These notes develop the diagnostics of chaos from their classical definitions through to holography, in six lectures given at the ST4 Workshop, Chennai Mathematical Institute, in 2026. No prior exposure to chaos theory is assumed. Lectures 1 and 2 define classical chaos geometrically and build the instruments that measure it, every one of which reads a phase-space trajectory. Quantization removes that trajectory, and Lectures 3 and 4 rebuild the instruments from spectra and eigenstates instead: level statistics, random matrix theory, the spectral form factor and eigenstate thermalization, as much the working toolkit for thermalization in many-body physics as for holography. What they return is a classification and a count rather than a rate. Lecture 5 recovers the rate from the squared commutator, treats scrambling as operator growth, follows the commutator to the out-of-time-order correlator, marks where that diagnostic stops being reliable, and reaches the butterfly velocity and the chaos bound. Lecture 6 constructs the eternal black hole, the thermofield double and the shock-wave geometry behind that bound, then follows two probes whose exponents discriminate between backgrounds where the horizon result cannot, one tracking a thermodynamic phase transition, the other resolving an anisotropy induced by a background field. That last part is research rather than review, and every figure throughout is computed from the scripts supplied with the source rather than redrawn from the literature. Every lecture closes with exercises. Claude Opus 5, an AI assistant developed by Anthropic, worked as a research assistant under the author's direction.
These notes develop the diagnostics of chaos from their classical definitions through to holography, in six lectures given at the ST4 Workshop, Chennai Mathematical Institute, in 2026. No prior exposure to chaos theory is assumed. Lectures 1 and 2 define classical chaos geometrically and build the instruments that measure it, every one of which reads a phase-space trajectory. Quantization removes that trajectory, and Lectures 3 and 4 rebuild the instruments from spectra and eigenstates instead: level statistics, random matrix theory, the spectral form factor and eigenstate thermalization, as much the working toolkit for thermalization in many-body physics as for holography. What they return is a classification and a count rather than a rate. Lecture 5 recovers the rate from the squared commutator, treats scrambling as operator growth, follows the commutator to the out-of-time-order correlator, marks where that diagnostic stops being reliable, and reaches the butterfly velocity and the chaos bound. Lecture 6 constructs the eternal black hole, the thermofield double and the shock-wave geometry behind that bound, then follows two probes whose exponents discriminate between backgrounds where the horizon result cannot, one tracking a thermodynamic phase transition, the other resolving an anisotropy induced by a background field. That last part is research rather than review, and every figure throughout is computed from the scripts supplied with the source rather than redrawn from the literature. Every lecture closes with exercises. Claude Opus 5, an AI assistant developed by Anthropic, worked as a research assistant under the author's direction.
In quantum systems with a classical limit, advanced semiclassical methods provide the crucial link between phase-space structures, reflecting the distinction between chaotic, mixed or integrable classical dynamics, and the corresponding quantum properties. Well established techniques dealing with ergodic wave interference in the usual semiclassical limit ℏ→0, where the classical limit is given by Hamiltonian mechanics of particles, constitute a now standard part of the toolkit of theoretical physics. During the last years, these ideas have been extended into the field theoretical domain of systems composed of N indistinguishable particles, aka quantum fields, displaying a different type of semiclassical limit ℏeff=1/N→0 and accounting for genuine many-body quantum interference. The foundational concept behind this idea of many-body interference, the many-body version of the van Vleck-Gutzwillers semiclassical propagator, is explained in detail. Based on this the corresponding semiclassical many-body theory is reviewed. It provides a unified framework for understanding a variety of quantum chaotic phenomena addressed, including random-matrix spectral correlations in many-body systems, the universal morphology of many-body eigenstates, interference effects kin to mesoscopic weak localization, and the key to the scrambling of many-body correlations characterized by out-of-time-order correlators.
Through semiclassical methods the subject of quantum chaos motivates and depends on Hamiltonian chaos research. Presented here is a selection of some of the most important Hamiltonian chaos topics that in this way get directly related to any of a variety of quantum chaos research problems. The overall emphasis is on intuitive explanations and illustrations of various ideas with the references containing more mathematically rigorous expositions. The chapter begins with a description of various useful theoretical and computational tools of chaos research, e.g.~surfaces of section, paradigms of chaos, stability analysis, and symbolic dynamics... This is followed by discussions regarding the geometry of chaos, how chaotic systems respond to perturbations, and the complexification of Hamiltonian dynamics. Most of the contents concern topics that are long established and well known by now. The exceptions involve complications that arise from the complexification of phase space variables and time. There are relatively recent results summarized regarding the appropriate branches of determinant phases and how to distinguish complex trajectories that are associated with classically allowed processes versus those associated with non-allowed ones.
