Time as Generalized Entropy Optimal Path:\\ Reconstruction of Time Arrow on Causally Consistent History Space

Time as Generalized Entropy Optimal Path: Reconstruction of Time Arrow on Causally Consistent History Space Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 20, 2025 Abstract This paper proposes a unied framework that reconstructs time as the optimal path of a generalized entropy functional. We no longer view time as a predetermined one-dimensional parameter, but dene the time arrow as an extendable path on the causally consistent history space that makes a certain class of generalized entropy functional take an extremum (under appropriate constraints, a minimum), along with its parametrization equivalence class. Specically, on a given causal structure and observable algebra, we construct the following three levels: (1) Structural Level : View world history as a curve family γ:I→ C on conguration space, where C is the state space satisfying eld equations and constraint conditions; introduce the causally consistent subspace Cons ⊂Paths(C) , composed of paths satisfying local causality, record extendability, and conservation laws. (2) Functional Level : On Cons , dene the generalized entropy functional Sgen[γ] = αSth[γ] + βSent[γ] + γDrel[γ] + λB[γ], where Sth is coarse-grained thermodynamic entropy, Sent is entanglement entropy or generalized entropy, Drel is relative entropy-type divergence, and B is a boundary term from boundary geometry or extrinsic curvature. The coecients α, β, γ, λ are determined by physical scenarios and scale choices. (3) Time Level : Dene the time arrow as the path family γ⋆ that makes Sgen satisfy the extremum principle on Cons with non-negative local entropy production rate, and dene the time scale equivalence class as all monotonic reparametrizations t7−→ f(t), f ∈Diff1 +(I), under orbits. Thus, time is no longer an external parameter, but the solution to a causal consistency + generalized entropy optimization problem. At the scattering and spectral theory end, we introduce the unied scale mother ruler κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), where S(ω) is the scattering matrix, Q(ω) = −iS(ω)†∂ωS(ω) is the WignerSmith delay operator, φ(ω) = 1 2arg det S(ω) is the total half-phase, and ρrel is the relative state density. We prove that in a well-posed scatteringgeometryinformation 1 setting, κ(ω) can be used to concretize the time cost of the generalized entropy functional as a spectral integral, thereby obtaining an observable time scale proxy. At the information and causal end, taking relative entropy monotonicity and QNEC/QFC-type inequalities as consistency constraints, we prove: if local ux and entropy ow satisfy a set of natural convexity and positivity conditions, then under given causal structure and boundary data, the causally consistent history that minimizes Sgen is unique under monotonic reparametrization, thereby reconstructing the time arrow as the causally extendable path with minimum generalized entropy cost. This framework provides a unied variational interpretation for thermodynamic second law, entanglement entropy growth, scattering group delay, and cosmological redshift. Keywords: Time Arrow; Generalized Entropy; Causal Structure; Relative Entropy; Scattering Phase; WignerSmith Time Delay; Scale Unication; Causally Consistent History  1 Introduction 1.1 Restating the Problem of Time In classical and quantum theory, time is traditionally viewed as a predetermined parameter: in general relativity it is the proper parameter of timelike curves or Killing/ADM time, in quantum theory it is the evolution parameter of the Schrödinger equation, and in statistical physics it is the time index of Markov processes. However, once we simultaneously consider the following three classes of facts: 1. Irreversibility and arrow of time in thermodynamics and information theory; 2. Generalized entropy conditions in general relativity (such as generalized entropy monotonicity, quantum null energy condition, quantum focusing condition); 3. Scale identity among phasedelaystate density in scattering theory and spectral theory, we nd: time is more like some selected structure rather than a background parameter written in the world equations from the beginning. The starting point of this paper is: given causal structure and observable algebras, can we characterize time as the solution to an optimization problem among all causally consistent history paths, select the extremal path of a certain class of generalized entropy functionals and its monotonic parametrization equivalence class as the essence of time arrow and time scale? 