Canonical and noncanonical variables, Baxter’s Q-operator and the XXX model

Baxter’s Q-operator for the XXX model is analysed to study the different roles played by the canonical and noncanonical variables using the formalism of Sklyanin. In this approach to the study of Bäcklund transformation (BT) and Baxter’s Q-operator one requires two Lax operators obeying the same Poisson algebra (i.e. having the same classical r matrix). Usually the nonlinear variables in the Lax operator are canonically conjugate quantities. In this communication we have shown that even when the variables in one Lax operator are not canonical, it is still possible to construct the BT and Q-operator by a proper representation of the corresponding nonlinear variables. The price for this is that, the BT is not easily interpretable as a canonical transformation in the conventional sense of the term. However the Q-operator for the XXX chain turns out to be similar to that obtained by Derkachov [J Phys A 32 (1999) 316] using a similar representation but in a different manner. It is important to note that although, in contrast to the results obtained by Kuznetsov et al. [J Phys A 33 (2000) 171], the Q-operator depends on two adjacent sites (xi,xi−1), its relevant properties can be explicitly established.