Matrix Quantization of Turbulence

Abstract

Based on our recent work on Quantum Nambu Mechanics \cite{af2}, we provide an explicit quantization of the Lorenz chaotic attractor through the introduction of Non-commutative phase space coordinates as Hermitian $N \times N$ matrices in $R^{3}$. For the volume preserving part, they satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction it violates the quantum commutation relations. We demonstrate that the Heisenberg-Nambu evolution equations for the Matrix Lorenz system develop fast decoherence to N independent Lorenz attractors. On the other hand there is a weak dissipation regime, where the quantum mechanical properties of the volume preserving non-dissipative sector survive for long times.Comment: 14 pages, Based on invited talks delivered at: Fifth Aegean Summer School, "From Gravity to Thermal Gauge theories and the AdS/CFT Correspondance", September 2009, Milos, Greece; the Intern. Conference on Dynamics and Complexity, Thessaloniki, Greece, 12 July 2010; Workshop on "AdS4/CFT3 and the Holographic States of Matter", Galileo Galilei Institute, Firenze, Italy, 30 October 201