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The Journal of Chemical Physics

On the Feynman path centroid density as a phase space distribution in quantum statistical mechanics

Rigoberto Hernandez, Jianshu Cao, Gregory A. Voth

Published Sep 22, 1995• 75 citations
Publisher

Abstract

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.

DOI
10.1063/1.470588

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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.
A formulation of quantum statistical mechanics is discussed in which the Feynman path centroid density in Feynman path integration is recast as the central statistical distribution used to average equilibrium and dynamical quantities. In this formulation, the path integral centroid density occupies the same role as the Boltzmann density in classical statistical mechanics. Therefore, the statistical ensemble of imaginary time path centroid configurations provides the distribution which is used to average the appropriately formulated effective operators and imaginary time correlation functions. An accurate renormalized diagrammatic perturbation theory for the centroid density and centroid-constrained imaginary time propagator will also be described with particular emphasis given to the mathematical advantages arising from the centroid-based formulation. The present paper is concerned with the calculation of equilibrium properties from the centroid perspective, while the companion paper describes a centroid-based formalism for calculating dynamical time correlation functions.
A formalism is presented in this paper which, for the first time, establishes the theoretical basis for the quantum time evolution of path integral centroid variables and also provides clear motivation for using these variables to study condensed phase quantum dynamics. The equilibrium centroid distribution is first shown to be a well-defined distribution function which is specific to the canonical density operator. A quantum mechanical quasi-density operator (QDO) is associated with each value of the distribution so that, upon application of the standard quantum mechanical formalism, the QDO can be used to provide a rigorous definition of both static and dynamical centroid variables. Various properties of the dynamical centroid variables are derived, including the perspective that the centroid constraint on the imaginary time paths introduces a nonstationarity in the equilibrium ensemble which, in turn, can be shown to yield information on the correlations of spontaneous fluctuations. The analytic solution for the harmonic oscillator and a numerical solution for a double well system are provided which illustrate the various aspects of the theory. The theory contained herein provides the basis for a derivation of Centroid Molecular Dynamics, as well as the systematic improvements of that theory.
This paper is devoted to the development of an operator formulation of the recent extension of the centroid molecular dynamics method [J. Chem. Phys. 110, 3647 (1999); 111, 5303 (1999)] to boson and fermion systems. An operator calculus is used to rederive the basic equations of centroid dynamics. The following generalization to the case of systems of many indistinguishable particles is based on the use of a projection operator. Two different definitions of the quasi-density operator for bosonic and fermionic systems are suggested. The first definition allows an exact evaluation of equilibrium properties for systems with exchange effects using classical-like molecular dynamics calculations. The second one provides a formal justification of Bose–Einstein/Fermi–Dirac centroid dynamics with the same set of approximations as for Boltzmann statistics, and can be used to extract quantum dynamical information. In this case, the corresponding centroid correlation function can be related to a double Kubo transformed quantum mechanical one.
Expressions are derived for the Feynman path centroid density of systems of identical particles, namely, Bosons and Fermions. The resulting expressions are applied to a system of two independent particles in a quartic oscillator and the corresponding centroid densities are computed, plotted and compared to the distinguishable particle case. In regions where the particle centroids are close to each other, the Boson centroid density displays an enhanced amplitude in comparison to the distinguishable case. This behavior is attributed to the attractive correlations in Boson systems. The Fermion centroid density, however, can have the peculiar property of being negative in those regions where the particle centroids are in close proximity. This feature is related to the Pauli exclusion principle. This property of the Fermion centroid density rules out its strict interpretation as a probability distribution. Equilibrium properties such as the canonical partition function and the average position were accurately computed using both densities. The Boson density was also used to compute the position autocorrelation function using the Centroid Molecular Dynamics method and the results are in excellent agreement with those of an exact quantum calculation.
Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 (2016) and Phys. Rev. A 96, 022117 (2017)] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function---the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics---extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. First we provide a brief modernized discussion of the general framework of phase-space quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space---putting it for the first time on a truly equal footing to Schr\"odinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of ``parity'' and ``displacement'' in a natural broadening of previously developed phase-space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.
This is a text on quantum mechanics formulated simultaneously in terms of position and momentum, i.e. in phase space. It is written at an introductory level, drawing on the remarkable history of the subject for inspiration and motivation. Wigner functions density -- matrices in a special Weyl representation -- and star products are the cornerstones of the formalism. The resulting framework is a rich source of physical intuition. It has been used to describe transport in quantum optics, structure and dynamics in nuclear physics, chaos, and decoherence in quantum computing. It is also of importance in signal processing and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: it furnishes a third, alternative way to formulate and understand quantum mechanics, independent of the conventional Hilbert space or path integral approaches to the subject. In this logically complete and self-standing formulation, one need not choose sides between coordinate or momentum space variables. It works in full phase-space, accommodating the uncertainty principle; and it offers unique insights into the classical limit of quantum theory. The observables in this formulation are c-number functions in phase space instead of operators, with the same interpretation as their classical counterparts, only composed together in novel algebraic ways using star products. This treatise provides an introductory overview and supplementary material suitable for an advanced undergraduate or a beginning graduate course in quantum mechanics. Readership: Advanced undergraduates and graduates studying quantum mechanics.