The functional integral representation of quantum statistical mechanics by means of the Feynman--Kac formula leads to a classical-like description of the system. Point quantum particles are then described in terms of random loops (closed Brownian paths), and all techniques of classical statistical mechanics become available. One advantage of this formalism is that it is not perturbative with respect to the interaction strength, in contrast to the standard many-body perturbative treatment. We apply these ideas to the Coulomb gas by constructing an effective potential (the quantum analogue of the Debye potential) that incorporates both long distance collective screening effects as well as the short range quantum mechanical binding. For Bose systems, we show that mean field theory corresponds to summing all tree-graphs and investigate how to go beyond the mean field description.

PACS numbers: 05.30.--d, 71.45.--d