The aim of this report is to introduce the reader to the field of integrability in classical and quantum physics. The report begins with a reminder of Hamiltonian mechanics and explains the notion of Liouville integrability. The introduction of the Lax formalism is then used to show a method for generating conserved quantities that lends itself well to a generalisation to classical field theory. In this context, the fundamental concepts of r-matrix, monodromy and transfer matrix are presented. To close the section on classical integrability, the classical inverse scattering method is presented on the example of the Korteweg - de Vries equation. Solving the spin chain XXX 1/2 using the coordinate Bethe ansatz is then used to open the door to the field of quantum integrability. The generalisation of the classical concepts introduced previously to the quantum domain will allow us to implement the algebraic Bethe ansatz method, which is a more general approach. Using this procedure, the examples of the spin chains XXX s and XXZ s are treated, clearly highlighting the importance of symmetries in the field of integrability. Finally, a brief overview of the Gauge/Bethe correspondence is presented.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:uu-529626 |
| Date | January 2024 |
| Creators | Morand, Samuel |
| Publisher | Uppsala universitet, Teoretisk fysik |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Student thesis, info:eu-repo/semantics/bachelorThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | FYSAST ; FYSPROJ1342 |
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