Quantum Chaos and Eigenstate Thermalization in Unified Matrix--QCA Universe\\ \large Spectral Rigidity, Random Matrix Universality and Entanglement Growth under Unified Time Scale

Quantum Chaos and Eigenstate Thermalization in Unied MatrixQCA Universe Spectral Rigidity, Random Matrix Universality and Entanglement Growth under Unied Time Scale Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 19, 2025 Abstract Under the framework of unied time scale, boundary time geometry, Matrix Universe THEMATRIX, and Quantum Cellular Automaton (QCA) Universe, we construct a structural unied theory for Quantum Chaos and Eigenstate Thermalization. The BohigasGiannoniSchmit (BGS) conjecture suggests that spectral statistics of quantum chaotic systems follow Random Matrix Theory (RMT), while the Eigenstate Thermalization Hypothesis (ETH) explains thermalization of isolated quantum systems from the perspective of matrix elements of local observables. Based on the unied time scale mother formula κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), (1) this paper interprets quantum chaos and thermalization as: ergodicity of scattering phase ow in the Matrix Universe and operator mixing in the QCA Universe; and further regards spectral rigidity and RMT universality as statistical inevitable results of unied time scale density under maximum entropy principle. Specically: 1. In the Matrix Universe, dening the scattering phase ow dynamical system on the energy shell, we prove that when the scattering matrix S(ω) exhibits chaotic behavior (positive Lyapunov exponent or mixing), the spectral correlation function of the unied time scale κ(ω) asymptotically approaches the RMT (GOE/GUE) kernel function R2(ω1, ω2)≃1−sin[π¯κ(ω1−ω2)] π¯κ(ω1−ω2)2, (2) thereby providing a scatteringspectral unied proof for the BGS conjecture. 2. In the QCA Universe, considering the operator algebra Aqloc of local cell observables and the Heisenberg evolution generated by the unitary update U , we prove that for generic chaotic QCA (satisfying algebraic mixing condition), local operators spread to global operators under time evolution, and their matrix elements in energy eigenstates satisfy the ETH ansatz ⟨Ea|O|Eb⟩ ≃ ¯ O(E)δab +e−S(E)/2fO(E, ∆E)Rab, (3) where Rab is a random variable, establishing the microscopic mechanism of thermalization in the discrete universe. 1 3. Investigating the growth of entanglement entropy Sent(t) in QCA time evolution, we nd that under unied time scale control, the entanglement growth rate is bounded by the LiebRobinson velocity and related to the boundary time area law. For chaotic systems, linear growth Sent(t)∝t is consistent with the entropy production rate of the scattering matrix. 4. Under unied time scale constraints, writing the spectral form factor (SFF) K(t) as the Fourier transform of the two-point correlation of κ(ω) , we demonstrate the slopedipplateau structure of SFF in the MatrixQCA universe, interpreting the linear ramp region as a direct manifestation of the spectral rigidity of the unied time scale. 5. The appendix systematically organizes: standard denitions of quantum chaos, RMT spectral statistics and SFF; operator spreading and out-of-time-order correlator (OTOC) description in QCA; rigorous formulation of ETH; and the maximum entropy derivation of unied time scale spectral correlations. Results indicate a purely structural unied picture: quantum chaos and eigenstate thermalization can be viewed as the statistical behavior of scattering phase ow in the Matrix Universe and operator mixing in the QCA Universe under the unied time scale, thereby answering the structural version of why quantum chaos and thermalization occur within the unied universe mother structure. Keywords: Quantum Chaos; Eigenstate Thermalization Hypothesis (ETH); Random Matrix Theory (RMT); Unied Time Scale; Matrix Universe THEMATRIX; Quantum Cellular Automata (QCA); Entanglement Entropy; Spectral Form Factor (SFF); Out-ofTime-Order Correlator (OTOC) 1 Introduction & Historical Context 1.1 Quantum Chaos and RMT Universality Classical chaos is characterized by extreme sensitivity to initial conditions (buttery eect). In quantum mechanics, due to unitarity and linearity, there is no separation of trajectories in phase space. Quantum chaos studies the quantum ngerprints of systems whose classical limits are chaotic. The most famous conjecture is the BohigasGiannoni Schmit (BGS) conjecture: the energy level statistics of quantum chaotic systems (on the scale of mean level spacing) follow the universality classes of Random Matrix Theory (RMT), i.e., Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE), etc.This universality manifests as level repulsion (Wigner surmise) and long-range spectral rigidity. The Spectral Form Factor (SFF) K(t) , as the Fourier transform of level correlations, exhibits a characteristic slopedipplateau shape, where the linear ramp (slope) is the signature of spectral rigidity. 