Research activity into field theory has been recently focused on classical field theory. The reason for this is that modern experiments with heavy ion scattering have shown that the quantum effects are killed by the effects of temperature and pressure, and so only the classic fields survive. For classical field theory, however, the modern tools of Feynman diagrams had not been developed, so we did this by using a path-integral formulation of classical mechanics developed in Trieste over the last twenty years. These diagrams are actually "super-diagrams" because they provide the evolution not only of the basic fields, but also of the response fields and their associated first variations. By means of these "super-diagrams", we were able to see that several cancellations took place, simplifying the perturbative calculations enormously. We then introduced temperature, developed the relative Feynman diagrams for that version and built the classical versions of quantum formulation with temperature, namely that of Matsubara, of the closed time path, and of thermofield dynamics. The superdiagrams above appeared because of a universal non-relativistic supersymmetry present in this formulation of classical field theory. Actually, this supersymmetry turned out to be none other than the modern version in field theory of Cartan's well-known equivariant cohomology, associated with the Hamiltonian flow. We think that this supersymmetry can be made relativistic by using the De Donner Weyl formulation and not the Hamiltonian one to describe the evolution of the system. If this were the case, it would be very nice to have discovered such a supersymmetry in a very general context.
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