[en] 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

[en] The grand partition function Z (α,β) of a quantum system is studied, using diagrammatic representations of the perturbation expansion. For a fermions system, it is possible to show, by proper resummation, without approximations but under some 'regularity hypothesis', that Log Z (α,β) takes a form where, besides trivial dependences, α and β only appear through a statistical factor F_k^- = [1 + e^{-α+βε_k0-βW_k}]^-1. W_k is a (real) self-consistent potential, generalized to all orders and can be defined by a stationary condition on Log Z (α,β) under variations of F_k^-. The thermodynamical quantities take a form analogous to the expressions Landau introduced for the Fermi liquids. The zero temperature limit (for isotropic systems) gives back Goldstone expressions for the ground state of a system. (author)

[en] A short account is given of some recent work on the perturbation expansion of the Gibbs potential of quantum statistical mechanics. (author)

[en] The dynamics of critical phenomena as is described by stochastic equations of the Landau-Ginzburg type with or without conservation law, is studied by the technique of field renormalization. The case of mode coupling systems is briefly touched upon

[en] Published in summary form only

[en] Strong infrared divergences prevent standard spin glass field theory to be meaningful, at least below d=6. A mechanism that softens infrared divergences is displayed, that, via the existence of a sea of zero or near zero modes, relates softening in infrared divergences and in the samll x behavior of Parisi order parameter q(x). This mechanism may, hopefully, allow the building of a meaningful spin glass field theory in the condensed phase. (orig.)

[en] In this work we show that a class of Ising spin glass models with finite connectivity exhibits instability when replica symmetry is not broken in the condensed phase at low temperatures. These models are relevant for complex optimization problems and may also give important insights about real spin-glasses

[en] We develop a field theory for spin glasses using replica Fourier transforms (RFT). We present the formalism for the case of replica symmetry and the case of replica symmetry breaking on an ultrametric tree, with the number of replicas n and the number of replica symmetry breaking steps R generic integers. We show how the RFT applied to the two-replica fields allows one to construct a new basis which block-diagonalizes the four-replica mass-matrix, into the replicon, anomalous and longitudinal modes. The eigenvalues are given in terms of the mass RFT and the propagators in the RFT space are obtained by inversion of the block-diagonal matrix. The formalism allows one to express any i-replica vertex in the new RFT basis and hence enables one to perform a standard perturbation expansion. We apply the formalism to calculate the contribution of the Gaussian fluctuations around the Parisi solution for the free-energy of an Ising spin glass. (paper)