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SMSTC Supplementary Module: Classical and …SMSTC Supplementary Module: Classical and Quantum...
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SMSTC Supplementary Module:Classical and Quantum Integrable Systems
Bernd SchroersDepartment of MathematicsHeriot-Watt University, UK
Perth, 2 October 2019
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What is this module about?
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The oldest classical integrable system: Kepler’s problem
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Classical integrability in a nutshell
We haveI A symplectic manifold M of dimension 2n: physically the phase
space or space of ‘positions and momenta’I Poisson brackets {f ,g} for f ,g ∈ C∞(M)
I A Hamiltonian H ∈ C∞(M) which generates time evolutionaccording to
f = {H, f}.
We want:I n − 1 further functions f1, . . . , fn−1 so that
{H, fi} = 0, and {fi , fj} = 0, i , j = 1, . . . , (n − 1).
I An understanding of the geometrical flow generated by H.
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Contents
1. Foundations: Review of manifolds, differential forms, vectorfields and Lie derivatives; Lie groups and Lie algebras.
2. Introduction to symplectic geometry and mechanics:Hamiltonian mechanics; Poisson brackets; symmetry andNoether’s theorem in Hamiltonian mechanics; moment(um)maps; Liouville theorem; examples
3. Poisson-Lie structures I: Poisson manifolds and symplecticleaves; co-adjoint orbits; Poisson-Lie algebras and Poisson-Liegroups; examples.
4. Poisson-Lie strcutures II: Co-boundary Poisson-Lie algebrasand the classical Yang-Baxter equations; classical doubles;Sklyanin brackets; dressing action and symplectic leaves;examples.
5. Classical integrable systems: Lax pairs and classicalr-matrices; construction of integrable systems out of co-boundaryLie bi-algebras; applications (e.g Toda chain)
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Quantum integrability: the hydrogen atom
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Quantum integrability in a nutshell
We haveI A Hilbert space H: the space of quantum states.I Self-adjoint operators A : H → H: the observables of the theory.I A particular self-adjoint operator, called the Hamiltonian H, which
generates time evolution according to
A = i~[H,A].
We want:I A (possibly infinite) family operators Ai , i = 1,2 . . . which satisfy
[Ai ,Aj ] = [Ai ,H] = 0, i , j = 1, . . .
I The ground state of HI The discrete spectrum of H and the Ai , i = 1,2 . . .
I Other quantum properties of the system such as correlationfunctions.
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Contents
1. Statistical lattice models & combinatorics: Motivatingexamples from enumerative combinatorics; partition functions;transfer matrices; quantum R-matrices; the quantumYang-Baxter equation;
2. The Yang-Baxter Algebra & the Bethe Ansatz: Monodromymatrices and the Yang-Baxter algebra; diagonalising the transfermatrix; Bethe equations; guiding example: the 6-vertex model onthe cylinder and the torus
3. Quantum Groups I: Axioms and examples; the Yang-Baxteralgebra as a quantum group; the 6-vertex model and XXZspin-chain as motivating example
4. Quantum Groups II: Representation theory; Uq(sl2); theR-matrix; short-exact-sequences and the XXZ fusion hierarchy
5. Quantum integrability at the boundary: The reflectionequation and K-matrices; examples of diagonal andnon-diagonal boundary conditions; the XXZ Hamiltonian andBethe equations in the presence of boundaries;
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Bibliography
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Bibliography
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Bibliography
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Bibliography
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Bibliography
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Who should take this module?
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Prerequisites
I Some understanding of differentiable manifolds and differentialforms, group theory
I Some familiarity with Lie algebras and Lie groups would behelpful but will not be assumed.
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Take this course if you are interested in ...
I A rapidly evolving field which connects algebra, geometry,topology and physics
I A geometrical formulation of classical mechanics in the languageof symplectic and Poisson structures
I The classical Yang-Baxter equation and how it gives rise toPoisson-Lie structures and integrable systems
I The role of quantum groups in quantum integrable systemsI A toolkit for solving problems in quantum field theory, string
theory and condensed matter physics which are not (easily)amenable to other techniques
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Who is teaching this module?
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The team
José Figueroa O’Farrill U of Edinburgh
Bernd Schroers Heriot-Watt
Robert Weston Heriot-Watt
Christian Korff U of Glasgow