1.2 Core Idea of This Paper The core idea of this paper can be briey summarized as a variational principle: On a given causally consistent history space, the real world corresponds to the path that makes a certain class of generalized entropy functional Sgen take an extremum (under natural assumptions, a minimum); the so-called time 2 arrow is precisely the monotonic parametrization equivalence class on these extremal paths where the local entropy production rate is non-negative. Unlike the traditional time arrow = entropy increase narrative, here:  We do not presuppose entropy must inevitably increase with time, but dene time itself as the path parameter that extremizes the generalized entropy functional;  We are not limited to thermodynamic entropy, but introduce generalized entropy : including thermal entropy, entanglement entropy, relative entropy, and boundary geometric terms;  We require that paths not only satisfy dynamical equations, but also meet causal consistency , record extendability , and information monotonicity constraints. More specically, this paper will: 1. Construct the causally consistent history space and generalized entropy functional; 2. Prove that under natural convexity and bound constraints, the minimal history is unique under monotonic reparametrization; 3. Through the scale mother ruler in scattering theory κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), unify the abstract time scale with observable phase derivatives, group delays, and state densities; 4. Discuss the relationship among local observers, measurement records, and time perception. 1.3 Article Structure The structure of the full text is as follows: Section 2 gives the formalization of causal structure, history space, and generalized entropy functional, and proposes the axiomatic reconstruction of time. Section 3 presents the main theorem: under well-posed conditions, the generalized entropy functional's minimal causally consistent history is unique under monotonic reparametrization, thus dening the time arrow and time scale equivalence class. Section 4 introduces the scale mother ruler κ(ω) in the scattering and spectral theory context, embeds it into the generalized entropy framework, and gives the observable realization of time scale. Section 5 discusses the relationship among local observers, records, and subjective time. Section 6 provides simplied models to illustrate how the framework works. Appendices give strict proofs of main theorems and technical details.  3 2 Causally Consistent History Space and Generalized Entropy Functional 2.1 Spacetime, Causality, and Observable Algebra Let (M, g) be a Lorentzian manifold with globally hyperbolic structure, having the standard causal structure J±(·) . Let A be the observable algebra associated with M (e.g., a net of C∗ -algebras satisfying HaagKastler axioms, or restricted algebra on scattering channels), and ω a state on it. We consider the structural triple satisfying the following properties: Data = (M, g;A, ω;C), where C is the eective state space, composed of states (or state equivalence classes) satisfying eld equations and constraint conditions, which can be viewed as an appropriate completion of some conguration space or phase space. 2.2 World History as Path: History Space and Causal Consistency Denition 2.1 (History) . A continuous curve γ:I→ C, I ⊂R is an interval , is called a world history or history path . Denote all such curves as Paths(C) . Denition 2.2 (Causally Consistent History) . Given Data , a history γ∈Paths(C) is called causally consistent if the following structure exists: 1. For each point γ(t) , there exists a corresponding spatial slice or Cauchy section Σt⊂M such that t1< t2⇒Σt1⊂J−(Σt2) ; 2. The state evolution γ(t) induces restricted states ωt on A(Σt) satisfying local causality (e.g., satisfying Einstein causality, microlocality conditions); 3. There exists a family of record subalgebras Rt⊂ A(Σt) such that for any t1< t2 , the record distribution in Rt1 can be traceback-reconstructed from records in Rt2 (record extendability). Denote the set of all causally consistent histories as Cons(C)⊂Paths(C). The above denition abstracts the basic requirements of world history, causality, and recording: not only must evolution itself conform to causal structure, but it must also allow robust reconstruction of past records, which is crucial when dening the time arrow. 4 2.3 Generalized Entropy Functional and Time Cost We introduce a generalized entropy functional dened on the causally consistent history space. Denition 2.3 (Generalized Entropy Functional) . For γ∈Cons(C) , dene Sgen[γ] = αSth[γ] + βSent[γ] + γDrel[γ] + λB[γ], where: 1. Sth[γ] = RIσth(γ(t),˙γ(t)) dt is the integral of coarse-grained thermodynamic entropy density along the path; 2. Sent[γ] = RIσent(γ(t)) dt can be taken as the integral of entanglement entropy or generalized entropy density; 3. Drel[γ] = RId(ωt|ω(0) t) dt , where d is the relative entropy density and ω(0) t is a reference state; 4. B[γ] is a functional of boundary geometric behavior, typically written as B[γ] = Z∂M[γ] Lbdy(h, K) dΣ, where h is the induced metric and K is the extrinsic curvature. The coecients α, β, γ, λ are determined by physical scenarios and scale choices (e.g., unied time scale). Hypothesis 2.4 (Positivity and Convexity) . 