Hamiltonian and Lagrangian structures in an integrable hierarchy: space-time duality
Speaker: Vincent Caudrelier, City University of London
Abstract: In the context of integrable systems, the object known as the classical r-matrix emerged from two remarkable discoveries: 1) integrable PDEs can be written as a Hamiltonian system by introducing an appropriate Poisson bracket of the (infinite dimensional) phase space; 2) combined with the Lax pair formulation and motivated by the question of canonical quantization of such Hamiltonian systems, certain fundamental Poisson brackets can be written as a commutator involving a special matrix (the r matrix). Understanding these observations on a fundamental level led to the discovery of Poisson-Lie groups and the modern theory of classical integrable systems. However, from the point of view of the independent variables (x,t) involved in the original PDEs, the entire theory is asymmetric. It is based only on "one half" of the Lax pair: the matrix describing the space evolution in the auxiliary problem at fixed time. In this talk, I will introduce some motivations to treat both independent variables x and t on equal footing. Using ideas from covariant field theory, the following observation will be made: it is possible to construct a new Poisson bracket on the phase space of an integrable hierarchy for which the fundamental Poisson brackets of the other half of the Lax pair have exactly the same structure as the usual one, with the same r matrix. Perhaps ironically, this observation actually raises more questions than it answers and I will mention some of them.
Room Reservation Information
Room Number: 106 McAllister
Time: 2:30pm - 3:30pm