Time as Density of States: Unified Time Identity \kappa = \rho and Microscopic Scattering Mechanism of Gravitational Redshift

Time as Density of States: Unied Time Identity κ=ρ and Microscopic Scattering Mechanism of Gravitational Redshift Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract In conventional physics, time plays incompatible roles in quantum theory and general relativity: in quantum mechanics it is an external evolution parameter generated by a Hamiltonian, whereas in general relativity proper time is a path-dependent functional of the spacetime metric. Within a quantum cellular automaton (QCA) and optical-path conservation framework, this work proposes a microscopic denition of time in terms of the density of quantum states. Building on EisenbudWignerSmith (EWS) time-delay theory and Krein FriedelLloyd (KFL) trace formulas, we show that the trace of the WignerSmith time-delay operator Q(E) = −iℏS(E)†∂ES(E) is universally related to the relative density of scattering states via the spectral shift function. This leads to a unied time identity κ(E) = 1 2πℏtr Q(E)=∆ρ(E), where ∆ρ(E) is the change of density of states (DOS) induced by an interaction with respect to a reference background. Physically, κ(E) quanties the rate at which a system traverses quantum states in Hilbert space per unit energy. We interpret κ(E) as an intrinsic time density, and argue that macroscopic clock rates are governed by the local DOS of the underlying microscopic degrees of freedom. On this basis we develop a microscopic picture of gravitational redshift: deep in a gravitational potential well, matter and radiation experience an enhanced local DOS due to modied phase space volume and eective refractive index of the vacuum. The increased DOS yields a larger WignerSmith delay per unit energy, thereby slowing local proper time relative to distant observers. We show how, in the weak-eld limit, the DOS-based time density reproduces the standard gravitational redshift relation ∆ν/ν ≃∆Φ/c2 when normalized to asymptotically at regions, and argue that the metric component g00 can be viewed as an emergent functional of the microscopic DOS eld. The timeDOS identity is embedded into Dirac-type QCA models where relativistic dynamics emerge from discrete, strictly causal unitary updates, demonstrating that relativistic time dilation and gravitational redshift can be reinterpreted as collective properties of state-traversal rates on a discrete quantum network. Finally, we connect the proposed time density with thermodynamic time in KMS equilibrium states and the thermal time hypothesis, showing that the DOS-weighted EWS time density denes a natural arrow of time compatible with modular ow in algebraic quantum eld theory. We outline experimental and engineering proposals in microwave cavities, wavechaotic scattering systems and optical clock networks to test the quantitative link between WignerSmith delays, DOS measurements and gravitational redshift. Keywords: time delay; density of states; EisenbudWignerSmith operator; spectral shift function; KreinFriedelLloyd formula; gravitational redshift; quantum cellular automaton; optical metric; KMS state; thermal time 1 1 Introduction & Historical Context The problem of time pervades the development of physical theory. Special and general relativity take the four-dimensional spacetime manifold (M, gµν) as the fundamental object, describing proper time τ as a line element integral along worldlines. Quantum theory, by contrast, is built on Hilbert space and the Hamiltonian H , where time t is an external continuous parameter and evolution is generated by the unitary operator U(t) = exp(−iHt/ℏ) . While the two frameworks are compatible in the semiclassical limit, a unied microscopic ontology of time is lacking. Pauli's theorem indicates that in systems with an energy spectrum bounded from below, there exists no self-adjoint time operator satisfying canonical commutation relations with H , seemingly ruling out attempts to operatorize time within traditional quantum mechanics. On the other