Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2012-01-27
Nonlinear Sciences
Chaotic Dynamics
This paper has been withdrawn by the author due to a misinterpretation that invalidates some of the conclusions
Scientific paper
Here it is shown that the most general Parisi-Sourlas-Wu stochastic quantization procedure applied to any stochastic differential equation (SDE) leads to a Witten-type topological field theory - a model with a global topological Becchi-Rouet-Stora-Tyutin supersymmetry (Q-symmetry). Q-symmetry can be dynamically broken only by (anti-)instantons - ultimately nonlinear sudden tunneling processes of (creation)annihilation of solitons, e.g., avalanches in self-organized criticality (SOC) or (creation)annihilation of vortices in turbulent water. The phases with unbroken Q-symmetry are essentially markovian and can be understood solely in terms of the conventional Fokker-Plank evolution of the probability density. For these phases, Ito interpretation of SDEs and/or Martin-Siggia-Rose approximation of the stochastic quantization are applicable. SOC, turbulence, glasses, quenches etc. constitute the "generalized turbulence" category of stochastic phases with broken Q-symmetry. In this category, (anti-)instantons condense due to the non-potential Novikov-type driving that can be interpreted as an external energy reservoir. This dynamically mixes the probability distribution and the wavefunctions of non-trivial Fadeev-Popov ghost content. Even for white noises, the stochasticity can not be described in terms of the probability distribution only. Such systems may be said to possess "genuine" non-Markovianity. Representing instanton modulii and having the meaning of Ruelle-Pollicott resonances, gapless goldstinos explain stochastic self-similarity of generalized turbulence, e.g., algebraic statistics of avalanches in SOCs and algebraic power spectrum of turbulent water. It is pointed out that stochastic quantization is closely related to the concept of negative probabilities that represent the freedom of the stochastic system to chose among various possible solutions of SDE.
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