Quantum--Classical Bridge on Time Scale:\\ Phase, Time Delay, Redshift, and Gravity--Entropy Geometry

QuantumClassical Bridge on Time Scale: Phase, Time Delay, Redshift, and GravityEntropy Geometry Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 20, 2025 Abstract Under the unied time scale perspective, this paper systematically constructs a set of equivalence relations among quantum phase, proper time, scattering group delay, cosmological redshift, and boundary entropy evolution, organizing them into an axiomatizable quantumclassical bridge framework. On the geometric end, proper time dτ=p−gµνdxµdxν and generalized entropy Sgen on causal boundaries are taken as fundamental objects; on the quantum end, path integral phase ϕ=−S/ℏ , total phase Φ(ω) of scattering matrix S(ω) , and WignerSmith group delay operator Q(ω) = −iS(ω)†∂ωS(ω) are taken as fundamental objects. We propose and adopt the unied scale identity φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), where φ(ω) is normalized total phase, ρrel is relative state density. We prove: under semiclassical limit and appropriate regularity assumptions 1. When single-particle wave packet propagates along classical geodesic, its phase ϕ is equivalent to a linear function of proper time accumulated along that worldline ϕ=−mc2Rdτ/ℏ , thus proper time scale can be viewed as geometric parameter of phase; 2. In static or asymptotically at gravitational elds, gravitational time delay of photons or matter waves equals derivative of scattering phase with respect to frequency, i.e., equals trace of WignerSmith group delay, thus gravitational time dilation can be interpreted as curvature of phasefrequency geometry; 3. In FRW cosmological background, redshift 1 + z=a(t0)/a(te) can be equivalently written as ratio of phase frequency (dϕ/dt) on same photon worldline at emission and detection events, thus cosmological redshift becomes phase expression of cosmic time scale shear; 4. In local causal diamonds, taking extremality and monotonicity of generalized entropy Sgen =A/(4Gℏ) + Sout (along null generator parameter λ ) as axioms, Einstein equations with cosmological constant can be derived in semiclassical holographic window, viewing gravitational geometry as eective equations for how entropy organizes along time scale on causal boundaries. 1 This yields a unied timephaseentropygeometry correspondence diagram. Appendices provide: derivation of scale identity between WignerSmith group delay and spectral shiftstate density; semiclassical proof of phaseproper time equivalence under worldline path integral; renement of redshiftphase expression in FRW cosmology; and detailed proof outline of deriving Einstein equations from generalized entropy extremality and quantum energy conditions on local causal diamonds. Keywords: Time Scale; WignerSmith Group Delay; Proper Time; Cosmological Redshift; Generalized Entropy; Entropic Interpretation of Einstein Equations MSC (2020): 83C45, 81T20, 81U40, 83C57  1 Introduction Time plays dierent roles in classical physics and quantum theory: in general relativity, time and space together form four-dimensional spacetime manifold, whose proper time scale dτ is determined by metric gµν ; in quantum mechanics and quantum eld theory, time is more manifested through phase factor exp(−iEt/ℏ) and unitary evolution operator U(t) . WignerSmith group delay operator Q(ω) introduced in scattering theory provides an operational scale of time as phase derivative with respect to frequency; cosmological redshift reects macroscopic shear of time rhythm through changes in frequency or wavelength due to cosmic scale factor evolution. On the other hand, black hole thermodynamics, Jacobson-type entropic interpretation, and holographicentanglement geometry research show: gravitational geometry can be viewed as macroscopic equations of certain entropyinformation organization, especially on causal boundaries and local horizons, extremality and