Classical and Quantum Integrable Systems
From Liouville's Theorem to the Bethe Ansatz. Each lecture's core is what is taught in its two-hour slot; reading and tutorial material is marked as such. The PDF of every lecture is the same text as its page.
Lecture 0
Why integrability? Where exact solvability shows up in nature, what it costs, and the map of the whole course.
Part I — Classical Integrable Systems
The classical structures, built properly: phase space, the Poisson bracket and differential forms, the symplectic form and canonical maps, the Liouville–Arnold theorem and action–angle variables, what fails without integrability, and Lax pairs with a spectral parameter — ending with a proof that the Toda ring is integrable.
- Lecture1
Phase space and flows
4 definitions · 1 worked example · 7 problems
→ - Lecture2
Two languages: the Poisson bracket and differential forms
8 definitions · 3 worked examples · 6 problems
→ - Lecture3
The exterior derivative, Stokes, and the symplectic form
6 definitions · 3 worked examples · 7 problems
→ - Lecture4
One structure: canonical maps and Lagrangian submanifolds
4 definitions · 5 worked examples · 8 problems
→ - Lecture5
The Liouville–Arnold theorem
5 definitions · 4 worked examples · 5 problems
→ - Lecture6
Action–angle variables
4 definitions · 7 problems
→ - Lecture7
When integrability fails
3 definitions · 1 worked example · 7 problems
→ - Lecture8
Lax pairs, and the spectral parameter
1 definition · 7 problems
→ - Lecture9
Why the integrals commute: the Toda ring is integrable
6 definitions · 3 worked examples · 7 problems
→
Part II — Quantum Integrable Systems
Exact solution in quantum mechanics: the coordinate Bethe ansatz for the Heisenberg chain and the Lieb–Liniger gas, scattering without diffraction and the Yang–Baxter equation, the quantum inverse scattering method and the algebraic Bethe ansatz, the thermodynamic limit, free fermions, and why integrable systems do not thermalise — ending with generalised hydrodynamics.
- Lecture10
The coordinate Bethe ansatz: the Heisenberg chain
7 problems
→ - Lecture11
The Lieb–Liniger gas
1 definition · 7 problems
→ - Lecture12
Scattering without diffraction: the open Toda chain
7 problems
→ - Lecture13
Three-body consistency: the Yang–Baxter equation
7 problems
→ - Lecture14
The quantum inverse scattering method, and its classical limit
7 problems
→ - Lecture15
The algebraic Bethe ansatz
7 problems
→ - Lecture16
The thermodynamic limit: densities and ground states
7 problems
→ - Lecture17
Dressing, excitations and the thermodynamic Bethe ansatz
2 definitions · 7 problems
→ - Lecture18
Free fermions: the transverse-field Ising chain
2 definitions · 7 problems
→ - Lecture19
Why integrable systems do not thermalise
3 definitions · 7 problems
→ - Lecture20
Generalised hydrodynamics, and the quantum Newton’s cradle
7 problems
→
Optional lectures
Three lectures off the path, each building on it and none needed by it: quantum chaos and level statistics, the Calogero–Moser system classical and quantum, and nesting and the Hubbard model.