Classical and Quantum Chaos from Foundations to Holography

Abstract

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.