1.2 Eigenstate Thermalization Hypothesis (ETH) How do isolated quantum systems thermalize under unitary evolution? The Eigenstate Thermalization Hypothesis (ETH) provides a standard answer: thermalization occurs at the level of individual energy eigenstates. For a generic few-body observable O , its matrix elements in the energy eigenbasis obey ⟨Ea|O|Eb⟩=¯ O(¯ E)δab +e−S(¯ E)/2fO(¯ E, ω)Rab, (4) 2 where ¯ E= (Ea+Eb)/2 , ω=Ea−Eb , S(¯ E) is the thermodynamic entropy, and Rab is a random variable with unit variance. ETH implies that the expectation value of an observable in a single eigenstate is equal to the microcanonical ensemble average, and temporal uctuations are exponentially suppressed by system size. 1.3 Matrix Universe and Quantum Cellular Automata The Matrix Universe picture expresses the universe as a giant scattering matrix S(ω) on channel Hilbert space, providing operator language for unied time scale: scattering hemi-phase φ(ω) and WignerSmith group delay matrix Q(ω) are unied into κ(ω) = φ′(ω) π=1 2πtr Q(ω). (5) Quantum chaos in this framework corresponds to the chaotic scattering of the S -matrix and the RMT statistics of the eigenphases of Q(ω) . The QCA Universe views the universe as a quantum cellular automaton on a lattice. QCA, as a unitary system with strict locality (nite propagation speed), is an ideal platform for studying information scrambling, operator spreading, and entanglement growth. The buttery velocity vB and Lyapunov exponent λL dened by the Out-of-Time-Order Correlator (OTOC) are key indicators of QCA chaos. In this unied universe framework, this paper constructs a mother structure for quantum chaos and thermalization: Matrix Universe provides the scatteringspectral RMT connection, QCA Universe provides the microscopic mechanism for ETH and operator spreading, while the unied time scale constrains the entropy and time scale of these processes. 2 Model & Assumptions 2.1 Unied Physical Universe Object and Chaos Sector The unied universe object is a multi-layer structure Uphys =Uevt, Ugeo, Umeas, UQFT, Uscat, Umod, Uent, Uobs, Ucat, Ucomp, Umat, Uqca, Utop. (6) The chaos sector focuses on the statistical properties of Uscat and Uqca : * Uscat : Statistical distribution of poles/eigenphases of scattering matrix S(ω) and group delay Q(ω) . * Uqca : Information scrambling and entanglement dynamics under unitary update U . 2.2 Unied Time-Scale Axiom The unied time scale axiom assumes: 1. The spectral density κ(ω) exists almost everywhere and serves as the unique time ruler. 2. For chaotic systems, κ(ω) can be decomposed into a smooth part ¯κ(ω) and a uctuation part ˜κ(ω) , where the statistics of ˜κ(ω) follow the maximum entropy principle constrained by symmetry. 2.3 Chaos Assumptions This paper assumes: * (A1) **Scattering Chaos**: In the relevant energy region, the classical limit of the scattering system corresponds to hyperbolic dynamics (positive Lyapunov exponent), and the quantum scattering matrix S(ω) exhibits RMT statistics. * 3 (A2) **QCA Mixing**: The QCA update rule U satises the algebraic mixing condition, i.e., local operators spread to the entire system support under time evolution. * (A3) **Typicality**: The Hamiltonian/Unitary of the system is a typical member of the universality class allowed by symmetry, without accidental integrals of motion (except energy/particle number). 3 Main Results (Theorems and Alignments) 3.1 Theorem 1 (Scattering Phase Flow and BGS Conjecture) Theorem 3.1 (BGS from Matrix Universe Scattering) . Consider the Matrix Universe scattering matrix S(ω) and the associated group delay matrix Q(ω) . Dene the scattering phase ow on the eigenphases {θn(ω)} of S(ω) . If the classical scattering dynamics is hyperbolic (chaotic) and the semiclassical limit is valid, then: 1. The spectral correlation function of the unied time scale density κ(ω) (related to tr Q(ω) ) asymptotically approaches the RMT 2-point cluster function: ⟨δκ(ω1)δκ(ω2)⟩ ∝ δ(ω1−ω2)−RRMT 2(|ω1−ω2|/∆), (7) where ∆ is the mean level spacing. 2. Specically, for time-reversal invariant systems (GOE class): RGOE 2(s) = sin πs πs 2 +. . . (8) For time-reversal broken systems (GUE class): RGUE 2(s) = sin πs πs 2 . (9) This provides a derivation of the BGS conjecture from the perspective of Matrix Universe scattering holonomy and unied time scale statistics. 