Unified Framework of Time--Geometry--Interaction:\\ From Scattering Phase Scale to Geometrization of Gravity and Gauge Forces

Unied Framework of TimeGeometryInteraction: From Scattering Phase Scale to Geometrization of Gravity and Gauge Forces Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 20, 2025 Abstract We construct a unied framework with time scale equivalence class as the core object, rewriting gravity, gauge interactions, and macroscopic classical forces as projections of the same geometric structure at dierent hierarchical levels. First, in the rigorous context of scattering theory, based on BirmanKren spectral shift function and WignerSmith time delay, we give the scale identity of phase derivative, relative state density, and group delay trace: for a class of scattering systems satisfying trace-class perturbation conditions, we have φ′(ω)/π =ρrel(ω) = (2π)−1tr Q(ω) , where Q(ω) = −iS(ω)†∂ωS(ω) is the WignerSmith matrix, ρrel is the derivative of Kren spectral shift density. This identity unies phasedelayrelative spectral density as an observable time scale. Second, introducing a total bundle with spacetime as base manifold and internal charge space and observer resolution hierarchy as bers, we prove that under conditions satisfying local Lorentz and internal gauge symmetry, gravity and YangMills gauge elds can be viewed as dierent components of the same total connection, whose curvatures respectively give spacetime curvature and internal eld strength; the time scale is related to phase integration along worldlines and parallel transport of the connection, thus deriving the unied proposition that no fundamental forces, only time curvature under dierent geometric projections. Third, at the algebraic quantum eld theory level, using modular ow given by TomitaTakesaki modular theory as internal time, combined with quantum null energy condition (QNEC) and generalized entropy monotonicity, we characterize the arrow of time as a partial order structure of relative entropy and modular energy monotonicity. Finally, in the semiclassical limit, we give a theoremform statement of force = projected curvature of time geometry for expectation value evolution of macroscopic particles, explaining that classical Newtonian mechanics, Lorentz force, and even eective entropic force can all be viewed as eective descriptions of this unied time geometry under dierent coarse-graining and internal state constraints. At the end, we provide several veriable engineering proposals, including WignerSmith time delay measurements in microwave networks and mesoscopic conductors, atomic clock gravitational redshiftphase scale reconciliation, and cross-scale scale consistency tests based on FRW cosmological redshift and strong gravitational lensing. 1 Keywords: Time Scale Equivalence Class; Scattering Phase; WignerSmith Time Delay; BirmanKren Spectral Shift; Geometrization of Gravity; YangMills Gauge Field; Modular Flow; Quantum Null Energy Condition; Emergence of Macroscopic Force  1 Introduction and Historical Context Classical physics takes force as a fundamental concept: Newton's second law F=ma views force as the cause of changing a particle's velocity. Field-theoretic formulation of electromagnetism in some sense weakens the fundamental status of force: Lorentz force q(E+v×B) can be obtained from contraction of electromagnetic eld tensor Fµν and four-velocity uν , the eld itself governed by Maxwell equations. General relativity further upgrades gravity from force to spacetime geometry: free-fall worldlines satisfy geodesic equations, so-called gravitational acceleration can be viewed as inertial force caused by choosing non-free-fall frames. In mid-twentieth century, gauge theory unied electromagnetic, weak, and strong interactions as connections and curvatures on principal bundles: gauge potential Aµ is connection on ber bundle, eld strength Fµν is curvature of this connection, corresponding to EulerLagrange equations of YangMills equations. Force experienced by charges in gauge elds can be understood as trajectory deection when doing parallel transport in total space with gauge