The renormalized and Renormalization-Group invariant Hartree-Fock approximation

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We study the renormalization problem for the Hartree--Fock approximation of the O(N)-invariant ϕ⁴ model in the symmetric phase and show how to systematically improve the corresponding diagrammatic resummation to achieve the correct renormalization properties of the effective field equations, including Renormalization--Group invariance with the one--loop beta function. These new Hartree--Fock dynamics is still of mean field type but includes memory effects which are generically nonlocal also in space.