Boundary as Clock: Time as Unified Translation Operator of Phase--Spectral Shift--Modular Flow

Boundary as Clock: Time as Unied Translation Operator of PhaseSpectral ShiftModular Flow Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 24, 2025 Abstract Against background of general C∗ -algebras and operator scattering theory, construct framework of time = boundary translation. Time not viewed as pre-given ow parameter in bulk domain but dened as unique translation scale generated by boundary spectral data, maintaining self-consistency among phasespectral shift modular ow triple readings. Specically: First, in scattering systems satisfying BirmanKrein conditions, take total scattering phase Φ(ω) = arg det S(ω) , spectral shift function ξ(ω) , relative state density ∆ρ(ω) , WignerSmith operator Q(ω) = −iS(ω)†∂ωS(ω) as core; establish scale identity φ′(ω) π=ρrel(ω) = 1 2πTr Q(ω), φ(ω) := 1 2Φ(ω), interpreting as spectral ruler of scaling time by boundary phase gradient. Second, for given boundary observable algebra A∂ and faithful state ω 's GNS representation, introduce TomitaTakesaki modular operator ∆ and modular ow σω t ; under scatteringKMS consistency assumption prove: time parameter t inferred from S(ω) and Q(ω) identical with modular time parameter under appropriate normalization. Finally propose time equivalence principle and scale identity axiom: any physical evolution of bulkexterior data pairs equivalently rewritable as translation U(t) = e−itH∂ generated by boundary generator H∂ , where t uniquely determined by boundary spectral measure readout function T . Under natural monotonicity and regularity assumptions, prove time scale satisfying these axioms unique in sense of additive and proportional transformations. Thus at purely operatorgeometric and scattering information level, time characterized as boundary translation parameter with phasespectral shiftmodular ow triple reading self-consistent, providing testable theoretical basis for reconstructing spacetime and dynamics from boundary information. Keywords: Time Essence; Scattering Phase; Spectral Shift Function; WignerSmith Operator; Modular Flow; Boundary Algebra; KMS State; Time Equivalence Principle MSC 2020: 81U40, 81Q10, 46L55, 58J40  1 1 Introduction and Historical Context Classical mechanics views time as absolutely uniformly owing external parameter; in general relativity, time embedded as coordinate function with causal cone structure; standard quantum theory mostly uses internalexternal parameter split, taking time as continuous parameter in Schrödinger equation. In contrast, development of operator algebras and quantum statistical mechanics shows: given observable algebra A and state ω , can construct intrinsic automorphism group family σω t via TomitaTakesaki modular theory, naturally interpreted as modular time. This structure plays central role in KMS conditions and equilibrium state theory. On other hand, in scattering theory, WignerSmith time delay concept interprets total scattering phase frequency derivative as average residence time particle experiences in potential eld; this concept extensively generalized and veried in random media, chaotic scattering, electromagnetic, acoustic systems. In rigorous operator scattering framework, BirmanKrein formula connects scattering determinant with spectral measure using spectral shift function ξ(λ) , giving det S(λ) = exp(−2πi ξ(λ)), while Krein trace formula connects spectral shift function between two operators with test function dierence trace. These chains unify phasespectral shiftrelative state density as dierent aspects of same object. ConnesRovelli thermal time hypothesis further proposes: in generally covariant quantum theories, physical time ow shouldn't be given by preset external parameter but jointly determined by system's statistical state and observable algebra; time ow realized by state's modular automorphism group. This makes time = modular ow powerful candidate answer. Above three threads respectively answer how to read out time from scattering phase, how to scale state density by spectral shift function, how to construct time ow from state and algebra. This paper's goal: within single, geometrically minimal-structure framework, unify these three; give rigorous existence and uniqueness conclusions. Core idea: introduce boundary algebra A∂ as insideoutside information interface, requiring: 1. All observable outputs ultimately land on A∂ ; 2. Given faithful state ω and scattering data S(ω) , exists unique (up to ane) time translation group αt such that: • αt generated by self-adjoint operator H∂ in GNS representation; •αt consistent with σω t ; • Time ruler under αt normalized by scale identity. From this perspective, time no longer ow variable in bulk domain but characterized as unique translation parameter realizing alignment between boundary spectral data and modular ow. This characterization maintains spiritual continuity with thermal time hypothesis but requires additional observable scattering data as scale anchor, making time have direct experimental readout.  