Nambu quantum mechanics on discrete 3-tori

Abstract

We propose a quantization of linear, volume preserving, maps on the discrete and finite 3-torus represented by elements of the group . These flows can be considered as special motions of the Nambu dynamics (linear Nambu flows) in the three-dimensional toroidal phase space and are characterized by invariant vectors a of . We quantize all such flows, which are necessarily restricted on a planar two-dimensional phase space, embedded in the 3-torus, transverse to the vector a. The corresponding maps belong to the little group of , which is an subgroup. The associated linear Nambu maps are generated by a pair of linear and quadratic Hamiltonians (Clebsch–Monge potentials of the flow) and the corresponding quantum maps realize the metaplectic representation of on the discrete group of three-dimensional magnetic translations, i.e. the non-commutative 3-torus with a deformation parameter the Nth root of unity. Other potential applications of our construction are related to the quantization of deterministic chaos in turbulent maps as well as to quantum tomography of three-dimensional objects.