hand, time delay has long appeared as an observable in scattering theory. Eisenbud, Wigner, and Smith dened the WignerSmith group delay matrix from the energy derivative of the scattering matrix S(E) as Q(E) = −iℏS(E)†∂ES(E), whose diagonal elements Qαα(E) give the time delay in each channel, while the trace tr Q(E) characterizes the total temporal response of the system to a family of scattering states. In parallel, Krein, Friedel, Lloyd and others established the celebrated KreinFriedelLloyd formula in scattering spectral theory, relating the energy derivative of scattering phase shifts to the change in DOS ∆ρ(E) . In nite-box normalization, this formula can be written simply as ∆ρ(E) = 1 2πitrS(E)†∂ES(E)=1 π∂Eδtot(E), where δtot(E) is the total scattering phase shift. Modern spectral shift function theory further shows that the derivative of the spectral shift function ξ(E) is precisely the DOS dierence induced by the interaction, and ξ(E) can itself be characterized by the determinant phase of S(E) . These results hint at a profound fact: the time delay and the density of states in scattering systems are not two independent quantities, but rather dierent projections of the same spectral structure. Building on this foundation, the present work proposes a unied time identity that strictly equates the time ow rate κ(E) with the DOS ρ(E) . On the other hand, general relativity interprets gravitational redshift as the spatial variation of the g00 component in static spacetime metrics. In the weak-eld limit, the static gravitational potential Φ(x) and the metric satisfy g00(x)≃ −1 + 2Φ(x)/c2, and the proper time of stationary observers satises dτ=√−g00dt , so that for two positions x1,x2 at dierent potential values we have ∆ν ν≃Φ(x2)−Φ(x1) c2, a relation that has been precisely veried in the PoundRebka γ -ray experiment, the Hafele Keating around-the-world atomic clock ight, and recent optical lattice clock height comparison experiments. However, the geometric description of gravitational redshift still leaves the origin of time ow in the continuous eld of the metric tensor, lacking a direct connection to the structure of quantum states. Quantum cellular automaton and discrete quantum walk studies have shown that under appropriate symmetry conditions, the Dirac and Weyl equations can emerge as the continuum limit of local unitary discrete dynamics, thus providing rigorous models for the discrete picture of the universe as quantum computation. 2 In such a discrete ontology, the time step is a discrete parameter of the local update rule, rather than an external continuous variable. The natural question is: how can we reconstruct continuous proper time within this discrete framework, and relate it to gravitational redshift? The basic viewpoint of this paper is: time is an alias for density of states . More precisely, through the EWS time-delay operator and the KFL trace formula, we prove that under quite general scattering settings, the following unied time identity holds: κ(E) = 1 2πℏtr Q(E)=∆ρ(E) where κ(E) is interpreted as the time ow density per unit energy interval, i.e., the eective time delay experienced by the system per unit energy in the neighborhood of energy E . We further show that, under appropriate coarse-graining and normalization choices, the spatial variation of this quantity can restate gravitational redshift in general relativity. 2 Model & Assumptions 2.1 Scattering System and Density of States Let H0 be the free Hamiltonian and H=H0+V the interacting Hamiltonian containing a local potential V . We make the following assumptions: 1. H0, H are self-adjoint operators on the same Hilbert space H , and V is a suciently rapidly decaying bounded or relatively H0 -bounded perturbation, so that the wave operators Ω±= s-lim t→±∞ eiHt/ℏe−iH0t/ℏ exist and are complete. 2. The scattering operator S= (Ω+)†Ω− decomposes in the energy representation into a family of unitary matrices S(E) , acting on the channel space HE on each energy shell. 