Information scrambling, the process by which quantum information spreads and becomes effectively inaccessible, is central to modern quantum statistical physics and quantum chaos. These lecture notes provide an introduction to information scrambling from both static and dynamical perspectives. The spectral properties of reduced density matrices arising from Haar-random states are developed through the geometry of the unitary group and the universal results of random matrix theory. This geometric framework yields universal, model-independent predictions for entanglement and spectral statistics, capturing generic features of quantum chaos without reference to microscopic details. Dynamical diagnostics such as the spectral form factor and out-of-time-ordered correlators further reveal the onset of chaos in time-dependent evolution.
The notes are aimed at advanced undergraduate and graduate students in physics, mathematics, and computer science who are interested in the connections between quantum chaos, information dynamics, and quantum computing. Concepts such as entanglement growth, Haar randomness, random-matrix statistics, and unitary t-designs are introduced through their realization in random quantum circuits and circuit complexity. The same mathematical framework also underpins modern quantum-device benchmarking, where approximate unitary designs and random circuits translate geometric ideas into quantitative tools for assessing fidelity, noise, and scrambling efficiency in real quantum processors.
Chaos in classical systems can be characterized by Lyapunov exponents that measure the exponential divergence of nearby trajectories, but directly extending this framework to quantum mechanics has been a persistent challenge. The wavelike nature of quantum states and the non-commutative geometry of quantum phase space obstruct a straightforward generalization of classical chaos theory. Here we develop a rigorous approach to quantum chaos by leveraging quantum optimal transport theory, which provides the missing geometric foundation for measuring distances between extended quantum distributions. We define quantum Lyapunov exponents that naturally avoid divergences encountered when naïvely generalizing classical exponents, and show that in the semiclassical limit they recover the classical global expansion rate, and hence the usual maximal Lyapunov exponent when they coincide. Our framework provides a tight connection between the divergence of classical trajectories and semiclassical phase space evolution, and additionally clarifies the role of out-of-time-order correlators as diagnostics of quantum chaos. These results establish quantum optimal transport as a unifying mathematical foundation for quantum chaos theory, providing new tools to characterize dynamical behavior across the full range of quantum dynamics from simple few-body models to complex many-body systems.
Émergence du chaos dans la dynamique des systèmes à plusieurs corps classiques et quantiques S.Cette thèse étudie l'émergence du chaos dans la dynamique des systèmes à plusieurs corps, tant classiques que quantiques, à travers trois études interconnectées, aboutissant à plusieurs résultats novateurs. La recherche explore initialement les corrélations dans les circuits symplectiques duaux, fournissant une analyse approfondie des flux hamiltoniens et des systèmes symplectiques. Une contribution significative est l'introduction du modèle Ising-Swap au sein des circuits symplectiques classiques duaux, qui révèle des corrélations dynamiques en utilisant des portes symplectiques et duaux-symplectiques. Une méthode générale est proposée, permettant le calcul exact des fonctions de corrélation dynamique à deux points, qui ne s'annulent que le long des bords des cônes de lumière. Ces résultats sont validés par des simulations de Monte Carlo, montrant une excellente concordance avec les prédictions théoriques pour divers observables.L'étude suivante aborde le chaos et les conceptions unitaires, commençant par un examen des conceptions unitaires, des k-conceptions et de la mesure de Haar, et progressant vers la distribution de Porter-Thomas. Cette recherche fait progresser la compréhension des distributions universelles des chevauchements issus de la dynamique unitaire en utilisant des modèles tels que les circuits en brique et le modèle de phase aléatoire. Notamment, l'étude parvient à la diagonalisation des matrices de Toeplitz généralisées et analyse leur spectre, ce qui permet un calcul exact du potentiel de cadre, essentiel pour comprendre l'universalité de notre théorie.La dernière partie de la thèse se concentre sur la dynamique universelle hors équilibre des systèmes quantiques critiques, en utilisant la théorie des champs conformes (CFT) pour étudier les champs et les fonctions de corrélation. L'étude traite de la dynamique hors équilibre des systèmes quantiques perturbés par le bruit couplé à l'énergie. Les résultats clés incluent des analyses détaillées des corrélations à deux points, des distributions de l'entropie d'intrication et des fluctuations de la densité d'énergie, qui sont montrées être directement liées à un ensemble d'équations différentielles stochastiques (SDE). Il est démontré que l'on peut étudier ces SDE et prouver analytiquement l'existence de distributions stationnaires non triviales avec des queues de −3/2. La validation de ces résultats avec un modèle de fermions libres souligne l'universalité et la robustesse du cadre théorique présenté.Dans l'ensemble, cette thèse intègre des modèles théoriques et des cadres mathématiques pour améliorer la compréhension du chaos dans les systèmes classiques et quantiques. En reliant les résultats des circuits symplectiques, des conceptions unitaires et de la dynamique hors équilibre, elle offre un récit complet qui met en lumière les caractéristiques universelles du comportement chaotique dans la dynamique des systèmes à plusieurs corps.
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.