Improved Gaussian Approximation

In a recently developed approximation technique [1] for quantum field theory the standard one-loop result is used as a seed for a recursive formula that gives a sequence of improved Gaussian approximations for the generating functional. In this paper we work with the generic φ3+φ4 model in d = 0 dimensions. We compare the first, and simplest, approximation in the above sequence with the one-loop and two-loop approximations, as well as the exact results (calculated numericaly). The central object in quantum field theory is the generating functional Z[J ]. Functional derivatives of Z[J ] with respect to the external fields J(x) give the Green’s functions of the theory. The generating functional is determined from the (Euclidian) action S[φ] through the path integral Z[J ] = ∫ [dφ] e( ∫ dx J(x)φ(x)) . (1) The integration measure is, formaly, simply [dφ] = ∏ x∈R dφ(x), where d is the dimension of space-time. In this paper we will work with models in d = 0 dimensions. In d = 0 functionals become functions, and the path integral reverts to a single definite integral over the whole real line Z(J) = ∫ dφ e φ) . (2) Two further important objects are W (J) — the generator of connected diagrams (or free energy) Z(J) = Z(0) e (J) , (3) and the quantum average of the field φ = 〈φ〉 = − ∂ ∂J W (J). In the Gaussian approximation, we Taylor expand the action in the path integral around some reference point φref , and keep terms that are at most quadratic in φ− φref . The integral in (2) is now a Gaussian and we find WGauss(J, φref) = S(φref)− J φref + 1 2 lnS ′′(φref)− 1 2 (S ′(φref)− J) 2 S ′′(φref) . (4)