3. The spectral measures of both free and interacting systems are absolutely continuous plus a nite number of bound states, so that the well-dened state-counting functions exist: N0(E)=#{λn(H0)≤E}, N(E)=#{λn(H)≤E}, whose derivatives give the densities of states ρ0(E), ρ(E) . The change in DOS induced by the interaction is ∆ρ(E) = ρ(E)−ρ0(E). The spectral shift function ξ(E) is dened by the Krein trace formula, via trf(H)−f(H0)=Z+∞ −∞ f′(E)ξ(E) dE for suciently smooth functions f . An approximation to the indicator function yields ξ(E) = N(E)−N0(E) , whose derivative satises ξ′(E) = ∆ρ(E). On the other hand, the determinant of the scattering matrix and the spectral shift function are related by det S(E) = exp−2πiξ(E), so that 1 2πi∂Eln det S(E) = ξ′(E) = ∆ρ(E). 3 2.2 WignerSmith Time-Delay Operator In the energy representation, the WignerSmith group delay matrix is dened as Q(E) = −iℏS(E)†∂ES(E), whose diagonal element Qαα(E) gives the time delay of channel α , and the trace tr Q(E) = −iℏtrS(E)†∂ES(E) characterizes the total group delay of all channels. The self-adjointness and observability of Q(E) can be veried by dierentiating the unitarity of the scattering matrix S(E)†S(E) = I . Taking the trace and comparing with the spectral shift relation from the previous section, we immediately obtain ∆ρ(E) = 1 2πi∂Eln det S(E) = 1 2πi 1 det S(E)∂Edet S(E) = 1 2πitrS†∂ES=1 2πℏtr Q(E). This is the spectral-theoretic foundation of the unied time identity in the present work. 2.3 QCA Continuum Limit and Scattering Quantum cellular automata are a class of strictly local, discrete-time, unitary evolutions U de- ned on a lattice Λ , which in the single-particle subspace are equivalent to discrete-time quantum walks. A considerable body of work has shown that, in the limit of suciently small wavevector and mass, appropriately constructed QCA evolutions can approximate Dirac equations in various dimensions and their generalizations in curved spacetime. In the QCA framework, time is essentially the discrete step number n∈Z , and energy is given by the quasi-energy spectrum e−iε(k) of the single-step evolution operator U . Local defects or external potentials can be implemented by modifying the local update operator in a nite region, corresponding to Floquet scattering scenarios where the scattering matrix S(ε) is closely related to continuous-time scattering. Through the continuum-limit mapping ε7→ E , the FloquetEWS time delay in QCA can be unied with Q(E) in continuous scattering theory. We make the following model assumptions: 1. The universe can be described microscopically by a class of Dirac-type QCA models, whose single-particle subspace approximates the standard Dirac equation in the long-wavelength limit. 2. Macroscopic gravitational elds correspond to slow spatial variations of the QCA background cell update rules, i.e., gradual changes in local propagation velocity and phase response. This can be likened to the construction of quantum walks in curved spacetime. 3. In this framework, all macroscopic time observations can ultimately be reduced to phase dierences and group delay measurements in some kind of scattering or interference experiments, and therefore can be uniformly described by the EWSKFL structure. 3 Main Results (Theorems and Alignments) This section presents the unied time identity and its main consequences for gravitational redshift, the thermodynamic arrow of time, and the QCA continuum limit. Denition 3.1 (Time Density) . For a scattering system satisfying the above assumptions, dene the time density at energy E as κ(E)≡1 2πℏtr Q(E), where Q(E) = −iℏS(E)†∂ES(E) is the WignerSmith group delay operator. 4 Theorem 3.2 (Unied Time Identity) . Under the assumptions that the KreinFriedelLloyd formula and spectral shift function exist and are dierentiable, the time density and DOS change satisfy κ(E)=∆ρ(E) = ρ(E)−ρ0(E). In other words, the time density equals the relative DOS introduced by the interaction . This theorem unies the following three objects as dierent expressions of the same function: 1. The energy derivative of the total scattering phase φ(E) = arg det S(E) : κ(E) = 1 2π∂Eφ(E). 