language Triple Denition 2.3 (Boundary Language) . A boundary language is the triple LΣ= (A∂, ω, F), where A∂, ω are as described above, and F ⊂ A∗ ∂ is a family of real-valued linear functionals called ux functional family, used to characterize boundary readings of exchangeable quantities such as energy, charge, entropy, or information. Typical examples include:  In scattering theory, functionals related to probability current, energy ow, or time delay;  In gravitational theory, functionals related to quasilocal energy, momentum, angular momentum, and generalized entropy;  In algebraic quantum theory, functionals related to modular Hamiltonian, relative entropy, and information ow. 2.3 Axiom A1: Conservation and Flux Axiom 2.4 (Conservation and Flux) . For a bulk action Sbulk and boundary action Sbdry satisfying appropriate regularity conditions, there exists a ux functional F∈ F such that for any compactly supported bulk variation δΦ, δg , δ(Sbulk +Sbdry) = (volume integral) +F(δXΣ), where δXΣ∈ A∂ is the corresponding boundary source variation. It is required that for all boundary condition-satisfying physical variations, when the bulk equations hold, F(δXΣ)=0 and F is linear for the allowed boundary variation family, so that boundary variation can be interpreted as an expression of the ux conservation condition. Intuitively, A1 requires that any residual term of bulk variation on the boundary can be identied as a ux functional acting on boundary data variation, thereby restating variational completeness as the condition that ux can be attributed to boundary language. 4 2.4 Axiom A2: Time Generation Axiom 2.5 (Time Generation) . On the boundary observable algebra A∂ , there exists a one-parameter ∗ -automorphism group {αt}t∈R⊂Aut(A∂), whose generator is a closed unbounded derivation δ , satisfying: 1. There exists a family of time observables T ⊂ A∂ such that for each T∈ T , t7→ ω(αt(T)) is continuously dierentiable; 2. The ux functionals satisfy appropriate invariance or conservation properties under αt , e.g., for energy functional FE∈ F , FE◦αt=FE; 3. The corresponding generator δ can be represented through a certain boundary Hamiltonian H∂∈ A′′ ∂ as d dtω(αt(A)) = i ω([H∂, αt(A)]) at least on a dense domain. We call the parameter t∈R the time scale generated by the boundary language. It is conjugate to frequency ω in the scattering realization, conjugate to ADM/proper time in the gravity realization, and conjugate to the modular parameter in the modular ow realization. 2.5 Axiom A3: Monotonicity and Consistency Axiom 2.6 (Monotonicity and Consistency) . Given boundary language LΣ , for any two boundary states ω, ω′ , there exists a non-negative function Srel(ω′∥ω) called relative entropy or generalized entropy, satisfying: 1. Non-negativity: Srel(ω′∥ω)≥0 , with equality if and only if ω′=ω ; 2. Monotonicity: for any subalgebra inclusion A∂,1⊂ A∂,2⊂ A∂ , S(1) rel (ω′∥ω)≤S(2) rel (ω′∥ω); 3. Time consistency: under appropriate physical conditions (such as energy conditions or KMS conditions), if evolving along the ow ωt, ω′ t of the time generation axiom A2, then d dtSrel(ω′ t∥ωt)≤0 at least holds in one direction (dening the time arrow). A3 characterizes causal consistency and information cannot increase as intrinsic properties of the boundary language, providing a basis for the boundary geometric algebraic denition of the time arrow.  5 3 Boundary Language Realization in Scattering Theory 3.1 Short-Range Scattering and S-Matrix Consider self-adjoint operators H0 and H=H0+V on Hilbert space H , where V is a short-range perturbation satisfying standard assumptions such that the wave operators Ω±= slim t→±∞ eitHe−itH0 exist and are complete. The S-matrix is dened as S= Ω∗ +Ω−, which can be decomposed in the energy representation as S=Z⊕ S(ω) dµ(ω), where S(ω) is a unitary matrix acting on the channel space. Here we take the causal cut surface to be the asymptotic innite in/out boundary, with the boundary algebra being A∂=B(Hin)∨B(Hout) or an appropriate subalgebra (e.g., algebra generated by asymptotic behavior). The boundary state ω can be taken as an incident wave packet or thermal equilibrium state. Denition 3.1 (Total Scattering Phase and Group Delay) . Let Φ(ω) = arg det S(ω), φ(ω) = 1 2Φ(ω). Dene the WignerSmith group delay operator as Q(ω) = −iS(ω)†∂ωS(ω). Denote the trace over the channel space as tr Q(ω) . Also let ξ(ω) be the BirmanKren spectral shift function, and dene the relative state density as ρrel(ω) = d dωξ(ω) (sign convention see Appendix A). 