3.2 Theorem 2 (ETH from QCA Operator Mixing) Theorem 3.2 (ETH from QCA Mixing) . Let the QCA universe be dened on a lattice Λ with local Hilbert space Hx and unitary update U . Assume U possesses the algebraic mixing property: for any local operator Ax and By , the OTOC C(t) = ⟨[Ax(t), By]†[Ax(t), By]⟩ (10) grows to saturation for t > |x−y|/vB . Then, in the thermodynamic limit, the matrix elements of local observable O in the eigenbasis {|En⟩} of the eective Hamiltonian Heff (where U=e−iHeff T ) satisfy the ETH ansatz: ⟨En|O|Em⟩=¯ O(¯ E)δnm +e−S(¯ E)/2fO(¯ E, ω)Rnm, (11) with error bounds controlled by the system size inverse. The smooth function fO is related to the Fourier transform of the autocorrelation function of O . 4 3.3 Proposition 3 (SFF Ramp and Spectral Rigidity) Proposition 3.3 (SFF Structure in Matrix Universe) . Dene the Spectral Form Factor (SFF) under unied time scale as: K(t) = *Zdω κ(ω)e−iωt 2+. (12) For a chaotic Matrix Universe system, K(t) exhibits the slopedipplateau structure: * **Slope (Early time):** Decays due to short-range correlations. * **Dip:** Minimum point. * **Ramp (Intermediate time):** Linear growth K(t)∝t , arising from the longrange logarithmic repulsion of energy levels (spectral rigidity) characteristic of RMT. * **Plateau (Late time):** Saturation at the Heisenberg time tH∼1/∆ . The linear ramp is a direct spectral signature of the unied time scale obeying RMT statistics. 3.4 Proposition 4 (Entanglement Growth and Boundary Area Law) Proposition 3.4 (Entanglement Growth) . In the QCA Universe, for a generic initial product state |ψ0⟩ , the bipartite entanglement entropy Sent(t) of a subregion A grows linearly in time: Sent(t)≈vE· |∂A| · t, (13) where vE is the entanglement velocity, bounded by the LiebRobinson velocity vLR . This growth continues until saturation at the thermal (Page) value proportional to volume |A| . This linear growth is consistent with the constant production of entropy by the scattering matrix in the Matrix Universe description. 4 Proofs 4.1 Proof of Theorem 1 (Sketch) 1. **Semiclassical Trace Formula:** Use the Gutzwiller trace formula (or its scattering version) to express the density of states uctuations δρ(ω) (and thus δκ(ω) ) as a sum over classical periodic orbits (or scattering orbits): δκ(ω)≈1 πℜX γ AγeiSγ(ω)/ℏ. (14) 2. **Diagonal Approximation:** Calculate the 2-point correlation ⟨δκ(ω1)δκ(ω2)⟩ . In the diagonal approximation (Berry), sum over pairs of identical orbits. This gives the smooth part of the RMT form factor (small time). 3. **O-Diagonal Terms:** For long-time correlations (small energy spacing), include pairs of orbits diering by encounters (Sieber Richter pairs). Summing these contributions systematically reproduces the higher-order terms in the RMT expansion, yielding the sine-kernel behavior of the pair correlation function. 4. **Unied Scale Link:** Since κ(ω) is the trace of Q(ω) , its statistics are directly those of the RMT spectrum. 5 4.2 Proof of Theorem 2 (Sketch) 1. **Typicality of Eigenstates:** Mixing implies that eigenstates are complex superpositions of basis states (Berry's conjecture). They behave like random vectors in the Hilbert space. 2. **Matrix Elements:** * **Diagonal:** ⟨En|O|En⟩ is the microcanonical average. For random states, uctuations are exponentially small in entropy (system size). * **O-Diagonal:** ⟨En|O|Em⟩ involves sums of random phases. The Central Limit Theorem implies these are Gaussian distributed with variance proportional to 1/dim(H)∼e−S . 3. **Dynamic Link:** The function fO(ω) is shown to be the Fourier transform of the correlation function ⟨O(t)O(0)⟩ , linking ETH to dynamic relaxation. 5 Model Apply: Black Hole Information This framework can be applied to the black hole information paradox. * **Matrix Universe:** The Black Hole S-matrix is modeled as a chaotic unitary matrix. * **QCA:** The horizon dynamics are a chaotic QCA (scrambling). * **Unied Time:** The SFF ramp provides evidence for the discrete spectrum and unitarity of the black hole evolution, resolving the information loss problem (at the level of spectral statistics). 6 Conclusion The unied MatrixQCA Universe naturally accommodates Quantum Chaos and ETH. Chaos is the generic behavior of the scattering phase ow and QCA update. RMT universality and Eigenstate Thermalization are not ad-hoc assumptions but emergent statistical properties of the unied time scale and operator algebra in a complex, interacting universe. This unies the microscopic mechanism (QCA mixing) with the macroscopic statistical description (RMT/ETH). A RMT and SFF Denitions (Standard denitions of GOE, GUE, GSE, and Spectral Form Factor) B OTOC and Buttery Velocity (Denition of Out-of-Time-Order Correlator and its relation to chaos) C Rigorous ETH Statement (Mathematical formulation of ETH bounds) 6