2. The DOS change ∆ρ(E) induced by the interaction. 3. The normalized trace of the WignerSmith delay matrix (2πℏ)−1tr Q(E) . See Appendix A for the detailed proof. Proposition 3.3 (Time Flow Rate as Function of DOS) . Suppose a macroscopic clock is driven by a family of scattering states concentrated near energy E0 , and its output period T is determined by the energy dependence of the total phase φ(E) . Then in the narrow-band approximation, the proper time increment of the clock satises dτ dt≃κref(E0) κ(E0)=∆ρref(E0) ∆ρ(E0), where ref denotes a xed reference position (e.g., the asymptotically at region at innity). This shows that within a given energy window, the time ow rate is inversely proportional to the local DOS . Theorem 3.4 (Gravitational Redshift in the Weak-Field Limit) . Consider a weak gravitational eld with static potential Φ(x) satisfying |Φ|/c2≪1 . Suppose that the Hamiltonian of some quantum eld in a local inertial frame is approximately H(x,p)≃pm2c4+c2p2+mΦ(x), and the local DOS ρ(E, x) is given by the semiclassical Weyl formula. Then for lightly bound states in the nonrelativistic limit E≃mc2 , we have ρ(E, x)≃ρ∞(E)1−αΦ(x) c2, where ρ∞(E) is the DOS at innity, and the constant α depends on the dimension and the specic model (in the three-dimensional nonrelativistic gas model α= 3/2 ). Adopting the time-density normalization dτ(x) dt=κ∞(E) κ(E, x)=ρ∞(E) ρ(E, x), a rst-order expansion yields dτ(x) dt≃1 + αΦ(x) c2. By appropriately choosing an information weight for dening the DOS (e.g., considering energy-weighted DOS or optical mode DOS), one can set α= 1 , thereby obtaining a time dilation factor consistent with the weak-eld limit of general relativity: dτ(x) dt≃1 + Φ(x) c2, 5 and recovering the gravitational redshift formula ∆ν ν≃Φ(x2)−Φ(x1) c2. See Appendix B for the detailed derivation. Proposition 3.5 (Time Density and KMS Thermal Time) . Let (A, αt) be a C∗ dynamical system, and ωβ a KMS state at temperature T= 1/(kBβ) . The energy-weighted DOS is ρβ(E) = Z−1ρ(E)e−βE, Z =Zρ(E)e−βEdE. Dene the average time density in the thermal state as ¯κβ=Zκ(E)ρβ(E) dE=Z∆ρ(E)ρβ(E) dE. Then ¯κβ increases monotonically with energy density, is one-to-one correlated with the imaginary time period β of the KMS ow, thereby providing a DOS-time realization of the thermodynamic arrow of time compatible with the thermal time hypothesis and the UnruhKMS structure. Proposition 3.6 (Time Density in QCA Continuum Limit) . In a Dirac-type QCA model, the single-step evolution operator U has quasi-energy spectrum ε(k) , and scattering defects introduce the Floquet scattering matrix S(ε) . Dene the FloquetEWS operator QF(ε) = −iUeff(ε)†∂εUeff(ε). In the continuum limit ε→E , its trace converges to the continuous scattering time delay, satisfying κ(E) = 1 2π∂Eφ(E) = 1 2πℏtr Q(E) = lim ε→E 1 2πtr QF(ε), so that the time density of discrete steps in QCA is consistent with the DOS denition in continuous theory. 4 Proofs This section provides proof outlines for the unied time identity and its main corollaries; complete technical details are placed in the appendices. 4.1 Proof of the Unied Time Identity (Theorem 3.2) The proof proceeds in two steps. Step one: KreinFriedelLloyd formula and DOS. Consider the one-dimensional case. Place the system in a nite box of length L with appropriate boundary conditions. For the free system, the momentum quantization condition is knL=nπ , and the number of states is N0(k)≃L πk, ρ0(k) = dN0 dk=L π. After adding the potential V(x) , scattering boundary conditions give the phase-shift-corrected quantization condition knL+δ(kn) = nπ, so that N(k) = L πk+1 πδ(k),∆ρ(k) = ρ(k)−ρ0(k) = 1 π∂kδ(k). 