monotonicity of generalized entropy Sgen impose mandatory constraints on spacetime curvature. Behind these seemingly disparate phenomena is a common structure: they all use time scale as bridge, connecting quantum phase, classical clocks, scattering delay, redshift, and entropy ow. The goal of this paper is to systematically, clearly, and axiomatically characterize this structure, providing rigorous equivalence or correspondence relations. This paper develops around the following main questions: 1. How to view quantum phase, group delay, proper time, and cosmological redshift as dierent cross-sections of the same time geometry under unied scale? 2. Under what assumptions can we equate how generalized entropy organizes along time on causal boundaries with macroscopic gravitational equations (Einstein equations)? 3. How do these equivalence relations naturally connect quantum with classical, microscopic with macroscopic in semiclassical limit? To this end, Section 2 gives notations and axiomatic scale identity; Sections 35 successively construct precise correspondences of phaseproper time, time delaygravitational time dilation, redshifttime scale shear; Section 6 gives entropic geometry theorem of generalized entropy evolutiongravitational equations in causal diamond framework. Section 7 summarizes these bridges as a geometric picture. Appendices provide detailed proofs of main technical results.  2 2 Notations and Basic Structures 2.1 Geometric End: Spacetime, Proper Time, and Causal Diamonds Let (M, gµν) be geodesically complete Lorentzian spacetime manifold, metric signature (−+ ++) . Proper time of timelike curve γ(λ) is dτ=r−gµν dxµ dλ dxν dλdλ. In local discussions, choose point p∈M and small parameter r≪Lcurv , dene small causal diamond at that point Dp,r := J+(p−)∩J−(p+), where p± are points oset by proper time ±r along some chosen timelike direction. Boundary of Dp,r is generated by two families of null geodesics, forming local causal boundary. 2.2 Quantum and Scattering End: S-Matrix and WignerSmith Group Delay Let H0 be free Hamiltonian, H be Hamiltonian with interaction term. Under standard scattering assumptions, wave operators Ω±= slim t→±∞ eiHte−iH0t exist and are complete, scattering operator dened as S:= (Ω+)†Ω−. In energy or frequency representation, absolutely continuous spectrum ω∈I⊂R gives for each ω nite-dimensional channel space Hω≃CN(ω) , with unitary matrix S(ω) . Dene total scattering phase Φ(ω) := arg det S(ω). Denition 2.1 (WignerSmith Group Delay Operator (Denition 2.1)) . Under frequency dierentiability assumption, dene Q(ω) := −i S(ω)†∂ωS(ω), called WignerSmith group delay operator. Q(ω) is self-adjoint matrix, whose eigenvalues denoted τj(ω) can be interpreted as group delays of respective channels. Trace gives total group delay Tr Q(ω) = ∂ωΦ(ω). 3 2.3 Spectral Shift Function and Relative State Density Under suitable traceable perturbation assumption, let ξ(ω) be Kren spectral shift function, ρrel(ω) be relative state density. Classical result gives Φ(ω) = −2π ξ(ω), ρrel(ω) = −∂ωξ(ω). Thus ∂ωΦ(ω) = 2π ρrel(ω). 2.4 Unied Scale Identity Combining Sections 2.22.3, we obtain the following scale identity. Axiom 2.2 (Time Scale Unication (Axiom 2.2)) . For all considered scattering congurations and energy windows, there exist well-dened generalized phase φ(ω) and relative state density ρrel(ω) such that φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), where tr is trace on channel space, Q(ω) := Q(ω) . This identity unies three types of objects on same time scale: 1. Phase derivative φ′(ω) : curvature of phasefrequency geometry; 2. Relative state density ρrel(ω) : correction to spectralstate density; 3. Group delay trace tr Q(ω) : arrival time oset relative to free propagation. All time scale-related quantities in what follows will be written back to this identity as much as possible. 