Toward a mean field theory of spin glasses: the TAP route revisited

After a sketchy derivation of TAP equations, we show first how one can recover two known results (Sherrington Kirkpatrick, and Sommers) by direct perturbation resommation a la Martin Siggia Rose. Going beyond perturbation, it is recalled that TAP equations can be reduced to a one site self consistent problem involving several order parameters. The existence of a large number of solutions to TAP equations at low T forces us to choose a weight to be attributed to each solution. Also we are forced to reintroduce replicas to compute averages over the ensemble of solutions. In contradistinction to what usually occurs it is argued that here replicas are rather innocent. The uniform (white) average and the microcanonical average are briefly discussed