On quantum integrability of the Landau–Lifshitz model

Abstract

Positive quantization in the presence of a variable magnetic field J. Math. Phys. 52, 112101 (2011) All the groups of signal analysis from the (1+1)-affine Galilei group J. Math. Phys. 52, 103504 (2011) We investigate the quantum integrability of the Landau-Lifshitz ͑LL͒ 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 ͓Lett. Math. Phys. 15, 357 ͑1988͔͒ 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.

Quantum integrability of the Alday-Arutyunov-Frolov model

“…Even though the quantum inverse scattering method is a relatively straightforward procedure, in the AAF model the main difficulty lies in the type of the interaction Hamiltonian, namely, in the singular behavior of the quantum mechanical Hamiltonian and all other conserved charges. Let us note, that in the Landau-Lifshitz model one encounters exactly the same type of singularity, and the development of the quantum inverse scattering method required considerable effort and careful analysis of operator products and their regularization, as well as the construction of the correct Hilbert space [16][17][18]32]. It has also been shown that the self-adjointness of the operators is essentially equivalent to the S-matrix factorization.…”

“…Therefore the n-loop scattering amplitude corresponds to the product of n of these modified vertices: 15) where in the last step, we took the outcoming pseudo-particles k 1 and k 2 on-shell. We are now in the position to obtain the full scattering amplitude: 16) from which, by comparison with (4.1), we read off the S-matrix for the scattering of twoparticle,…”

“…Since we assumed the scattering pseudo-particles to have positive energy, the condition ∆ 0 + ω(∆) cannot be satisfied for any p and k, and therefore, we can disregard the second delta function above. We can conveniently rewrite the positive-energy delta function as follows: 16) so that the delta function for overall energy-momentum conservation can be further simplified, allowing us to use the identity (4.7) and our ordering of initial momenta to trivially compute the spinorial products, along the same lines as with the two-particle case. After long but straightforward calculations the delta term contribution reduces to:…”

“…Such singularities in the context of integrable systems are hard to handle. One example where such difficulties are present is the Landau-Lifshitz model [16][17][18], which is generated by a string in the su(2) subsector. It has been shown that the complete understanding of the quantum inverse scattering method is only possible upon careful analysis of the singularities, the associated quantum operators, and the reconstruction of the correct Hilbert space.…”

Anisotropic Landau-Lifshitz sigma models from q-deformed AdS5×S5 superstrings

“…The next is to consider larger subsectors, or directly the full sector (For a supersymmetric generalization of the undeformed case, see [76,77]). It would also be nice to consider the LLSMs obtained here at the quantum level, for example, by following [78][79][80][81].…”