VARIATIONAL APPROACH TO MULTITIME CORRELATION-FUNCTIONS

A variational method is proposed to evaluate the generating functional phi which gives the multi-time correlation functions in equilibrium and non-equilibrium statistical mechanics or field theories. Its definition, phi = In Tr DA, 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 phi 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.