About the non vanishing of boundary terms in correlation functions as origin of phase transitions. A microscopic proof of a Goldstone theorem in classical statistical mechanics

We give a general microscopic proof that the well known Goldstone theorem connected with spontaneous symmetry breaking in quantum statistical mechanics and quantum field theory has a counterpart in classical statistical mechanics. Our approach is mainly based on the replacement of commutators by Poisson brackets as infinitesimal generators of symmetries and on the Fourier transformed version of the so called Kubo-Martin-Schwinger property of equilibrium states. Especially we show that the phase transition is related to the non vanishing of certain boundary contributions in Poisson brackets which from the naive point of view should vanish.