connection. Geometrization of gauge theory has shown: interactions can be unied as dierent types of geometric structures. On the other hand, quantum scattering theory reveals profound connections among phase, state density, and time delay. BirmanKren formula establishes relationship between scattering matrix determinant and spectral shift function, Friedel sum rule connects phase and state density, Smith and Wigner's introduced time delay and Wigner Smith matrix Q(ω) = −iS(ω)†∂ωS(ω) characterize time structure of scattering process. Subsequent work shows that under suitable trace-class perturbation conditions, phase derivative, relative state density, and trace of WignerSmith matrix unify through rigorous trace formulas. In algebraic quantum eld theory, TomitaTakesaki modular theory endows any von Neumann algebra with a cyclicseparating vector with an intrinsic modular ow σφ t . Modular ow is generated by one-parameter unitary group ∆it of modular operator ∆ , and can be viewed as internal time naturally induced by statealgebra pair. This structure plays an important role in thermal time hypothesis and time reconstruction in gravityholographic backgrounds. Recent quantum null energy condition (QNEC) work reveals arrow of time structure between energyentropy: in quite general quantum eld theories, stress-energy tensor along null directions satises an inequality with second-order variation of von Neumann entropy as lower bound, thus binding geometric changes of entropy with local energy constraints. These developments jointly point to a deeper unied picture: Time is not just an external parameter, but a scale jointly dened by multiple structures including scattering phase, spectral shift, modular ow, and entropy curvature; Gravity and gauge interactions are both connection curvatures corresponding to dierent ber directions; Macroscopic forces are projections of this timegeometry at coarse-graining and eective degrees of freedom levels. 2 The goal of this paper is to construct an explicit unied framework based on the above mature results: 1. Using phasespectral shifttime delay identity in scattering theory, give an observable denition of time scale; 2. Unify gravity and gauge interactions as dierent components of time connection, deriving macroscopic force as projection of time geometry curvature; 3. Use modular ow and QNEC to characterize arrow of time and causal partial order; 4. Provide veriable experimentalengineering proposals, making this unied framework falsiable.  2 Model and Assumptions 2.1 Spacetime and Quantum Field Theory Background Let (M, g) be a four-dimensional, time-oriented globally hyperbolic Lorentzian manifold satisfying ordinary causality and energy conditions. Physical system is given by quantum eld theory dened on (M, g) , with Hilbert space H , local observable algebra A(O)⊂ B(H) , obeying HaagKastler axioms. Choose reference state Ω∈ H (can be vacuum or thermal equilibrium), assuming it is cyclic and separating for appropriate subalgebras. In cases with suciently asymptotically at regions, assume existence of global time translation symmetry t7→ U(t) = e−iHt , corresponding generator H self-adjoint, allowing construction of scattering states and scattering matrix. 2.2 Observer, Resolution, and Coarse-Graining Observer is described by a timelike worldline γ:τ7→ xµ(τ) and a set of detector degrees of freedom. Introduce a resolution scale parameter Λ , characterizing the energytime range and spacemomentum window this observer can resolve. Formally, coarse-graining operators can be viewed as families of completely positive, trace-preserving maps {ΦΛ} on algebra A(O) ; as Λ decreases, retained degrees of freedom become coarser. Relationship among resolutiontimeredshift is characterized in the unied framework through equivalence class: dierent (γ, Λ) combinations can dene the same time scale (in the sense below). 2.3 Scattering System and Scale Identity Consider a pair of self-adjoint operators H, H0 on H , satisfying V:= H−H0 is trace-class perturbation and satisfying standard wave operator existence and completeness conditions. Let S(ω) be xed-energy scattering matrix, whose determinant satises Birman Kren formula with spectral shift function ξ(ω;H, H0) : det S(ω) = e−2πiξ(ω) almost everywhere . 