2.5 Cosmological Time and Redshift Consider spatially homogeneous isotropic FRW metric (taking k= 0 temporarily) ds2=−dt2+a(t)2dx2. For photon propagating along comoving coordinate x(t) , null geodesic condition ds2= 0 gives dx dt=±1 a(t)ˆ n. Cosmological redshift dened as 1 + z:= λ0 λe =νe ν0 =a(t0) a(te), where subscripts e, 0 denote emission and observation events respectively. 4 2.6 Boundary Generalized Entropy and Time Parameter In quantum eld theory with gravity, consider boundary Σ of some causal region and its external (or internal) quantum eld degrees of freedom, dene generalized entropy Sgen(Σ) := Area(Σ) 4Gℏ+Sout(Σ), where Sout is von Neumann entropy relative to Σ . When deforming Σ along some null generator family γ(λ) , taking ane parameter λ as boundary time, study extremality and monotonicity properties of Sgen(λ) , giving how entropy organizes with time. Axiom 2.3 (Boundary Entropy Time Evolution Axiom (Axiom 2.3)) . Under appropriate energy conditions and semiclassical assumptions, for small causal diamond Dp,r at any point p , the boundary admits a family of local cuts {Σλ} such that 1. Under appropriate constraints (xed local energy or eective volume), Sgen(λ) takes extremum at λ= 0 ; 2. For extrapolation along any null direction, Sgen(λ) satises appropriate monotonicity or convexity conditions (such as QNEC/QFC type inequalities) under physical evolution. We will use this axiom and scale identity as basis to give entropic geometric interpretation of gravitational equations.  3 Equivalence of Phase and Proper Time This section discusses semiclassical propagation of single particle or narrow wave packet in curved spacetime, explaining that essence of quantum phase is proper time accumulated along worldline, thereby connecting quantum and classical on time scale. 3.1 Worldline Action and Path Integral For point particle of mass m , classical worldline action can be taken as S[γ] = −mc2Zγ dτ=−mc2Zr−gµν dxµ dλ dxν dλdλ. Quantum amplitude in worldline path integral framework is formally written as A(xf, xi)≃Zγ:xi→xf Dγexpi ℏS[γ]. In semiclassical limit ℏ→0 , main contribution comes from stationary phase trajectories, i.e., worldlines γcl satisfying geodesic equation. 5 3.2 PhaseProper Time Theorem Theorem 3.1 (PhaseProper Time Equivalence (Theorem 3.1)) . Let narrow wave packet propagate in curved spacetime, whose center trajectory γcl is timelike geodesic for particle of mass m . Then in semiclassical approximation, phase evolution of wave packet center is ϕ=−1 ℏS[γcl] = mc2 ℏZγcl dτ, thus instantaneous phase frequency satises dϕ dτ=mc2 ℏ. Proof. Construct narrow wave packet initial state in some local Fermi normal coordinate system, concentrating it in phase space at classical initial condition (xi, pi) . Path integral can be expanded using stationary phase approximation in semiclassical limit. Let γcl be unique geodesic satisfying given boundary conditions, then 1. For each path γ , amplitude phase is ϕ[γ] = S[γ]/ℏ ; 2. In limit ℏ→0 , trajectory with maximum weight is that with δS = 0 , i.e., geodesic γcl ; 3. On that trajectory S[γcl] = −mc2Rdτ . Therefore, total phase of dominant state is ϕ=−1 ℏS[γcl] = mc2 ℏZdτ. Taking derivative with respect to proper time gives dϕ dτ=mc2 ℏ, completing the proof. □ Corollary 3.2 (Quantum Time Scale (Corollary 3.2)) . For particle of mass m , its phase rotation frequency on proper time scale is constant mc2/ℏ . Therefore, proper time dτ is equivalent to quantum phase dierence dϕ , diering only by constant factor: dϕ=mc2 ℏdτ. This shows: on geometric end, proper time is intrinsic scale along worldline; on quantum end, phase is angular coordinate under that scale. Linear equivalence between them provides unied background for subsequent time delaygroup delay, redshift phase rhythm.  6 4 Scattering Time Delay and Gravitational Time Dilation This section examines propagation of light or matter waves in static or asymptotically at gravitational elds, explaining that gravitational time delay equals derivative of scattering phase with respect to frequency, thus can be scaled by WignerSmith group delay. 