On quantum integrability of the Landau-Lifshitz model

Abstract

We investigate the quantum integrability of the Landau-Lifshitz model and solve the long-standing problem of finding the local quantum Hamiltonian for the arbitrary n-particle sector. The particular difficulty of the LL model quantization, which arises due to the ill-defined operator product, is dealt with by simultaneously regularizing the operator product, and constructing the self-adjoint extensions of a very particular structure. The diagonalizibility difficulties of the Hamiltonian of the LL model, due to the highly singular nature of the quantum-mechanical Hamiltonian, are also resolved in our method for the arbitrary n-particle sector. We explicitly demonstrate the consistency of our construction with the quantum inverse scattering method due to Sklyanin, and give a prescription to systematically construct the general solution, which explains and generalizes the puzzling results of Sklyanin for the particular two-particle sector case. Moreover, we demonstrate the S-matrix factorization and show that it is a consequence of the discontinuity conditions on the functions involved in the construction of the self-adjoint extensions.

17 pages

References in corpus (4)

• Landau-Lifshitz sigma-models, fermions and the AdS/CFT correspondence
• On S-Matrix factorization of the Landau-Lifshitz model
• The S-matrix of the Faddeev-Reshetikhin Model, Diagonalizability and PT Symmetry
• Quantum Landau-Lifshitz model at four loops: 1/J and 1/J^2 corrections to BMN energies

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• On S-Matrix factorization of the Landau-Lifshitz model
• On the equivalence theorem for integrable systems
• The r-matrix of the Alday-Arutyunov-Frolov model
• Thermodynamics of the quantum $su(1,1)$ Landau-Lifshitz model
• Higher charges and regularized quantum trace identities in su(1,1) Landau-Lifshitz model
• Exceptional solutions to the eight-vertex model and integrability of anisotropic extensions of massive fermionic models