Hirota equation and Bethe ansatz

At present it is gradually realized that quantum and classical integrable systems have a deeper connection than the common assertion that the latter are obtained from the former
5.2 Generalized Baxter's relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.3 Factorization formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6 A1-models
33
6.1 General solution to discrete Liouville equation . . . . . . . . . . . . . . . . . . 33
6.2 Di erent forms of Baxter's relation . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6 On elliptic and trigonometric cases . . . . . . . . . . . . . . . . . . . . . . . . 15
Joint Institute of Chemical Physics, Kosygina str. 4, 117334, Moscow, Russia and ITEP, 117259, Moscow, Russia
2.2 Quantum monodromy matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Quantum transfer matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2
scattering method has a very di erent meaning in classical and quantum contexts, to say nothing about more speci c tools having no analogues like the nite-gap integration technique (from the classical side) or the Bethe ansatz (from the quantum side).
7 Conclusion
39
1 Introduction
Modern developments of the theory of integrable systems are characterized by substantial interaction between methods speci c for classical and quantum cases which so far did not intersect. For a long time classical and quantum integrable models were studuied on their own grounds that was quite natural since the very questions used to be asked in the both cases usually did not have common points. An advantage of exploring interrelations between classical and quantum methods became clear already in the early days of the quantum inverse scattering method (QISM) 1]. As a result of its classical limit, the notion of r-matrix was introduced. The r-matrix plays a fundamental role in the advanced hamiltonian theory of classical integrable equations, providing a universal frame of quadratic Poisson bracket algebras.
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Nested Bethe ansatz as a chain of Backlund transformations
20
4.1 Zero curvature representation and auxiliary linear problems . . . . . . . . . . . 20
4.2 Functions Qt(u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Analytic conditions in u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
well. This phenomenon was "experimentally" observed in many di erent examples. The earliest ones are non-linear equations for correlation functions in the Ising model obtained in 3]. In particular, it has been shown 4], 5] that the pair spin correlation function in the Ising model on square lattice in the self-dual point obeys the di erence analogue of the well known Toda chain equation. The list of examples of a similar type was substantially enlarged about ten years ago 6, 7], when it was discovered that correlation functions in many other integrable models of quantum eld theory obey integrable (in general di erential-di erence) classical equations. Examples of other types include thermodynamic Bethe ansatz equations in the model-independent form 8] and functional relations for transfer matrices in quantum integrable models 9, 10]. They can be represented in the form of the purely classical di erence Hirota equation 11] (a di erence analogue of the two-dimensional Toda chain). Other aspects of the classical-quantum connections can be found in 12]{ 16].
2.4 Functional relations and determinant formulas . . . . . . . . . . . . . . . . . . 13
2.5 Bilinear form of the fusion rules for rectangular diagrams . . . . . . . . . . . . 14
4.3 Bethe equations as a dynamical system in discrete time . . . . . . . . . . . . . 24
5 Di erence equations for the functions Qt
26
5.1 General solution to the bilinear relations . . . . . . . . . . . . . . . . . . . . . 27
In other respects, theories of classical and quantum integrable systems so far developed separately and looked like two di erent branches of mathematical physics with their own notions, methods and traditions. For instance, even such a universal approach as the inverse