2 Model and Assumptions Give model structure and basic assumptions in abstract framework. Goal: obtain minimal condition family sucient for applying BirmanKrein formula, spectral shift function, 2 modular theory without relying on specic spacetime geometry. 2.1 Boundary Algebra and State Let A∂ be separable C∗ -algebra representing boundary observables. Select faithful state ω:A∂→C ; GNS representation denoted (πω,Hω,Ωω) satisfying ω(A) = ⟨Ωω, πω(A)Ωω⟩, A ∈ A∂, Ωω is cyclic and separating vector. Assume strongly continuous C∗ -automorphism family exists: αt:A∂→ A∂, t ∈R, realized on GNS space by unitary group U(t) : πω(αt(A)) = U(t)πω(A)U(t)−1, U(t)Ωω= Ωω. Generator H∂ is self-adjoint operator satisfying U(t) = e−itH∂ . Call (A∂, ω, αt) **boundary dynamical system**. 2.2 Scattering System and BirmanKrein Setting Let H0, H be self-adjoint operators on separable Hilbert space H satisfying typical scattering assumptions: 1. V:= H−H0 is trace-class perturbation; 2. H0 's absolutely continuous spectrum dominates on energy axis I⊂R ; 3. Wave operators W± exist making W±= s-lim t→±∞ eitHe−itH0Pac(H0); 4. Scattering operator S:= W∗ +W− is unitary on Pac(H0)H . In energy representation, S decomposable as xed-energy scattering matrix family S(λ) : K(λ)→ K(λ), λ ∈I, where K(λ) is channel space at each energy. Assume λ7→ S(λ) suciently smooth on I . Under these assumptions, spectral shift function ξ(λ)∈L1 loc(I) exists satisfying Krein trace formula Tr(f(H)−f(H0)) = ZI f′(λ)ξ(λ)dλ for suciently large function class. Simultaneously, BirmanKrein formula gives scattering determinantspectral shift function relation: det S(λ) = exp(−2πi ξ(λ)) almost everywhere on I. 2.3 Relative Density of States, Scattering Phase, WignerSmith Operator Dene total scattering phase Φ(λ) := arg det S(λ), φ(λ) := 1 2Φ(λ). 3 From BirmanKrein formula: Φ(λ)≡ −2πξ(λ) (mod 2π), thus on locally continuous representative Φ′(λ) = −2πξ′(λ). Dene relative state density (DOS dierence) ∆ρ(λ) := ρ(λ)−ρ0(λ), where ρ, ρ0 are state density functions of H, H0 . Under standard setting, ∆ρ and spectral shift function derivative satisfy ∆ρ(λ) = −ξ′(λ)⇒1 2πΦ′(λ) = ∆ρ(λ), yielding φ′(λ) π= ∆ρ(λ). On other hand, for each energy, dene WignerSmith delay operator Q(λ) := −iS(λ)†∂λS(λ) on K(λ) . Q(λ) is self-adjoint operator; trace Tr Q(λ) equals total scattering phase derivative under standard scattering framework: Φ′(λ) = Tr Q(λ). Merging above relations gives scale identity φ′(λ) π= ∆ρ(λ) = 1 2πTr Q(λ). To avoid notation confusion, uniformly denote energy variable as ω ; write scale identity as φ′(ω) π=ρrel(ω) = 1 2πTr Q(ω), where ρrel(ω) := ∆ρ(ω) . 2.4 Modular Flow, KMS Condition, Thermal Time On GNS representation (πω,Hω,Ωω) , dene Tomita operator S0πω(A)Ωω=πω(A)∗Ωω, A ∈ A∂. Closure denoted S ; polar decomposition S=J∆1/2 gives antilinear unitary conjugation J and modular operator ∆ . TomitaTakesaki theorem asserts modular automorphism family exists: σω t(A) := ∆itA∆−it, t ∈R, 4 forming one-parameter automorphism group on A∂ ; ω satises KMS condition for σω t . Formally write modular generator Kω:= −log ∆, σω t(A) = eitKωAe−itKω. ConnesRovelli thermal time framework proposes: in generally covariant eld theories, physical time ow determinable by given state and observable algebra, specically modular ow σω t . This paper restricts this idea to boundary algebra A∂ ; requires modular time consistent with time parameter scaled from scattering phasespectral shiftWignerSmith operator.  3 Main Results (Theorems and Alignments) Propose time equivalence principle, scale identity axiom, modular consistency axiom; give existence and uniqueness theorem for unied time scale. 