Classical and quantum inverse problem method and generalized Bethe ansatz

Leon Armenovich Takhtadzhyan

L A Takhtadzhan, Lyudvig Dmitrievich Faddeev

CONTENTS Introduction § 1. Classical statistical physics on a two-dimensional lattice and quantum mechanics on a chain § 2. Connection with the inverse problem method § 3. The six-vertex model § 4. Generating vectors and permutation relations § 5. The general Bethe Ansatz § 6. Integral equations Conclusion Appendix 1 Appendix 2 References

Evgeny Konstantinovich Sklyanin, Leon Armenovich Takhtadzhyan, Lyudvig Dmitrievich Faddeev

V. E. Korepin, N. M. Bogoliubov, Anatoli G. Izergin

The quantum inverse scattering method relates the Bethe Ansatz to the theory of classical completely integrable differential equations. These are sometimes called soliton equations. The modern way to solve them is called the classical inverse scattering method. In a sense this is a nonlinear generalization of the Fourier transform. In this Part the quantum inverse scattering method is expounded. The main statements of the classical inverse scattering method necessary for quantization are given in Chapter V where the Lax representation is introduced. The Hamiltonian structure of integrable models is also discussed along with the infinite number of integrals of motion. The most convenient method of analyzing the Hamiltonian structure relies on the classical r -matrix. Some concrete models will be considered. Chapter VI is devoted in particular to the quantum inverse scattering method. The R-matrix, which is the main object of this method, is introduced. The Yang-Baxter equation for the R-matrix is discussed. The main statements of the method are given and a number of examples are presented. The algebraic formulation of the Bethe Ansatz, one of the main achievements of the quantum inverse scattering method, is presented in Chapter VII. The notion of the determinant of the transition matrix in the quantum case is introduced in this chapter. (This is closely related to the concept of the antipode in quantum groups.)

Leon A. Takhtajan

Antal Jevicki

Progress in understanding and solving a large class of two-dimensional nonlinear quantum field theories is reviewed. The discovery and development of the inverse scattering method for solving partial differential equations, and development of new perturbative methods are discussed. The generalized Bethe-ansatz method and its application to exactly diagonalize a fermionic problem are covered. 52 references. (GHT)