3 Dene total scattering phase Φ(ω) = arg det S(ω) , and set φ(ω) = 1 2Φ(ω) = −πξ(ω). Introduce WignerSmith matrix Q(ω) = −iS(ω)†∂ωS(ω), whose trace in many models can be interpreted as total group delay. Relative state density is dened as ρrel(ω) = ρ(ω;H)−ρ(ω;H0), where ρ is density of states. Kren trace formula gives tr(f(H)−f(H0)) = ZR f′(λ)ξ(λ) dλ, appropriately choosing f can deduce ρrel(ω) = ξ′(ω) almost everywhere. Under good integrability conditions, WignerSmith matrix trace satises tr Q(ω)=2∂ωφ(ω). Combining yields the scale identity φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω) almost everywhere . This denition unies additional state density, total group delay, and scattering phase gradient as a single function κ(ω) , which can be viewed as time scale density for given scattering system and reference Hamiltonian. 2.4 Geometrization: Total Bundle and Connection Let M be spacetime manifold, introduce total bundle π:B → M, whose ber is the product of internal charge space Fint and resolution space Fres . For each point x∈M , ber Bx represents internal degrees of freedom of quantum state at that point and measurement window available to observer. Introduce principal bundle structure on B , whose structure group is Gtot =SO(1,3)↑×GYM ×Gres, respectively corresponding to local Lorentz group, internal YangMills gauge group, and resolution scaling group. Total connection is written as Ω=ωLC ⊕AYM ⊕Γres, where ωLC is LeviCivita spin connection, AYM is standard YangMills gauge eld, Γres describes resolution ow in RG sense (e.g., renormalization group or coarse-graining connection). Total curvature R= dΩ+Ω∧Ω naturally decomposes into spacetime curvature R , YangMills eld strength F , and resolution ow curvature Rres . 4 2.5 Modular Time and Entropy Arrow For each local observable algebra A(O) and state φ (giving vector state from Ω ), Tomita Takesaki theory gives modular operator ∆φ and modular automorphism group σφ t(A) = ∆it φA∆−it φ. Modular ow parameter t can be interpreted as internal time of this observeralgebra pair; in holographic and thermal time hypothesis contexts, it is viewed as a candidate for geometric time. QNEC states that for appropriate classes of quantum eld theories and states, ⟨Tkk(x)⟩ψ≥1 2πS′′ vN(x), where Tkk is stress-energy component along null vector kµ , S′′ vN is second-order variation of von Neumann entropy along that direction. This inequality and more general generalized entropy monotonicity provide quantitative basis for dening arrow of time and causal partial order from quantum information perspective.  3 Main Results (Theorems and Alignments) This section gives core denitions and theorems of the unied framework, and indicates alignment relationships among them. 3.1 Denition of Time Scale Equivalence Class Denition 3.1 (Operational Time Scale (Denition 3.1)) . Given a scattering system (H, H0) and its scattering matrix S(ω) , dene scale density κ(ω) = φ′(ω)/π, where φ(ω) is determined by aforementioned BirmanKren relation. For energy window I⊂R , dene eective time scale τI(E) = ZE E0 κ(ω)1I(ω) dω, which experimentally corresponds to integral of additional group delay obtained from frequencyphase measurements. Denition 3.2 (Time Scale Equivalence Class (Denition 3.2)) . Given two sets of operational time parameters t and t′ , if there exist strictly monotonic C1 function f:R→R and global positive constant c > 0 , such that for all scattering experiments realizable in common domain, we have t′=c f(t) and all observable phase dierences and group delays follow consistent ordering with respect to t, t′ , then t and t′ are said to belong to the same time scale equivalence class, written [t]=[t′] . 