Predrag Stojkov	Applications of Weyl-Moyal-Wigner approach to quantization on statistical and chaotic systems

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nce upon a time there was only a Classical Mechanics. He ruled the world in solitude until mighty Gods of Contradiction, Inspiration and There_Must_Be_Order_Attitude gathered to create new creature. And there he was, brave new beast, Quantum Mechanics. He was strong, fast and clever. He was predestined to be new ruler. And completely different than old one. At least he was thinking so. Only decades later, when he was seventy years old, he started to realize how similar he is the old ruler, Classical Mechanics.

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• Another research project that I have performed at Brown (under my own direction) is in the Weyl-Moyal's approach to quantization. This interest is the offspring of a third possible approach to quantum chaos, a initiated recently by W.H. Zurek. His merit is to insist on a description of the quantum system (in terms of the Weyl-Moyal prescription) as the dynamics of Wigner's distribution function in classical phase space of the system, where dynamics is defined by Moyal's bracket, instead of its classical limit, Poisson bracket. This description allowed him to gain an extremely intuitive picture on role of competition between quantum diffusion and tendency towards forming homoclinic tangle present in classically chaotic system. This makes possible a simple, analytic expressions for a time scale when the behavior of a quantum system breaks away from its classical counterpart (such a behavior is visible in numerical simulations). I am interested in a possible extension of such an intuitive picture to the case of infinitely many degrees of freedom (fields), where additional aspect of spatially chaotic motion (turbulence) appears.

• While learning Moyal's formalism, I have found some simple but intriguing results in its application to statistical systems. For example, for simplest systems that can be reduced to a set of harmonic oscillators, the quantum partition function is equivalent to a classical partition function for a classical system of oscillators with different temperature and nonzero chemical potential ("quantum temperature" and "quantum chemical potential"). These results can be extended in one direction to the case of Grassmann oscillators, and in other direction towards infinite many degrees of freedom (to noninteracting fields and to simple strings). A paper describing these results is in preparation.

• In this set of papers I am dealing with Weyl-Moyal's formulation of Quantum Mechanics.
  
  • What is Moyal's Approach to Quantization?
  • Why is it so interesting?
  • Quantum Statistical Function throught Moyal Quantization
    • General case:
    • Functional (Path-integral) representation: Nice but usless
    • Direct approach:
      • LHO
      • N-dim HO
      • N=1 & N=2 Fermionic HO
      • D=2, D=3 & D>3 Spin
  • Literature about Moyals Approach
  
  

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Updated on June 24th 1997
http://www.het.brown.edu/people/stojkov/research/moyal/moyal.index.html