Variational approach to multi-time correlation functions

Abstract

URL: http://www-spht.cea.fr/articles/T93/030International audienceA variational method is proposed to evaluate the generating functional $ \varphi $ which gives the multi-time correlation functions in equilibrium and non-equilibrium statistical mechanics or field theories. Its definition, $ \varphi = {\rm ln\ Tr} \ D\ A, $ involves the density operator $ D $ describing the initial state and an operator $ A $ depending on the observables of interest (in the Heisenberg picture), on the associated time-dependent sources and on the initial time. Regarded as function of this time, $ A $ is specified by a simple differential equation with the time running backward. Lagrangian multipliers are introduced to account both for this equation and for the one which characterizes $ D. $ Through this procedure $ \varphi $ is obtained as the stationary value of a functional depending on state-like and observable-like trial operators, with a complex time. Within a trial subspace, the resulting approximation optimizes both the initial state and the dynamics. The example of interacting fermions in many-body physics is worked out by restricting the trial objects to exponentials of single-particle operators. This leads to an extended mean-field approximation for the generating functional associated with any set of observables. Expansion in powers of the sources provides a variational approximation for the two-time causal or response functions. The result incorporates both the static and dynamic Hartree-Fock equations as well as the associated RPA equations. It is free of several inconsistencies occurring in the conventional mean-field approximations. ----- Les auteurs proposent une méthode variationnelle très souple pour évaluer la fonction génératrice des fonctions de corrélations d'observables arbitraires dans un système de mécanique quantique général, où l'état initial est une matrice densité arbitraire. L'un des aspects originaux est d'envisager la dépendance des quantités dans la variable temporelle où sont identifiées les représentations de Schrödinger et de Heisenberg. Les données et les équations du mouvement sont traitées comme des contraintes dans ce formalisme en introduisant des multiplicateurs de Lagrange appropriés. Les auteurs présentent à titre d'application l'exemple d'un système de fermions en interaction et trouvent des généralisations des diverses méthodes de champ moyen, statique ou dynamique (approximation de Hartree-Fock stationnaire ou dépendant du temps, méthode des phases aléatoires...). Ils montrent que leur nouvelle théorie est plus cohérente que celles envisagées jusqu'ici. \vskip 20mm \hfill{C. Itzykson

Understanding quantum measurement from the solution of dynamical models

“…The infinitesimal generator of this transformation being the HamiltonianĤ, the time-dependent observableÔ(t, t 0 ) is characterized either by the usual Heisenberg equation i ∂Ô(t, t 0 )/∂t = [Ô(t, t 0 ),Ĥ] with the boundary condition O(t 0 , t 0 ) =Ô or by the backward equation i ∂Ô(t, t 0 )/∂t 0 = [Ĥ,Ô(t, t 0 )] with the boundary conditionÔ(t, t) =Ô. The backward equation, more general as it also holds ifĤ orÔ depend explicitly on time, is efficient for producing dynamical approximations, in particular for correlation functions [306]. The interest of the backward viewpoint for the registration in a measurement is exhibited in § 7.3.1, Appendix F and § 13.1.3.…”

“…We will rely on this remark in subsection 13.1. The Heisenberg picture also allows to define correlations of observables taken at different times and pertaining to the same system [299,306]. Such autocorrelations, as the Green's functions in field theory, contain detailed information about the dynamical probabilistic behavior of the systems of the considered ensemble, but cannot be directly observed through ideal measurements.…”

BEC from a time-dependent variational point of view

“…This last quantity turns out to be of no interest in our particular case but it may play a major role in other situations especially when one intends to calculate correlation functions [11,12].…”

Variational Approximations for Correlation Functions in Quantum Field Theories

“…In spite of the Gaussian character of the variational ansätze, the approximations obtained in this way go beyond the usual mean-field approximations and include correlations between particles. This comes from the fact that the variational principle of Balian and Vénéroni [6] has been precisely constructed to provide directly the quantity of interest, namely the generating functional.…”

“…However, they are not suitable for multi-time correlation functions. In order to obtain approximations for these multi-time correlation functions, a variational principle was recently introduced in the context of non-relativistic fermions [6]. The action to minimize has for stationary value the generating functional for the correlation functions.…”