Sine-Gordon Solitons vs. Calogero-Moser Particles (Integrable Systems)
Solitons are particle-like solutions to nonlinear partial differential equations. In particular, the relativistically invariant sine-Gordon equation phi_xx-phi_tt=sin(phi) admits n-soliton solutions. There exist relativistic integrable n-particle systems of Calogero-Moser type that are intimately related to the sine-Gordon solitons. Specifically, the n-soliton solutions can be expressed in terms of the space- and time-translation generators of the n-particle systems. This yields soliton space-time trajectories that make it possible to follow the individual solitons during their collisions. At this classical level the particle-soliton correspondence is in great shape.
At the quantum level the corresponding particle-soliton equivalence has thus far only been established for the case n=2. This involves a new special function, which may be viewed as a 'relativistic' generalization of the Gauss hypergeometric function. Specializing to the sine-Gordon coupling, it yields a unitary eigenfunction transform for the n=2 particle Hamiltonians which encodes the bound-state spectrum and scattering. These are identical to those for the 2-body subspace of the sine-Gordon quantum field theory.
It is a major research project to establish the equivalence for n>2. To achieve this aim, various subproblems need to be tackled. They vary in difficulty, and for some of them there are very promising leads.
keywords: integrable systems, solitons, sine-Gordon equation, relativistic quantum mechanics, Calogero-Moser type systems, Lax matrices, action-angle maps
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