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Nambu quantum mechanics on discrete 3-tori. (English) Zbl 1168.81339

Summary: We propose a quantization of linear, volume preserving, maps on the discrete and finite 3-torus \(\mathbb{T}_N^3 \) represented by elements of the group \(SL(3,\mathbb{Z}_N) \). 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 \(\mathbb{T}_N^3 \). 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 \(a \in SL(3,\mathbb{Z}_N) \), which is an \(SL(2,\mathbb{Z}_N) \) 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 \(SL(3,\mathbb{Z}_N) \) on the discrete group of three-dimensional magnetic translations, i.e. the non-commutative 3-torus with a 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.

MSC:

81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
81R15 Operator algebra methods applied to problems in quantum theory
81R60 Noncommutative geometry in quantum theory
81T99 Quantum field theory; related classical field theories
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