3.2 Scale Identity and A2 Scattering Realization In the scattering context, we select time observables as observables generated by S(ω) in frequency space, and ux functionals include probability current, energy ow, and delay functionals. 6 Theorem 3.2 (ScatteringSpectral ShiftGroup Delay Scale Identity) . In a scattering system satisfying short-range and regularity assumptions, the scale function κ(ω) = φ′(ω) π satises the identity with the relative state density ρrel(ω) and the group delay trace (2π)−1tr Q(ω) : κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω). This identity unies the frequency derivative of the total phase, the spectral shift density, and the group delay into the same scale function κ(ω) , which can be viewed as the concrete expression of the time scale generated by the boundary language at the scattering end. Proof outline : Using the BirmanKren formula det S(ω) = e−2πiξ(ω), dierentiating with respect to ω and combining with det S(ω) = eiΦ(ω) and the denition of Q(ω) = −iS†∂ωS , we obtain tr Q(ω)=2πξ′(ω),Φ(ω) = −2πξ(ω), thus φ′(ω)/π =−ξ′(ω), ρrel(ω) = −ξ′(ω), synthesizing to yield the stated identity. Detailed proof in Appendix A. Corollary 3.3 (Scattering Time Scale) . For an incident wave packet frequency-localized in the neighborhood of ω0 , its average group delay τWS(ω0) gives an experimental reading of the boundary time scale: τWS(ω0) = Rdω|a(ω)|2tr Q(ω) Rdω|a(ω)|2≈2π κ(ω0). 3.3 Boundary Language Perspective on Spectral Flow and Topological Branches Consider a scattering system S(ω;λ) that varies continuously with parameter λ∈R . Under appropriate Fredholm and regularity conditions, the spectral shift function ξ(ω;λ) is a continuous function of (ω, λ) , and its integral over ω gives the spectral ow: Proposition 3.4 (Spectral Flow as Boundary Topological Branch Change) . Over the parameter interval [λ1, λ2] , the cumulative spectral ow produced by eigenvalue crossings of the spectral threshold within the energy interval [ω1, ω2] ∆N=Zω2 ω1 dω ρrel(ω;λ) is an integer and corresponds to a topological branch change of the boundary language. Its physical meaning is: parameter changes cause changes in the number of states allowed to cross the boundary, corresponding to transitions in topological classes or number of bound states.  7 4 Boundary Language in Gravity and GHY Term 4.1 Variational Problem of EinsteinHilbert Action Let (M, g) be a four-dimensional spacetime with boundary ∂M . The EinsteinHilbert bulk action is SEH[g] = 1 16πG ZM d4x√−g R. For metric variation δgµν , the scalar curvature variation can be written as δR =gµνδRµν +Rµνδgµν, and δRµν contains rst-order derivatives of δgµν . After integration by parts, δSEH includes volume and boundary terms, with boundary terms containing ∂nδg terms that cannot be expressed solely through variations of the induced metric hij =gij|∂M , meaning SEH alone cannot give a well-dened Dirichlet boundary variational problem. Proposition 4.1 (GHY Term as Necessary Condition for A1) . If only SEH is included without adding boundary action SGHY , then the boundary contribution to the metric variation contains terms that cannot be written as a ux functional F∈ F acting linearly on boundary source variation δhij , thus violating the boundary language axiom A1. Adding the GHY boundary term SGHY[g] = 1 8πG Z∂M d3xp|h|K, where hab is the boundary-induced metric, K is the trace of extrinsic curvature, ϵ=±1 depends on normal type. The boundary variation of the combined action Stot =SEH + SGHY can be written as δStot[g] = (volume integral) +1 16πG Z∂M d3xp|h|(Kij −Khij)δhij, where Kij is the extrinsic curvature, K=Kijhij . Therefore F(δXΣ) = 1 16πG Z∂M d3xp|h|(Kij −Khij)δhij constitutes a ux functional, implementing A1. Detailed derivation in Appendix B. 4.2 BrownYork Quasilocal Energy and Time Generation From the variation of the total action with respect to boundary metric, the BrownYork quasilocal energy-momentum tensor can be dened: TBY ij =2 p|h| δStot δhij =1 8πG(Kij −Khij) + (reference term) . For a given time slice Σt⊂∂M and its unit timelike vector eld ui , the quasilocal energy can be dened: EBY[Σt] = ZΣt d2x√σ uiujTBY ij , where σ is the induced two-dimensional metric on Σt . 8 Theorem 4.2 (Boundary Hamiltonian and Geometric Time Generation) . In spacetime with ADM decomposition, there exists a boundary Hamiltonian H∂ which, under appropriate boundary conditions, generates time evolution on the boundary algebra through Poisson brackets or commutators: for any boundary observable A∈ A∂ , d dtωt(A) = i ωt([H∂, A]), where ωt is the state along time slice Σt . Moreover, H∂ can be obtained from the integral of BrownYork quasilocal energy, so geometric time is completely generated by the boundary language, satisfying A2. In the static case (with timelike Killing vector), H∂ is equivalent to ADM mass or Komar energy, so Killing time, ADM time, and the time scale generated by boundary language belong to the same equivalence class.  