Quantum Chaos and Eigenstate Thermalization in Unified Matrix--QCA Universe\\ \large Spectral Rigidity, Random Matrix Universality and Entanglement Growth under Unified Time Scale
Haobo Ma, Wenlin Zhang
Under the framework of unified time scale, boundary time geometry, Matrix Universe THE--MATRIX, and Quantum Cellular Automaton (QCA) Universe, we construct a structural unified 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 unified time scale mother formula equation \kappa(\omega) =\varphi'(\omega){\pi} =\rho_{rel}(\omega) =1{2\pi}tr Q(\omega), equation this paper interprets quantum chaos and thermalization as: ergodicity of scattering phase flow in the Matrix Universe and operator mixing in the QCA Universe; and further regards spectral rigidity and RMT universality as statistical inevitable results of unified time scale density under maximum entropy principle. Specifically: 1. In the Matrix Universe, defining the ``scattering phase flow'' dynamical system on the energy shell, we prove that when the scattering matrix S(\omega) exhibits chaotic behavior (positive Lyapunov exponent or mixing), the spectral correlation function of the unified time scale \kappa(\omega) asymptotically approaches the RMT (GOE/GUE) kernel function equation R_2(\omega_1,\omega_2) \simeq 1-\Bigl(\sin[\pi\bar\kappa(\omega_1-\omega_2)]{\pi\bar\kappa(\omega_1-\omega_2)}\Bigr)^2, equation thereby providing a scattering--spectral unified proof for the BGS conjecture. 2. In the QCA Universe, considering the operator algebra \mathcal A_{qloc} 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 equation \langle E_a|\mathcal O|E_b\rangle \simeq \mathcal O(E)\delta_{ab} +e^{-S(E)/2}f_{\mathcal O}(E,\Delta E)R_{ab}, equation where R_{ab} is a random variable, establishing the microscopic mechanism of thermalization in the discrete universe. 3. Investigating the growth of entanglement entropy S_{ent}(t) in QCA time evolution, we find that under unified 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 S_{ent}(t)\propto t is consistent with the entropy production rate of the scattering matrix. 4. Under unified time scale constraints, writing the spectral form factor (SFF) K(t) as the Fourier transform of the two-point correlation of \kappa(\omega), 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 unified time scale. 5. The appendix systematically organizes: standard definitions 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 unified time scale spectral correlations. Results indicate a purely structural unified picture: quantum chaos and eigenstate thermalization can be viewed as the statistical behavior of scattering phase flow in the Matrix Universe and operator mixing in the QCA Universe under the unified time scale, thereby answering the structural version of ``why quantum chaos and thermalization occur'' within the unified universe mother structure.
•Nov 19, 2025•0 citations