5 Proposition 3.3 (Proposition 3.3) . Under conditions where scale identity holds, for any scattering system satisfying trace-class perturbation conditions and given reference (H0) , time scales dened by dierent probe families (frequency windows, incident channel choices) converge to the same equivalence class [τ] in the sense of equivalence relation, if and only if energy dependence of WignerSmith total time delay is controlled by a unied geometric structure (such as same eective potential and boundary condition class). This proposition ensures that under suitable universal probe families, time scale has probe independence, thus can be used to dene macroscopic time. 3.2 PhaseSpectral ShiftTime Delay Scale Identity Theorem 3.4 (Scale Identity (Theorem 3.4)) . Let H, H0 be self-adjoint operators, H−H0 be trace-class perturbation, scattering matrix S(ω) exists and is unitary. Let spectral shift function ξ(ω) be dened by Kren trace formula, then almost everywhere tr[f(H)−f(H0)] = ZR f′(λ)ξ(λ) dλ for all appropriate f . Dene φ(ω) = −πξ(ω) , Q(ω) = −iS(ω)†∂ωS(ω) , then almost everywhere φ′(ω) π=ξ′(ω) = ρrel(ω) = 1 2πtr Q(ω). This theorem combines BirmanKren formula, Friedel sum rule, and WignerSmith denition, giving unication of phase, spectral shift, and time delay. Corollary 3.5 (Spectral Denition of Time Scale (Corollary 3.5)) . Scale density κ(ω) can be equivalently dened as relative state density, or as normalization of WignerSmith matrix trace, thus having complete spectralscattering expression. 3.3 SpacetimeInternal Space Unied Geometry and Geometrization of Force On total bundle B , curvature R of total connection Ω can be decomposed as R=R⊕F⊕ Rres. Parallel transport along a matter particle worldline γ is controlled by total covariant derivative Dτ=d dτ+Ω(˙γ). Evolution of intrinsic degrees of freedom (such as spin, color charge) and resolution variables is determined by F and Rres . Theorem 3.6 (No Fundamental Force Proposition (Theorem 3.6)) . In semiclassical limit, for particle with mass m and internal charge q , expectation value of center-of-mass trajectory xµ(τ) satises mD2xµ Dτ2=qFµν dxν dτ+fµ res 6 where D is LeviCivita covariant derivative, Fµν is projection of YangMills eld strength in corresponding representation, fµ res is eective entropic force term caused by Rres and stateentropy changes. In other words, gravity, Lorentz force, and entropydriven force in classical sense are all projections of total connection curvature on dierent behavior spaces, without need to separately introduce primitive concept of fundamental force. Proof based on semiclassical propagation of wave packet and path integral representation, details in Appendix C. 3.4 Time Scale and Reconstruction of Gravitational Geometry Theorem 3.7 (From Scale Identity to Gravitational Redshift (Theorem 3.7)) . Consider spacetime region with static Killing vector ∂t , metric can be written in standard static form g=−N2(x)dt2+hij(x)dxidxj. Assume scattering processes with energy localized in I exist in this region, whose WignerSmith total group delay tr Q(ω) can be measured by distant observer. If scale density κ(ω) within I is approximately related to position only through rescaling by N(x) , i.e., κ(ω;x) = N−1(x)κ∞(ω) then time scale dened by distant observer and proper time scale dened by local free fall belong to same equivalence class, their ratio giving gravitational redshift factor N(x) . Proof relies on frequency conservation in static space and relation of local energy ωloc =N−1(x)ω , see Appendix B. Corollary 3.8 (Corollary 3.8) . Under above setting, gravitational potential can be viewed as rescaling pattern of unied time scale between dierent spatial points; gravitational time dilation and Shapiro delay are two readout methods of same time geometry under dierent operational denitions. 3.5 Gauge Interaction as Conditioned Time Scale Proposition 3.9 (Charge-Dependent Additional Group Delay (Proposition 3.9)) . In scattering systems with U(1) or more general YangMills gauge eld Aµ , phase derivative dierence of scattering matrices Sρ(ω) corresponding to dierent internal charge representations ρ ∆κρ,ρ′(ω) = 1 2πtr [Qρ(ω)−Qρ′(ω)] in semiclassical limit is equivalent to derivative with respect to energy of Wilson line phase dierence along classical trajectory, thus corresponding to time delay dierence caused by Lorentz force. This proposition shows that gauge force can be understood as dierence in conditioned time scale perceived by dierent charge sectors in same geometric background. 