1. σth and σent are convex and non-negative in velocity variables; 2. The relative entropy density d(·|·) is strictly convex in the rst variable and satises monotonicity; 3. The boundary functional B is lower semicontinuous on the allowed boundary variation space and has a good lower bound. Under these conditions, Sgen is a well-dened lower-bounded functional on Cons(C) . 2.4 Axiomatic Reconstruction of Time Arrow and Time Scale We now present this paper's axiomatic scheme for time. Axiom 2.1 (Causal Priority) . Physically allowed world histories must belong to the causally consistent history space Cons(C) . Axiom 2.2 (Generalized Entropy Optimization) . The real world corresponds to the history family that makes the generalized entropy functional Sgen :Cons(C)→R take an extremum under given boundary and initial state constraints, i.e., there exists γ⋆∈Cons(C),Sgen[γ⋆] = inf{Sgen[γ]|γ∈Admissible}, where Admissible ⊂Cons(C) is composed of initial states, constraint conditions, and energy/ux bounds. 5 Axiom 2.3 (Time Arrow Condition) . On the minimal history γ⋆ , there exists a monotonic parameter t such that the local entropy production rate ˙sloc(t) := d dt(αsth(t) + βsent(t) + γdt)≥0 almost everywhere , where sth, sent, dt are local densities along the history. This monotonic direction denes the time arrow. Denition 2.5 (Time Scale Equivalence Class) . Let γ⋆:I→ C be the minimal history. If f:I→I′ is a strictly monotonic dierentiable bijection, then ˜γ=γ⋆◦f−1 is another parametrization of the same history. Two parametrizations t and t′ are said to belong to the same time scale equivalence class if there exists f∈Diff1 + such that t′=f(t) . Denote this equivalence class as [t] . Thus, time is no longer a predetermined background axis, but a structure dened by the minimal history and its monotonic reparametrization equivalence class.  3 Existence of Minimal History and Geometric Uniqueness of Time Arrow The goal of this section is: under reasonable assumptions, prove that the minimal point of the generalized entropy functional Sgen on the causally consistent history space exists and is unique under monotonic reparametrization, thereby mathematically supporting the claim time = generalized entropy optimal path. 3.1 Variational Setting and Topological Structure Consider the function space Paths(C) = {γ:I→ C | γ absolutely continuous }, endowed with, e.g., a topology combining W1,1 or C0 with L1 . Assume: 1. C is a complete separable metric space; 2. Cons(C)⊂Paths(C) is closed in the above topology; 3. Admissible ⊂Cons(C) , given by boundary conditions and energy/ux constraints, is closed and has appropriate compactness (e.g., through ArzelàAscoli or Dunford Pettis type conditions). In this setting, the generalized entropy functional Sgen is lower semicontinuous and satises the following properties. Proposition 3.1 (Lower Bound and Compactness) . Under Hypothesis 2.3.2 and the above topological assumptions, there exists a constant C such that for all γ∈Admissible , Sgen[γ]≥ −C. Moreover, for any s∈R , the set {γ∈Admissible | Sgen[γ]≤s} is relatively compact in the chosen topology. 6 Proof outline : Using the lower bound and convexity of relative entropy and entropy density, provide energy-type estimates for velocity and state, then apply standard compactness theorems. 3.2 Existence of Minimal Point Theorem 3.2 (Existence of Generalized Entropy Minimal History) . Under the above assumptions, the generalized entropy functional Sgen attains a minimum on Admissible , i.e., there exists γ⋆∈Admissible such that Sgen[γ⋆] = inf{Sgen[γ]|γ∈Admissible}. Proof outline : Take a minimizing sequence (γn)⊂Admissible such that Sgen[γn]→ inf . By Proposition 3.1.1's compactness, there exists a subsequence (still denoted γn ) converging to γ⋆∈Admissible in the chosen topology. Using lower semicontinuity, Sgen[γ⋆]≤lim inf n→∞ Sgen[γn] = inf, thus γ⋆ is the minimal point. Complete rigorous proof in Appendix A. 3.3 Local EulerLagrange Equation and Entropy Production Rate On the minimal history γ⋆ , for local variation δγ (preserving initial/nal conditions and causal consistency), consider the rst-order variation δSgen[γ⋆;δγ] = 0. Formally, if writing the density as Lagrangian type L(γ, ˙γ) = ασth(γ, ˙γ) + βσent(γ) + γd(ω(γ)|ω(0)(γ)) + λ↕bdy(γ, ˙γ), the minimal path satises the EulerLagrange equation d dt∂L ∂˙γ−∂L ∂γ = 0, plus eective mechanical equations from causal constraints and record constraints. The entropy production rate can be written as ˙sloc(t) = α˙sth(t) + β˙sent(t) + γ˙ dt. In many physical scenarios (such as non-equilibrium thermodynamics consistent with local equilibrium, quantum channels consistent with complete positivity and conservation, and gravitational backgrounds satisfying QNEC/QFC conditions), it can be proven that ˙sloc(t)≥0 almost everywhere, thereby providing a variational interpretation for the time arrow. 7 3.4 Uniqueness and Time Scale Equivalence Class To extract the time arrow from the minimal path, we need to prove that the minimal path is unique under monotonic reparametrization. Hypothesis 3.3 (Strict Convexity and Topological Irreducibility) . 