Abstract
This Thesis covers research conducted at the University of Montpellier, during the three years of PhD contract financed by the doctorate school I2S, in thecontext of thermal quantum field theory. This theory is a framework for describing quantum fields in a medium, specifically at finite temperature and/or chemical potential (equivalently, density). A special focus is given to the renormalization group and its properties to define a better resummation scheme for the infrared divergences generically plaguing thermal field theory. We first define and apply a so-called renormalization group optimized perturbation theory (RGOPT), a resummation framework for the infrared divergences, for the specific case of the lambdaphi^4 model at next-to-next-to-leading order, where we found sound results concerning the improvement of the convergence of the series expansion as well as a drastic improvement with respect to the residual arbitrary renormalization scale dependence. Cold and dense quantum chromodynamics is also explored, where we derived an original all order resummation of the leading logarithms as well as the so-called soft next-to-leading logarithms. The discussion is extended so as to include the application of RG resummation in the quark sector at next-to-next-to-leading order, and first developments for the specific case of massive gluons in the context of the Hard Thermal loop perturbation theory. Finally, preliminary application of this resummation procedure to the determination of an equation of state for neutron stars are discussed.