3.1 Time Equivalence Principle and Boundary Generator Axiom 1 (Time Equivalence Principle) . Consider realizable bulkexterior data pairs (Din, Dout) as vectors or states in Hin,Hout . Boundary Hilbert space H∂ , self-adjoint operator H∂ , unitary group U(t) = e−itH∂, t ∈R exist such that for any realizable data pair, real number t exists satisfying Kout ∂Dout =U(t)Kin ∂Din. Axiom 2 (Boundary Generator Axiom) . C∗ -algebra A∂ and faithful state ω exist making above U(t) realize boundary dynamics on Hω : αt(A) = U(t)AU(t)−1, A ∈ A∂, and U(t)Ωω= Ωω . Call (A∂, ω, U(t)) **boundary as clock** data. 3.2 Gauge Fixing by PhaseSpectral-ShiftWS Trace Axiom 3 (Scale Identity Axiom) . Scattering system satises aforementioned Birman Krein and WignerSmith conditions. In energy window I , phase φ(ω) , relative state density ρrel(ω) , WignerSmith operator Q(ω) exist satisfying φ′(ω) π=ρrel(ω) = 1 2πTr Q(ω), ω ∈I. Dene time dierential as dt := 1 2πTr Q(ω)dω. Given reference point ω0, t0 , time scale determined by t(ω)−t0=Zω ω0 1 2πTr Q(˜ω)d˜ω. 5 Scale identity axiom transforms scattering spectral structure on frequency axis into boundary translation scale on time axis. 3.3 Modular Consistency and Unied Time Flow Axiom 4 (Modular Consistency Axiom) . For boundary dynamical system (A∂, ω, αt) , assume constant c > 0 exists such that for all A∈ A∂ αt(A) = σω ct(A), where σω t is TomitaTakesaki modular ow. Absorbing c into time unit, can losslessly rewrite as αt(A) = σω t(A). Thus boundary generator H∂ and modular generator Kω dier only by constant shift: H∂=Kω+λ1, λ ∈R. Denition 3.1 (Time Structure) . Call quadruple T= (A∂, ω, αt, S(ω)) time structure if satisfying: 1. ω is faithful normal state on A∂ ; 2. αt realized on GNS representation by U(t) = e−itH∂ , H∂ self-adjoint; 3. Scattering matrix family S(ω) and corresponding φ(ω), Q(ω), ρrel(ω) exist satisfying scale identity; 4. αt=σω t as automorphism groups consistent. Theorem 3.2 (Time Scale Existence) . Let T be time structure; assume in energy window I , ρrel(ω) is integrable continuous function nonzero on some interval. Then local bijection ω←→ t(ω) exists given by scale identity axiom such that: 1. For all A∈ A∂ , αt(ω)(A) = σω t(ω)(A)=∆it(ω)A∆−it(ω); 2. For scattering side, can view S(ω) as S(t) satisfying d dtφ(ω(t)) = π ρrel(ω(t)) = 1 2Tr Q(ω(t)), rewriting phase gradient, relative state density, WignerSmith trace as time derivatives. In other words, time parameter t simultaneously parametrizes modular ow and scattering time readouts, making latter observable scale of former. Theorem 3.3 (AdditiveProportional Uniqueness of Scale) . Under Theorem assumptions, further assume: 1. ρrel(ω) strictly positive or strictly negative in considered energy window; 2. Modular ow σω t non-trivial: no nonzero time t makes σω t identity. If another time parameter ˜ t and map ω7→ ˜ t(ω) exist such that: 1. α˜ t also realizes as modular ow: α˜ t=σω ˜ t ; 2. Scale identity holds under ˜ t in same energy window. Then constants a > 0 and b∈R exist making ˜ t=at +b. Time scale satisfying axiom system unique in ane transformation sense; time reversal (a < 0) excluded.  6 4 Proofs Provide proof structure of main theorems; concentrate technical operator scattering and modular theory details in appendices. 4.1 BirmanKrein Identity and PhaseSpectral-Shift Relation Under previous assumptions, spectral shift function ξ(λ) satises Krein trace formula. Taking smoothed approximation of f(λ) = χ(−∞,E](λ) yields ξ(E) = Tr(PH((−∞, E]) −PH0((−∞, E])), thus ξ′(λ) = −(ρ(λ)−ρ0(λ)) = −∆ρ(λ) holds in distributional sense. On other hand, BirmanKrein formula gives det S(λ) = exp(−2πi ξ(λ)) . Taking continuous branch and dierentiating with respect to λ : Φ′(λ) = −2πξ′(λ) = 2π∆ρ(λ), i.e., 1 2πΦ′(λ)=∆ρ(λ). With φ= Φ/2 obtain φ′(λ) π= ∆ρ(λ). 4.2 WignerSmith Trace and Relative Density of States WignerSmith delay operator dened as Q(λ) = −iS(λ)†∂λS(λ). In momentum or channel basis, Q(λ) is nite or countable-dimensional matrix satisfying Tr Q(λ) = −iTr(S(λ)†∂λS(λ)). On other hand, logarithmic derivative of det S(λ) satises ∂λlog det S(λ) = Tr(S(λ)−1∂λS(λ)) = Tr(S(λ)†∂λS(λ)), using S(λ) 's unitarity. Taking imaginary part yields ∂λΦ(λ) = Tr Q(λ), thus ∆ρ(λ) = 1 2πTr Q(λ). This directly veriable through spectral representation construction of H, H0 and S(λ) in rigorous scattering theory; widely used in multiphysics applications.  7 5 Model Applications Give several concrete models illustrating boundary as clock realization in dierent physical scenarios. 