Gaussian, Mean Field and Variational Approximation: the Equivalence

We start by clarifying the terms which appear in the title. For a given theory of selfinteracting quantum fields, by Gaussian approximation we mean the Gaussian part of the interacting measure. By mean field approximation we understand the leading term of the expansion given in [1, 2] and by variational approximation we understand the old variational technique performed with some particular trial states [3-5]. The novelty of this paper is the method of extracting the Gaussian peace of the interacting measure. Also we hope that the equivalence between the three methods will lead to a better understanding of the self-interacting field theories. The expansion around the mean field (MF) approximation have been successfully used in [1, 2] to λ ∑N n=1 αn (λφ) 2n interactions, for small λ. The key role of this approximation is that, after the translation to φ− φMF , it brings all the coupling constants near to zero where the cluster expansion [6] can be applied.