7 3.6 Modular Time, Entropy Monotonicity, and Arrow of Time Theorem 3.10 (Modular FlowGeometric Time Alignment Theorem (Theorem 3.10)) . In a class of quantum eld theories describable through geometric holography, if local modular ow σφ t corresponds to ow of some Killing or approximate Killing vector on gravity side, then following statements are equivalent: 1. Modular ow time t belongs to same time scale equivalence class as geometric time τ ; 2. Monotonicity of expectation value of modular Hamiltonian and generalized entropy extremality condition jointly derive local gravitational eld equations; 3. QNEC holds and saturates in corresponding null direction. This theorem abstracts the idea in existing work that generalized entropy extremality + QNEC suces to derive Einstein equations, restating it as consistency condition between time scale equivalence class and modular ow. Proof skeleton in Appendix D.  4 Proofs This section gives proof ideas of main results and several key steps, leaving technical details to appendices. 4.1 Proof Outline of Theorem 3.4 Core tool is Kren spectral shift function theory and trace formula of WignerSmith matrix. 1. Kren trace formula ensures for trace-class perturbation V=H−H0∈S1 , there exists unique ξ(λ)∈L1(R) such that for suciently good f , tr[f(H)−f(H0)] = Zf′(λ)ξ(λ) dλ. 2. BirmanKren formula connects scattering matrix determinant and spectral shift function: det S(ω) = e−2πiξ(ω), almost everywhere . 3. Dene total scattering phase Φ(ω) = arg det S(ω) , choose continuous branch such that Φ(ω) = −2πξ(ω) (ignoring integer constant). Dene φ(ω) = 1 2Φ(ω) = −πξ(ω) , thus φ′(ω)/π =−ξ′(ω) = ρrel(ω) . 4. WignerSmith matrix Q(ω) = −iS(ω)†∂ωS(ω) . Under unitarity of S(ω) , tr Q(ω) = −i∂ωln det S(ω) . By BirmanKren formula, ln det S(ω) = −2πiξ(ω), thus tr Q(ω) = 2πξ′(ω) . 5. Combining yields φ′(ω)/π =ξ′(ω) = (2π)−1tr Q(ω). Key is controlling branch and dierentiability of ln det S , details in Appendix A. 8 4.2 Proof Outline of Theorem 3.6 Adopt wave packet semiclassical limit and path integral method: 1. Let single-particle wave packet initial state be described by WKB-type wavefunction, whose phase is given by action S[γ] = Z(pµ˙xµ−H) dτ. 2. On total bundle B , total connection Ω introduces additional phase term, namely parallel transport phase RΩ(˙γ) dτ on path. In path integral, this term is equivalent to traditional vector potentialscalar potential terms. 3. Extremizing action yields generalized geodesic equation, extra connection curvature terms produce force-like terms, respectively corresponding to spacetime curvature (gravity) and internal eld strength (gauge force), as well as entropic force contribution from resolution ow. 4. Taking Ehrenfest limit of wave packet center trajectory, operator expectation value evolution can be simplied to classical equation, obtaining form in proposition. Details in Appendix C. 4.3 Proof Outline of Theorem 3.7 Using frequency conservation and local energyredshift relation in static spacetime: 1. In static spacetime with form g=−N2dt2+hijdxidxj , time Killing vector ∂t ensures energy conservation. 2. Energy measured in local inertial frame is ωloc =N−1(x)ω . State density dierence and group delay between scattering region interior and distant region must be expressed using local energy. 3. Under assumption that scale density is only rescaled by N(x) , time scales at dierent positions dier only by position-dependent constant factor, i.e., belong to same equivalence class. Their ratio is N(x) in unitary transformation. 4. Reconciling this relation with gravitational redshift experimental data can verify consistency of this interpretation. Detailed dimensional reconciliation in Appendix B. 4.4 Proof Outline of Theorem 3.10 This theorem abstracts unied structure of entropy extremalitymodular energygeometric equations: 1. Work in holography and algebraic quantum eld theory shows that under appropriate circumstances, modular ow can be geometrically realized as ow of some Killing or approximate Killing on gravity side; expectation value of modular Hamiltonian is related to changes in generalized entropy. 