5.1 One-Dimensional Schrödinger Scattering Consider 1D Schrödinger operator H0=−d2 dx2, H =−d2 dx2+V(x) on H=L2(R) ; assume V∈L1(R,(1 + |x|)dx) real-valued. Scattering theory completely solvable; reection, transmission amplitudes r(k), t(k) exist; energy E=k2 . Select boundary Hilbert space as momentum space channels H∂≃L2(Rk)⊕L2(Rk); boundary algebra A∂ as closure of bounded multiplication operators and nite-rank perturbations; ω as equilibrium state (e.g., FermiDirac or Boltzmann weight). Under appropriate thermal equilibrium limit, modular ow of A∂ and ω can correspond to Schrödinger evolution, realizing consistency between modular time and scattering time scale in energy window. Time readout t given by t(k)−t0=Zk k0 1 2πTr Q(˜ k)dE d˜ kd˜ k=ZE E0 ∆ρ(˜ E)d˜ E, transforming energy axis into boundary time axis. 5.2 Local Algebras and Rindler Wedge In algebraic quantum eld theory, von Neumann algebra A(W) associated with Minkowski space wedge region W 's modular ow in vacuum state given by BisognanoWichmann theorem as Lorentz boost preserving wedge. This means: for Rindler observer, proper time ow proportional to modular time on A(W) ; ConnesRovelli thermal time hypothesis generalizes this as time = modular ow paradigm in generally covariant eld theories. In this background, can view wedge boundary (or more generally double cone boundary) as this paper's boundary algebra A∂ ; scattering matrix constructed from far-region eld incident/outgoing modes; WignerSmith delay matrix characterizes eld residence time near wedge. Through scale identity, can align geometrically dened proper time with scattering phase derivative, realizing boundary as clock in concrete quantum eld theory models.  6 Engineering Proposals Propose several experimental and engineering schemes for testing key equations and scale identity construction of boundary as clock on controllable platforms. 8 6.1 Microwave Network with Vector Network Analyzer In microwave engineering, complex networks (waveguides, resonant cavities, couplers) commonly described by multi-port scattering matrix S(ω) , directly measurable by vector network analyzer (VNA). Construct multi-port network approximating dissipationless in working frequency band satisfying scattering theory regularity requirements: 1. Measure S(ω) with VNA; numerically dierentiate to get ∂ωS(ω) ; 2. Construct WignerSmith matrix Q(ω) = −iS(ω)†∂ωS(ω) ; compute Tr Q(ω) ; 3. Through appropriate energyfrequency normalization, map ω axis to time axis t(ω)−t0=Zω ω0 1 2πTr Q(˜ω)d˜ω; 4. View network as concrete realization of boundary algebra: port modes span H∂ ; network interior bulkexterior domain dynamics project onto ports giving S(ω) ; 5. Under statistical steady state, construct empirical state ωexp for port excitation and output; approximately recover eective modular ow through energy ow conservation and equilibrium conditions; test consistency with time translation dened by t(ω) in correlation functions. If measured Tr Q(ω) frequency integral and network interior average residence time plus energy storage rate satisfy scale identity, viewable as engineering-level verication of boundary phase gradient scales time.  7 Discussion (Risks, Boundaries, Past Work) Discuss applicability domain, potential risks, relation to existing work of boundary as clock framework. 1. **Dependence on scattering regularity**: Scale identity depends on BirmanKrein formula and well-dened spectral shift function, requiring H−H0 at least trace-class perturbation; scattering matrix smooth in energy window. For strong coupling, many-body pure point spectrum dominant systems, framework requires modication or generalization. 2. **Dynamical interpretation of modular ow**: Thermal time hypothesis criticism points out modular ow may not always obtain natural dynamical interpretation, especially lacking geometric background or equilibrium state assumptions. This paper by requiring modular ow consistent with scattering time scale actually selects family of states and algebras with good dynamical meaning; however, this selection itself requires additional physical input and experimental calibration. 3. **Locality and causal structure**: This paper doesn't explicitly introduce spacetime causal structure, only working at boundary algebra and scattering channel level. To elevate boundary as clock to complete time geometry, requires further introducing local subalgebras, causal embedding, macroscopic geometry reconstruction procedure. Closely related to algebraic quantum eld theory research on reconstructing spacetime structure through local algebras. 4. **Time arrow and irreversibility**: Scale identity only characterizes time parameter scale and direction, not directly explaining time arrow origin. Binding time arrow 9