A formally exact field theory for classical systems at equilibrium

We propose a formally exact statistical field theory for describing classical fluids with ingredients similar to those introduced in quantum field theory. We consider the following essential and related problems: (i) how to find the correct field functional (Hamiltonian) which determines the partition function, (ii) how to introduce in a field theory the equivalent of the indiscernibility of particles, and (iii) how to test the validity of this approach. We can use a simple Hamiltonian in which a local functional transposes, in terms of fields, the equivalent of the indiscernibility of particles. The diagrammatic expansion and the renormalization of this term are presented. This corresponds to a nonstandard problem in Feynman expansion and requires a careful investigation. Then a non-local term associated with an interaction pair potential is introduced in the Hamiltonian. It has been shown that there exists a mapping between this approach and the standard statistical mechanics given in terms of Mayer function expansion. We show on three properties (the chemical potential, the so-called contact theorem and the interfacial properties) that in the field theory the correlations are shifted on non-usual quantities. Some perspectives of the theory also are given.