On quantum integrability of the Landau-Lifshitz model

TL;DR

This paper establishes the quantum integrability of the Landau-Lifshitz model by constructing a well-defined quantum Hamiltonian for arbitrary particle sectors, resolving longstanding issues related to operator product singularities and Hamiltonian diagonalization.

Contribution

It introduces a novel regularization and self-adjoint extension method for the Landau-Lifshitz model, enabling systematic construction of solutions and demonstrating S-matrix factorization.

Findings

Successful quantization of the Landau-Lifshitz model for arbitrary n-particle sectors.

Explicit construction of the quantum Hamiltonian and its diagonalization.

Verification of S-matrix factorization as a consequence of the model's structure.

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.