as a phase space distribution

in quantum statistical mechanics

**Rigoberto Hernandez,
Jianshu Cao, and
Gregory A. Voth**

Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6323

The phase space formulation of quantum statistical mechanics using the Feynman path centroid density offers an alternative perspective to the standard Wigner prescription for the classical-like evaluation of equilibrium and/or dynamical quantities of statistical systems. The use of this formulation has been implicit in recent work on quantum rate theories, for example, in which the centroid density distribution replaces the classical Boltzmann distribution. In order to further understand the approximations involved in this and similar transcriptions, the present work elaborates and clarifies the issue of operator ordering in a rigorous centroid-based formulation. In particular, through the use of the Weyl correspondence, a precise definition of the centroid symbol of operators and their products is presented. Though we fall short of finding the algebraic structure tantamount to that found in the Weyl symbols - of which the Wigner distribution is an example - the resulting expressions have internal consistency and are amenable to approximate evaluation through cumulant expansions.

- i Introduction
- ii Phase space Feynman path centroid density
- iii Weyl Correspondence Rule
- iv Equilibrium Averages
- v Imaginary-Time Correlation Functions
- vi Cumulant Expansions
- vii Real-Time Correlation Functions
- viii Concluding Remarks
- References