1. For almost every t , the generalized entropy density (γ, ˙γ)7→ ασth(γ, ˙γ) + βσent(γ) + γd(ω(γ)|ω(0)(γ)) is strictly convex in ˙γ ; 2. The causally consistent history space Cons(C) is topologically irreducible under given initial/nal constraints: any two feasible paths that induce almost everywhere identical record distributions at each time slice are equivalent under monotonic reparametrization. Under this assumption we have: Theorem 3.4 (Monotonic Reparametrization Uniqueness of Minimal Causally Consistent History) . Under the above assumptions, if γ1, γ2∈Admissible are both minimal points of Sgen , then there exists a strictly monotonic dierentiable bijection f such that γ2(t) = γ1(f−1(t)), i.e., the two minimal histories dier only by a monotonic reparametrization, thus belonging to the same time scale equivalence class [t] . Proof key points : Strict convexity ensures that if two dierent minimal paths exist, the interpolated path between them will lower the functional value, contradiction; topological irreducibility ensures the freedom of dierent parametrizations is exactly the monotonic reparametrization group Diff1 + . See Appendix A for details. This shows that the time scale equivalence class is a structure uniquely selected by generalized entropy optimization, providing a geometricvariational denition of time.  4 Scale Mother Ruler in Scattering Theory and Observable Realization of Time Cost To connect the above abstract time structure with observables, we now turn to scattering spectral theory and introduce the scale mother ruler κ(ω) . 4.1 Scattering Matrix, Group Delay, and Relative State Density Consider a class of static or steady-state scattering systems whose scattering matrix S(ω) is a unitary matrix on frequency ω satisfying appropriate dierentiability. Dene the WignerSmith delay operator Q(ω) = −iS(ω)†∂ωS(ω), whose trace τ(ω) := tr Q(ω) 8 gives the total group delay. On the other hand, dene the total scattering phase Φ(ω) = arg det S(ω), φ(ω) = 1 2Φ(ω), then the half-phase derivative φ′(ω) π can be connected to the relative state density ρrel(ω) and the group delay trace. By BirmanKren type formulas and Friedel type relations, φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), holds rigorously within the applicable range. Denition 4.1 (Scale Mother Ruler) . Dene the frequency scale mother ruler κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω). On one hand, this quantity can be measured by scattering phase derivatives; on the other hand, it can be measured by group delay trace or state density dierence, thus having clear scale meaning in experiments and theory. 4.2 Time Cost and Coupling with Scale Mother Ruler Suppose a class of physical processes can be described in frequency space by a measure µγ induced by history γ : for each ω , let µγ(dω) describe the weight and ux of that frequency mode in history γ . Then we can dene a class of spectral time cost functionals T[γ] = Zκ(ω)µγ(dω). In many scattering or open system scenarios, it can be proven that T[γ] is equivalent to or bounded-controlled by some terms in the generalized entropy functional Sgen[γ] , e.g.:  If Drel[γ] is the relative entropy between incident/outgoing states, its density can be expressed through relative state density ρrel(ω) , thus Drel[γ]≈Zf(κ(ω)) µγ(dω), holds for some convex function f ;  If the boundary term B[γ] is related to scattering cross-section or reection phase, it can also be written as an integral over eigenfrequencies. This means that, at the scatteringspectral end, part of the generalized entropy functional can be written as a spectral integral weighted by the scale mother ruler, so time cost has directly observable scale proxies. 9 C Standard Derivation Outline of Scattering Scale Mother Ruler This appendix provides a standard derivation outline of the scale mother ruler κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω). C.1 BirmanKren Formula and Relative Spectral Shift Let H0 be the free Hamiltonian and H=H0+V the scattering Hamiltonian, satisfying appropriate trace class conditions. Dene the relative spectral shift function ξ(λ) whose derivative ξ′(λ) gives the relative state density ρrel(λ) = ξ′(λ). The BirmanKren formula gives det S(λ) = e−2πiξ(λ), thus Φ(λ) = arg det S(λ) = −2πξ(λ)+2πk, k ∈Z. Taking continuous branch and dening half-phase φ(λ) = 1 2Φ(λ) , φ′(λ) π=−ξ′(λ) = ρrel(λ), obtaining the rst equality under appropriate sign and convention choices. C.2 WignerSmith Delay Operator and Trace Formula The WignerSmith delay operator is dened as Q(λ) = −iS(λ)†∂λS(λ), if S(λ) is suciently smooth in λ , then tr Q(λ) = −itr S(λ)†∂λS(λ)=−i∂λlog det S(λ) = −i∂λ(iΦ(λ)) = Φ′(λ). By φ=1 2Φ , φ′(λ) = 1 2Φ′(λ) = 1 2tr Q(λ), thus φ′(λ) π=1 2πtr Q(λ). Combining C.1 and C.2 yields the scale mother ruler identity κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), providing a solid spectralscattering foundation for embedding the unied time scale into the generalized entropy optimal path framework in this paper. 16