Nambu Quantum Mechanics on Discrete 3-Tori
arXiv:0901.2638 · doi:10.1088/1751-8113/42/27/275201
Abstract
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.
13 pages, LaTeX2e
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Cited by in corpus (5)
- Hydrodynamics on non-commutative space --A step toward hydrodynamics of granular materials--
- Generalized Hamilton-Jacobi theory of Nambu Mechanics
- Hidden Nambu mechanics II: Quantum/semiclassical dynamics
- On the Structure Constants of Volume Preserving Diffeomorphism Algebra
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