6 Using E=ℏ2k2/(2m) and the chain rule, we convert to the energy representation to obtain ∆ρ(E) = 1 π∂Eδ(E). For multi-channel and higher-dimensional cases, this can be generalized via partial-wave decomposition and spectral shift function theory to ∆ρ(E) = 1 2πitrS†∂ES=1 2π∂Eφ(E), where φ(E) = arg det S(E) , i.e., the KreinFriedelLloyd formula. Step two: EWS time delay and the trace of Q(E) . The EWS operator is dened as Q(E) = −iℏS†(E)∂ES(E), whose trace is tr Q(E) = −iℏtrS†∂ES. Comparing this with the DOS expression we obtain ∆ρ(E) = 1 2πitr(S†∂ES) = 1 2πℏtr Q(E). This is precisely the unied time identity in the time density denition. Self-adjointness follows directly from S†S=I⇒(∂ES†)S+S†(∂ES)=0, which yields Q†(E) = +iℏ(∂ES†)S=−iℏS†(∂ES) = Q(E), so Q(E) is an observable. 4.2 Time Flow Rate and DOS (Proposition 3.3) Consider a narrow-band wavepacket whose energy distribution |a(E)|2 is concentrated near E0 , satisfying Z|a(E)|2dE= 1,⟨E⟩=E0,∆E≪E0. The scattered state in the far region can be written as ψout(t) = Za(E)e−iEt/ℏS(E)|E⟩dE. Expand S(E)≃eiφ(E)˜ S(E) , where ˜ S(E) has a slowly varying matrix structure, and the total phase is φ(E) = arg det S(E) . By comparing the translation to the free propagation state, we can dene the group delay to a certain far point as τ(E0) = ∂Eφ(E)E0. The reference state (e.g., distant at region) has a corresponding delay τref(E0) determined by the scattering matrix Sref(E) , with time density κ(E) = 1 2π∂Eφ(E), κref(E) = 1 2π∂Eφref(E). Within the same external coordinate time t , the internal subjective time increment dτ is proportional to the total phase increment scanned by the wavepacket, yielding dτ dt∝∂Eφref ∂Eφ=κref(E) κ(E)=∆ρref(E) ∆ρ(E). Under appropriate normalization, the proportionality constant can be taken as 1, and the proposition follows. 7 4.3 DOS and Gravitational Redshift (Theorem 3.4 Outline) Gravitational redshift essentially arises from Killing energy conservation in static spacetime and the spatial variation of proper time for stationary observers. We wish to show that in the weak-eld limit, this eect can be restated as a spatial variation of the local DOS. In the Newtonian limit, the nonrelativistic particle Hamiltonian is H(x,p) = p2 2m+mΦ(x), and the DOS is given by the phase space volume: ρ(E, x)≃1 (2πℏ)3ZδE−H(x,p)d3p. The integral can be computed explicitly to give ρ(E, x)∝E−mΦ(x)1/2. For E≃mc2 and |Φ|/c2≪1 , expanding yields ρ(E, x)≃ρ∞(E)1−1 2 Φ(x) c2. For three-dimensional relativistic models or systems with multiple degrees of freedom, the exponent changes, giving the general form ρ(E, x)≃ρ∞(E)1−αΦ(x) c2, where α depends on the specic system. Theorem 3.4 indicates that by introducing an appropriate information weight, dening the local time ow rate as dτ(x) dt=ρinfo,∞(E) ρinfo(E, x), and taking ρinfo(E, x) = ρ(E, x)β for some function class, one can adjust the exponent β so that the rst-order expansion coecient equals 1. Physically, this is the freedom to choose a time scale among dierent microscopic models, analogous to adopting dierent coordinate scales in generally covariant theories. Matching this denition to classical gravitational redshift experiments xes β and yields dτ(x) dt≃1 + Φ(x) c2. See Appendix B for the complete calculation and normalization procedure. 4.4 Time Density and KMS Thermal Time (Proposition 3.5) For a many-body system with a continuous spectrum, the KMS condition is ωβAαt(B)=ωβαt−iβ(B)A, where αt is the Heisenberg evolution. For operators AEE′ in the energy representation, the realization of the KMS condition relies on the Boltzmann weight e−βE weighting the DOS. 8 Dene the average time density under DOS weighting: ¯κβ=Zκ(E)ρβ(E) dE=Z∆ρ(E)ρβ(E) dE. Then ¯κβ is the average time delay density of the system at temperature T . As the energy density increases, the spectral weight shifts to higher energies, ¯κβ increases monotonically, corresponding to stronger information congestion and slower macroscopic time. This is compatible with the thermal time hypothesis view that time is the modular ow of the state on the algebra: when local energy densities dier, the ratio between the modular ow parameter and geometric time changes, corresponding to dierent gravitational time dilations and temperatures. 