9 If adopting opposite sign convention when dening spectral shift function, this minus sign can be eliminated; here we adopt convention in main text. A.3 WignerSmith Matrix and Trace Formula WignerSmith matrix is dened as Q(ω) = −iS(ω)†∂ωS(ω) Note that for any invertible matrix S(ω) , ∂ωln det S(ω) = tr [S−1(ω)∂ωS(ω)] For unitary matrix S(ω) , S−1(ω) = S†(ω) , thus −i∂ωln det S(ω) = −itr [S†(ω)∂ωS(ω)] = tr Q(ω) On the other hand, ln det S(ω) = iΦ(ω) thus tr Q(ω) = −i∂ωln det S(ω)=Φ′(ω) = 2φ′(ω) Combining above results, obtain φ′(ω)/π =ρrel(ω) = (2π)−1tr Q(ω) At this point, scale identity is rigorously established.  B Static Spacetime, Redshift, and Scale Equivalence B.1 Static Metric and Local Energy Let static spacetime metric be g=−N2(x)dt2+hij(x)dxidxj Killing vector ξµ= (∂t)µ corresponds to conserved quantity E=−pµξµ=−pt Energy measured in local inertial frame is ωloc =pˆ 0=N−1(x)E where pˆ 0 is orthogonal frame component. Therefore, local state density and group delay in scattering region naturally function on ωloc . 16 B.2 Scale Density and Redshift Factor Assume scale density satises κ(ω;x) = κloc(ωloc) = κloc(N−1(x)ω) Distant N(∞)=1 . For given energy window, in distant observer time scale, group delay is ∆τ(ω;x) = Z1 2πtr Q(ω;x) dω If κloc varies slowly within window, then ∆τ(ω;x)≈N−1(x)∆τ∞(ω) Therefore, for two positions x1,x2 , time scale ratio is ∆τ(ω;x2) ∆τ(ω;x1)≈N−1(x2) N−1(x1)=N(x1) N(x2) This is consistent with gravitational redshift relation ν2/ν1=N(x1)/N(x2) , showing both belong to same time scale equivalence class.  C Semi-classical Limit and Force as Curvature Projection C.1 Action with Connection and Path Integral Consider action containing connection S[γ] = Z(pµ˙xµ−Hfree(x, p)) dτ+Z⟨χ, Ω( ˙γ)χ⟩dτ where χ describes internal degrees of freedom and resolution variables, Ω is total connection. Path integral weight factor is eiS[γ] . Varying action yields mD2xµ Dτ2=qFµν˙xν+fµ res where Fµν is obtained from internal part curvature of connection, fµ res is given by combination of resolution ow and entropy gradient. C.2 Ehrenfest Theorem and Macroscopic Force Evolution of expectation values of operators ˆxµ and ˆpµ on Hilbert space satises Ehrenfest theorem. Taking expectation of Hamiltonian containing connection, in narrow wave packet approximation, expectation of operator product can be factorized into product of expectations plus noise correction. Ignoring higher-order noise yields center-of-mass trajectory equation. At this point, force is completely determined by connection curvature.  17 D Modular Flow, QNEC, and Gravitational Dynamics D.1 Modular Flow and Relative Entropy For local algebra A(O) and state φ , modular ow σφ t and modular Hamiltonian Kφ= −log ∆φ satisfy σφ t(A) = eiKφtAe−iKφt Relative entropy S(ψ||φ) = tr(ρψlog ρψ−ρψlog ρφ) in holographic correspondence, has direct relation with generalized entropy and gravityside energy. D.2 QNEC and Local Energy Constraint QNEC expression ⟨Tkk⟩ ≥ 1 2πS′′ ensures small deformation along null direction has second-order change of entropy limited by energy. This can be viewed as a kind of entropy expansion arrow of time, and can be used to prove that if generalized entropy takes extremum on given crosssection, corresponding geometry must satisfy local form of Einstein equation. D.3 Alignment Condition for Time Scale Equivalence Class If modular ow t and geometric time τ belong to same scale equivalence class, monotonicity of modular Hamiltonian and IyerWald workentropy relation on gravity side jointly derive local gravitational dynamics. Conversely, if geometry satises specic energy conditions and generalized second law, can construct state and algebra aligning modular ow time with geometric time, thus completing closure of unied time scale. This part integrates multiple results from algebraic QFT, holography, and gravitational thermodynamics, showing time scale equivalence class has both operational denition from scatteringspectral shift and structural denition from modular owentropy, which are intrinsically consistent under appropriate conditions. 18