Nambu Quantum Mechanics on Discrete 3-Tori

We propose a quantization of linear, volume preserving, maps on the discrete and finite 3-torus T_N^3 represented by elements of the group SL(3,Z_N). 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 T_N^3. 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 the vector a in SL(3,Z_N) which is an SL(2,Z_N) 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 SL(3,Z_N) on the discrete group of three dimensional magnetic translations i.e. the non-commutative 3-torus with deformation parameter the N-th 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.