  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.

High Energy Physics - Theory;Chaotic Dynamics