p,q-Duality and Hamiltonian Flows in the Space of Integrable Systems or Integrable Systems as Canonical Transforms of the Free Ones

Details p,q-Duality and Hamiltonian Flows in the Space of Integrable Systems or Integrable Systems as Canonical Transforms of the Free Ones p,q-Duality and Hamiltonian Flows in the Space of Integrable Systems or Integrable Systems as Canonical Transforms of the Free Ones

2001-07-12

arxiv.org/abs/hep-th/0107114v3

Phys.Lett. B524 (2002) 217-226

Physics

High Energy Physics

High Energy Physics - Theory

LaTeX, 16 pages (acknowledgements improved)

Scientific paper

10.1016/S0370-2693(01)01267-9

Variation of coupling constants of integrable system can be considered as canonical transformation or, infinitesimally, a Hamiltonian flow in the space of such systems. Any function $T(\vec p, \vec q)$ generates a one-parametric family of integrable systems in vicinity of a single system: this gives an idea of how many integrable systems there are in the space of coupling constants. Inverse flow is generated by a dual "Hamiltonian", $\widetilde T(\vec p, \vec q)$ associated with the dual integrable system. In vicinity of a self-dual point the duality transformation just interchanges momenta and coordinates in such a "Hamiltonian": $\widetilde T(\vec p, \vec q) = T(\vec q, \vec p)$. For integrable system with several coupling constants the corresponding "Hamiltonians" $T_i(\vec p, \vec q)$ satisfy Whitham equations and after quantization (of the original system) become operators satisfying the zero-curvature condition in the space of coupling constants: [ d/dg_a - T_a(p,q), d/dg_b - T_b(p,q) ] = 0. Some explicit formulas are given for harmonic oscillator and for Calogero-Ruijsenaars-Dell system.

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