4.5 Unied Time Identity in QCA Continuum Limit (Proposition 3.6) In a Dirac-type QCA, the single-step evolution U can be diagonalized in the momentum representation as U|k, σ⟩= e−iεσ(k)|k, σ⟩, where σ labels internal degrees of freedom. In the presence of defects, a Floquet scattering matrix S(ε) can be constructed in the long-time limit, with EWS operator QF(ε) = −iS(ε)†∂εS(ε), completely parallel to the continuous-time case. Combining the QCA continuum limit ε→E with the previous scatteringDOS theory, we obtain lim ε→E 1 2πtr QF(ε)=∆ρ(E), i.e., the time density denition is consistent between discrete and continuous descriptions. Related constructions can be found in studies of Dirac QCA and quantum walks in curved spacetime. 5 Model Apply This section discusses applications of the unied time identity in several physical scenarios. 5.1 Wave-Chaotic Cavities and Microwave Scattering In wave-chaotic cavities and multi-port electromagnetic scattering structures, the WignerSmith matrix Q(E) has long been used to characterize mode-averaged group delay and dwell time distributions, and its relation to DOS in the random matrix theory framework has been extensively veried. In these systems, experimental tests of the unied time identity can be realized in the following ways: 1. Measure the multi-port scattering matrix S(ω) , numerically dierentiate to obtain Q(ω) = −iS†∂ωS and tr Q(ω) . 2. Independently obtain DOS ρ(ω) and the relative ∆ρ(ω) with respect to the empty cavity through eigenfrequency statistics or Green's function measurements. The unied time identity predicts 1 2πtr Q(ω) = ∆ρ(ω). Deviations from this relation can be attributed to loss or non-conservative eects, requiring correction using a generalized (non-unitary) WignerSmith matrix. 9 so dτ(x) dt≃1 + βαΦ(x) c2. Choosing β= 1/α recovers the GR linear coecient 1. Physically, this corresponds to choosing a DOS denition proportional to the number of distinguishable microscopic states per unit energy in a many-body system, rather than simple single-particle phase space volume. Its specic form requires further derivation from relativistic eld theory and many-body correlation functions, which is beyond the scope of this paper. C Time Density, KMS States, and Thermal Time This appendix briey explains the relationship between time density and KMS modular ow. Let (A, αt) be a one-parameter group on a C∗ algebra, where αt is generated by a Hamiltonian H . The KMS condition is: for any A, B ∈ A , there exists an analytic function FA,B(z) satisfying FA,B(t) = ωβ(Aαt(B)), FA,B(t−iβ) = ωβ(αt(B)A). In the GNS representation, the KMS state corresponds to a special vector state on Hilbert space, whose modular ow is given by TomitaTakesaki theory. The ratio of the modular ow parameter s to physical time t can be understood as temperature or time scale. The unied time identity indicates that time density κ(E) is jointly determined by DOS and EWS delay. For a given temperature T , the system is in energy probability distribution Pβ(E) = Z−1ρ(E)e−βE, and the average time scale of the KMS ow can be related to ¯κβ=Zκ(E)Pβ(E) dE. The thermal time hypothesis holds that physical time is precisely the natural parameter of the modular ow in a given state, and temperature is the proportionality coecient between the modular ow and geometric time. In this framework, ¯κβ provides a way to directly dene this proportionality coecient using spectral quantities. D Time Density and Scattering in Dirac QCA Dirac QCA models provide discrete, local, unitary microscopic update rules, whose singleparticle subspace approximates the Dirac equation in the continuum limit. In the one-dimensional case, the single-step evolution can be written as U=X x|x+ 1⟩⟨x|⊗C++|x−1⟩⟨x|⊗C−, where C± are coin operators acting on internal degrees of freedom. Defects can be implemented by modifying C± or adding local phases in a nite region. In the quasi-energy representation, the scattering matrix S(ε) is similar to S(E) in continuous-time scattering, and the EWS operator is dened as QF(ε) = −iS(ε)†∂εS(ε). The unied time identity in the Floquet scenario becomes κ(ε) = 1 2πtr QF(ε)=∆ρF(ε), where ∆ρF(ε) is the quasi-energy DOS change induced by the interaction. 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