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Quantum \(2+1\) evolution model. (English) Zbl 0962.81023

Summary: A quantum evolution model in \(2+1\) discrete spacetime, connected with a 3D fundamental map \(\mathbb{R}\), is investigated. Map \(\mathbb{R}\) is derived as a map providing a zero curvature of a 2D linear lattice system called ‘the current system’. In a special case of the local Weyl algebra for dynamical variables the map appears to be canonical and it corresponds to the known operator valued \(\mathbb{R}\)-matrix. The current system is a type of the linear problem for the \(2+1\) evolution model. A generating function for the integrals of motion for the evolution is derived with the belp of the current system. Thus, the complete integrability in 3D is proved directly.

MSC:

81R12 Groups and algebras in quantum theory and relations with integrable systems
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