Quantum lattice KdV equation

A quantum theory is developed for a difference-difference system which can serve as a toy-model of the quantum Korteveg-de-Vries equation. Introduction This Letter presents an example of a completely integrable ‘discrete-space-time quantum model’ whose Heisenberg equations of motion have the form φ (τ, n) φ (τ, n− 1) + λ φ (τ, n− 1) φ (τ − 1, n− 1) = λ φ (τ, n) φ (τ − 1, n) + φ (τ − 1, n) φ (τ − 1, n − 1). (1) By ‘discrete...model’ we mean (i) an algebra ‘of observables’ Φ, whose generators φ n are labeled by integer numbers n which are regarded as a (discrete) spatial variable; together with (ii) an automorphism Q, whose sequential action x (0) ≡ x ∈ Φ Q : . . . 7→ x (τ − 1) 7→ x (τ) 7→ x (τ + 1) 7→ . . . is viewed as the (discrete) time evolution. Thus, we intend to produce a pair Φ&Q such that the evolution φ n(τ) of generators, in the natural notation φ (τ, n) ≡ φ n(τ), obeys the system (1). Complete integrability is understood as the existence of a commutative subalgebra ‘of conservation laws’ preserved under time evolution and spanning, in a sense, half of the algebra of observables: it is commonly believed that a Hamiltonian system may be either ‘completely’ nonintegrable possessing only a few conservation laws due to its manifest symmetries, or completely integrable enjoying a whole lot of conservation laws, one per degree of freedom. The commutative subalgebra which we encounter in this Letter definitely contains a lot of conservation laws but the question of how many is left to be answered elsewhere. Actually, we deal here not with a single model but rather with a family of them Φ&Q(λ), each model being related to a certain value of a complex parameter λ in (1). Moreover, all their evolution automorphisms Q(λ) turn out to be mutually commuting and sharing the common subalgebra of conservation laws. The order of presentation is as follows. In Section 1 we introduce an algebra of observables which is basically the same lattice U(1) exchange algebra which appeared already in [FV93, 94]. Naturally, the behaviour of that algebra depends on the value of a constant q involved in the commutation relations. For simplicity we shall assume that q is a root of unity. On leave of absence from Saint Petersburg Branch of the Steklov Mathematical Institute, Fontanka 27, Saint Petersburg 191011, Russia After some preliminaries (Sections 2 and 3) we pick in Section 4 from the algebra of observables a set of ‘Fateev-Zamolodchikov R-matrices’ R n(λ) which satisfy a chain of Yang-Baxter equations R n−1(λ) R n(λμ) R n−1(μ) = R n(μ) R n−1(λμ) R n(λ) amounting to mutual commutativity Q (λ) Q (μ) = Q (μ) Q (λ) of their properly defined ‘ordered products’. The family Q (λ) provides the demanded commuting (inner) evolution automorphisms φ (τ, n) ≡ Q φ n Q τ and doubles as their common conservation laws. In Section 6 we eventually establish that these evolutions do solve the equations (1). Prior to that, in Section 5, we discover one more face of the family Q (λ). As a function of λ it proves to satisfy the Baxter equation [Bax] q Nl Q (λ) t (λ) = α (λ) Q (qλ) + δ (λ) Q (qλ) which in turn makes Bethe ansatz equations to emerge in a purely algebraic context. While the quantum system (1) seems to be a recent invention (it has been looked at, albeit from a somewhat different angle, in [FV92, 94]), its classical counterpart has been around for quite a while. It was introduced (in a somewhat different form) by Hirota back in 1977 [H] as an integrable differencedifference approximation of the sine-Gordon equation but eventually proved far more universal making perfect sense as a lattice counterpart of numerous integrable equations including that of Korteveg and de Vries. To conclude the Introduction we shall list various continuous limits and alternative forms of the (classical) system (1). This should give some idea of where our model fits into the scheme of things accepted in Soliton Theory. For more of that and a comprehensive list of relevant references see [NC]. Pairs of integers (τ, n) may naturally be viewed as vertices of a plane lattice. In the shorthand notation A=(τ,n−1) B=(τ,n) C=(τ−1,n−1) D=(τ−1,n) t t t t t t t t t t t for a quartet of vertices enclosing some elementary cell of that lattice, with subscripts instead of parenthesized arguments and without bold letters reserved for the quantum case, the classical equations (1) read φ B φ A − φ D φ C − λ (φ B φ D − φ A φ C ) = 0. • At least one continuous limit is already quite apparent. Let us put the lattice on the coordinate plane (t, x) in such a way that vertices (τ, n) go to points (λ−1∆τ,∆n):