Generalized fluctuation-dissipation theorem for nonlinear response functions

A nonlinear generalization of the fluctuation-dissipation theorem (FDT) for the n-point Green functions and the amputated one particle irreducible vertex functions at finite temperature is derived in the framework of the closed time path formalism. We verify that this generalized FDT coincides with known results for $n=2$ and 3. New explicit relations among the 4-point nonlinear response and correlation (fluctuation) functions are presented.