5 Operator Algebras, Modular Flow, and Relative Entropy 5.1 Standard Form and Modular Flow Let M be a von Neumann algebra acting on Hilbert space H , and ω a faithful normal state on M . The GNS representation yields vector Ω∈ H such that ω(A) = ⟨Ω, AΩ⟩. The Tomita operator S is dened by SAΩ = A∗Ω, A ∈ M whose polar decomposition gives S=J∆1/2, where J is conjugation and ∆ is the modular operator. The TomitaTakesaki modular ow is dened as σω t(A) = ∆itA∆−it, t ∈R. In the boundary language framework, we take A∂=M, αt=σω t, ω given . Proposition 5.1 (Modular Flow as Canonical Realization of A2) . For any standard form (M,H,Ω) and faithful normal state ω , the TomitaTakesaki modular ow {σω t} is a one-parameter ∗ -automorphism group satisfying: 1. ω◦σω t=ω (i.e., ω is a KMS state for the modular ow); 2. For any A∈ M , the map t7→ ω(σω t(A)) is continuous. Therefore, (M, ω, {σω t}) naturally satises the time generation axiom A2, providing a rigorous mathematical realization of modular time. 9 C.2 Relative Modular Operator and Relative Entropy Given two states ω, ω′ , the relative modular operator ∆ω′,ω is constructed through the relative Tomita operator: Sω′,ωAΩω=A∗Ω′, whose polar decomposition is Sω′,ω =Jω′,ω∆1/2 ω′,ω . The Araki relative entropy is S(ω′∥ω) = −⟨Ω′,log ∆ω′,ω Ω′⟩. Relative entropy monotonicity can be proven through completely positive trace-preserving maps and Stinespring representation: if Φ : M → N is completely positive tracepreserving, then S(ω′◦Φ∥ω◦Φ) ≤S(ω′∥ω). Taking Φ as conditional expectation or subalgebra restriction yields Theorem 5.2.1. C.3 Inequality Form of Time Arrow In some cases, the time arrow statement can be strengthened to the following inequality: let ωt, ω′ t be states evolving along the modular ow, then d2 dt2S(ω′ t∥ωt)≥0, i.e., relative entropy is convex in modular time. This type of property is closely related to the second-order variation of generalized entropy in holographic and QNEC/QFC literature, corresponding to quantum focussing conditions on null boundaries.  Appendix D: Error Geometry of Finite-Order Euler Maclaurin and Poisson Discipline The scale function κ(ω) generated by boundary language and experimental readings are often connected through frequency integration and discrete sampling. To ensure rigorous theoryexperiment docking, an error geometry framework controlling aliasing and truncation errors is needed. D.1 Finite-Order EulerMaclaurin Formula For suciently smooth functions f , on interval [a, b] , the EulerMaclaurin formula is b X n=a f(n) = Zb a f(x) dx+f(a) + f(b) 2+ m X k=1 B2k (2k)! f(2k−1)(b)−f(2k−1)(a)+Rm, where B2k are Bernoulli numbers and Rm is the remainder. We enforce taking only nite order m and view the upper bound of Rm on a given function class as part of error geometry: its magnitude is controlled by high-order derivatives of f and principal singularities. In principle, we require: 16 1. Any polynomial or rational approximation tting the scale function κ(ω) controls the growth of high-order derivatives within physically relevant frequency intervals; 2. All errors Rm introduced by EulerMaclaurin truncation do not produce new singularities, i.e., singularity does not increase, poles = principal scales. D.2 Poisson Summation and Aliasing Control The Poisson summation formula X n∈Z f(n) = X k∈Z ˆ f(2πk) connects discrete sampling with frequency space. For numerical calculations of scale functions and related response functions, sampling step ∆ω and frequency domain support determine aliasing errors. In the boundary language framework, we require: 1. Sampling satises the NyquistShannon condition so that aliasing error can be upper bounded; 2. For each experimental sampling scheme, give explicit aliasing error estimates and prove they do not introduce new singularities, only changing weights or distributions; 3. Error analysis follows nite-order discipline: not relying on formally innite sums or innite dierentiation, but forming closed error geometry through nite-order truncation and rigorous error bounds. This framework makes the mapping from theoretical scale function κ(ω) to discrete measurement data mathematically controllable, both ensuring consistency of boundary language three axioms in numerical implementation and providing direct error budget tools for experimental design. 17