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One-dimensional chains, matrix groups and quantization of Poisson structures

Sławianowski J.

Journal of Technical Physics

1998

Vol. 39, no 2

163-187

Suggested are some toy models of one-dimensional multiparticle systems with internal parameters responsible for the interparticle interactions. Classical and quantum formulations are presented. The system consists of n particles moving along the straight-line or circle and of n(n-1) interaction channels, a kind of discretized field medium. Interaction parameters and particles positions satisfy a closed system of equations of motion. The model is essentially nonlinear, exhibits certain realistic features of intermolecular forces, and is, in principle, rigorously solvable. Mathematically, it is based on Hamiltonian systems on the matrix groups GL(n,R),U(n). There are some links with the theory of completely integrable lattices due to Cologero, Moser, Sutherland.