2009/10/31 by Minos Axenides, Emmanuel Floratos · 1 citation
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Attractor #Classical limit #Dissipation #Dissipative system #Noncommutative and Quantum Gravity Theories #Phase space #Quantization (signal processing) #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum dissipation #Quantum dynamics #Quantum field theory #hep-th #nlin.CD #quant-ph
paper · pdf · doi:10.1007/jhep04(2010)036
published as JHEP 1004:036,2010 · 35 pages, 4 figures, LaTex
arxiv created 2009/11/20 · openalex publication_date 2010/04/01 · arxiv updated 2014/11/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation in R3 phase space. We demonstrate that it accommodates the phase space dynamics of low dimensional dissipative systems such as the much studied Lorenz and Rössler Strange attractors, as well as the more recent constructions of Chen and Leipnik-Newton. The rotational, volume preserving part of the flow preserves in time a family of two intersecting surfaces, the so called \em Nambu Hamiltonians. They foliate the entire phase space and are, in turn, deformed in time by Dissipation which represents their irrotational part of the flow. It is given by the gradient of a scalar function and is responsible for the emergence of the Strange Attractors. Based on our recent work on Quantum Nambu Mechanics, we provide an explicit quantization of the Lorenz attractor through the introduction of Non-commutative phase space coordinates as Hermitian N × N matrices in R3. 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 Quantum Lorenz system give rise to an attracting ellipsoid in the 3 N2 dimensional phase space.