Correlations and fluctuations in static and dynamic mean-field approaches

A consistent approximation is proposed to treat the following general problem. Let the state of a many-body system at an initial time be specified, completely or partly; find the expectation values, correlations, and fluctuations of single-particle observables at a later time. To this aim the characteristic function of these observables is optimized within a general variational scheme. The trial objects are akin to an initial density operator, to a time-dependent one, and to a time-dependent operator associated with the observables of interest. We take an independent-particle form for the trial density operators and a similar ansatz for the trial operators. The variational equations couple the (approximate) initial conditions with the (approximate) evolutions of the operator and of the state. The expansion of the optimal characteristic function provides the same results as the conventional mean-field approaches for the thermodynamic potentials and the expectation values: for fermions the best initial state is then the Hartree-Fock (HF) solution and the evolution is described by the time-dependent Hartree-Fock (TDHF) equation. However, these approximations become inadequate for correlations and fluctuations, and new expressions come out from our mean-field variational treatment. Two special cases are investigated as preliminary steps. The first case deals with the evaluation of correlations (and Kubo correlations) for static problems, where the initial and final times coincide. Then the variational outcome deviates significantly from the trivial correlations given by the HF approximation, despite our restriction to independent-particle trial states. The result is analyzed perturbatively and shown to be related to the random-phase approximation (RPA) diagrams. It involves a matrix that is identified as a metric tensor generated by the HF entropy. In the second special case, the exact initial state is assumed to be an independent-particle one. The correlations between single-particle observables are then given by a simple formula where all quantities are brought back to the initial time; this requires solving a time-dependent RPA-like equation. Some deficiencies of the TDHF approaches are also cured: the single-particle conservation laws and the spreading of the wave-packet are restored. Systems in pure states are treated by taking a zero-temperature limit.