4.1 Static Metric and Refractive Index Perspective Consider static metric ds2=−V(x)c2dt2+gij(x) dxidxj, where V(x)>0 . Introduce time refractive index nt(x) := V(x)−1/2= (−gtt(x))−1/2. For wave eld of xed frequency ω , eikonal equation in geometrical optics limit can be written as gµν∂µϕ ∂νϕ= 0, whose solution ϕ gives wavefront phase. If taking ϕ=−ωt+S(x) , spatial part satises optical Fermat principle similar to refractive index nt(x) . 4.2 Time Delay and Phase Derivative Let there be two paths: one in gravitational eld γg , one in at background γ0 , corresponding to propagation times Tg(ω) and T0(ω) respectively. Dene time delay ∆T(ω) := Tg(ω)−T0(ω). On the other hand, gravitational eld correction to total phase ∆Φ(ω) is action dierence along classical path divided by ℏ . For xed-frequency wave, action dierence mainly comes from time refractive index correction, can be written as ∆Φ(ω)≃ −ω∆T(ω), where terms weakly dependent on frequency are ignored. Thus ∆T(ω) = −∂ω∆Φ(ω) In description with ω as spectral parameter, this is precisely denition of group delay. Combining scale identity ∂ωΦ(ω) = Tr Q(ω), we obtain: 7 Theorem 4.1 (Gravitational Time DelayGroup Delay Equivalence (Theorem 4.1)) . Under appropriate geometrical optics and semiclassical assumptions, for xed-frequency wave propagating in static gravitational eld background, macroscopically observable gravitational time delay ∆T(ω) is equivalent to derivative of total scattering phase Φ(ω) with respect to frequency, i.e., equivalent to trace of WignerSmith group delay operator Q(ω) : ∆T(ω) = ∂ωΦ(ω) = Tr Q(ω). Proof outline. In eikonal approximation, phase function ϕ satises HamiltonJacobi equation, phase accumulation on path proportional to classical action. For xed-frequency state, time direction action contribution is −ωT . Comparing congurations with/without gravitational eld, action dierence is −ω∆T , thus total phase dierence satises ∆Φ = −ω∆T . Dierentiating with respect to frequency gives ∆T=−∂ω∆Φ . Meanwhile, from scattering theory ∂ωΦ = Tr Q . Identifying both gives this equivalence relation. □ This theorem shows: macroscopic time dilation or time delay usually understood as caused by spacetime curvature, is completely equivalent to scattering group delay in frequency domain. Therefore, through scale identity, gravitational time eects can be described by unied time scale object ρrel(ω) .  5 Cosmological Redshift as Time Scale Shear This section considers light propagation and redshift in FRW universe, explaining that redshift can be viewed as shear of time scale on same photon worldline, expressible as ratio of phase frequency at dierent events. 5.1 FRW Geometry and Redshift Formula As in Section 2.5, for at FRW universe, metric is ds2=−dt2+a(t)2dx2. For photon propagating along comoving coordinate, using conformal time η dened by dη= dt/a(t) , metric written as ds2=a(η)2(−dη2+ dx2). Null geodesic satises dx dη=±ˆ n, i.e., photon propagates in conformal timespace as if in at Minkowski space. Consider measurement of frequency ν : for observer at rest in comoving coordinates, four-velocity is uµ= (1,0,0,0) , photon four-momentum kµ satises ν∝ −kµuµ . Obtain ν∝1 a(t), thus redshift 1 + z=νe ν0 =a(t0) a(te). 8 5.2 Geometric Interpretation of Phase Rhythm Let photon phase function be ϕ(x) . Along null geodesic γ with parameter λ have dϕ dλ=kµ dxµ dλ, where kµ is four-wavevector. For some observer, proper time is proper time τ . Frequency dened as ν:= 1 2π dϕ dτ. For comoving observer, τ=t . Thus ν(t) = 1 2π dϕ dt. From previous result, ν∝1/a(t) , therefore can write dϕ dtt=te dϕ dtt=t0 =νe ν0 = 1 + z. This gives: Proposition 5.1 (RedshiftPhase Rhythm Equivalence (Proposition 5.1)) . In FRW universe, ratio of phase time derivatives at emission event e and observation event 0 on same photon worldline equals cosmological redshift: 1 + z=νe ν0 = dϕ dte dϕ dt0 . Since ϕ also satises HamiltonJacobi equation of geometrical optics, it can also be understood as angular coordinate of certain cosmic time scale. Redshift is therefore ratio measurement of this scale at two epochs, i.e., macroscopic manifestation of time scale shear. 5.3 Relation to Unied Scale Identity If viewing cosmological propagation as certain eective scattering process, can formally introduce equivalent scattering matrix S(ω) , whose total phase Φ(ω) can be given by eikonal integral along conformal time path. Then redshift can be viewed as dierence in phase gradient of Φ(ω) at dierent cosmic time cross-sections. Through scale identity φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), can characterize redshift's eect on state density and group delay in frequency space, thus bringing cosmological observations into same time scale framework.  9 D Derivation Outline of Generalized Entropy Extremality and Einstein Equations This appendix gives detailed derivation framework of Theorem 6.2, focusing on showing how generalized entropy organization along boundary time constrains geometry. D.1 Small Causal Diamond and Null Generator Parameter In neighborhood of point p , choose timelike vector eld ξµ , construct small causal diamond Dp,r . Its boundary can be generated by two families of null geodesics, corresponding to future and past directions respectively. Choose one family of future null generators, parametrized by ane parameter λ , such that λ= 0 corresponds to cross-section passing through p . For each λ , dene transverse cut Σλ , whose area is A(λ) , generalized entropy is Sgen(λ) = A(λ) 4Gℏ+Sout(λ). D.2 First Variation: Extremality Condition First variation of generalized entropy is dSgen dλ=1 4Gℏ dA dλ+dSout dλ. Area variation satises dA dλ=ZΣλ θdA, where θ is expansion. Taking λ= 0 , Axiom 6.1 requires under appropriate constraint dSgen dλλ=0 = 0. On other hand, linear response of Sout to small deformation can be related to expectation value of modular Hamiltonian, i.e., dSout dλ= 2πZΣλ λ⟨Tkk⟩dA+· · · , taking appropriate limit near λ= 0 yields term proportional to ⟨Tkk⟩ . This step relies on local rst law and relative entropy linear response. Combining, rst-order extremality condition gives preliminary form of proportionality relation between Rkk and ⟨Tkk⟩ . D.3 Second Variation and Raychaudhuri Equation Consider second variation d2Sgen dλ2=1 4Gℏ d2A dλ2+d2Sout dλ2. Second variation of area uses Raychaudhuri equation: 16 dθ dλ=−1 2θ2−σµνσµν −Rµνkµkν. At λ= 0 can choose initial condition such that θ= 0 , shear contribution absorbed by higher-order terms, obtaining dθ dλλ=0 ≈ −Rkk, thus d2A dλ2λ=0 =ZΣ0 dθ dλλ=0 dA≈ − ZΣ0 Rkk dA. On other hand, d2Sout/dλ2 related to energy ow uctuations and quantum energy conditions (such as QNEC), latter gives d2Sout dλ2≤2πZΣ0 ⟨Tkk⟩dA, or equality under saturation condition. D.4 From Scalar Relation to Tensor Equation Substituting above expressions into generalized entropy second variation formula, combined with monotonicity or convexity requirement, obtain −1 4GℏZΣ0 Rkk dA+ 2πZΣ0 ⟨Tkk⟩dA≥0, equality under saturation or extremality condition. Since cut and direction kµ can be arbitrarily chosen locally, above formula holds for all null directions and small cuts, meaning at point p Rµνkµkν= 8πG ⟨Tµν⟩kµkν holds for all null vectors kµ . Thus tensor Eµν := Rµν −8πG ⟨Tµν⟩ satises Eµνkµkν= 0 for all null vectors, yielding Eµν = Λgµν , where Λ is some constant. Using Bianchi identity ∇µ(Rµν −1 2Rgµν) = 0 and energymomentum conservation ∇µTµν = 0 , can identify Λ as cosmological constant, obtaining Rµν −1 2Rgµν + Λgµν = 8πG ⟨Tµν⟩. Proof complete.  This paper, under unied time scale perspective, organizes quantum phase, proper time, scattering group delay, cosmological redshift, and generalized entropy evolution into self-consistent geometricentropicspectral framework, on this basis giving entropic geometric interpretation of macroscopic gravitational equations. 17