Converse and generalization of Wick's theorem

Question

Apologies if this comes off as too trivial. The question stems from my reading of Papadodimas and Raju's paper, 'An Infalling Observer in AdS/CFT' (hep-th/1211.6767), in which they introduce the notion of a 'generalized free field'. The n-point functions of such a field may be written as the sum over all possible pairings of n objects, with each pair corresponding to a 2-point function, up to leading order in some parameter 1/N (unrelated to the usual usage of N  as the number of flavors).

I was thus led to wonder whether it is possible to have distributions other than the Guassian in, say, a 0-dimensional QFT, which also satisfy Wick's theorem. Additionally, are there distributions (perhaps with the square in the exponent of the Gaussian replaced by an m^th power) for which the n-point functions are given by a sum over all possible m-groupings of n objects, so that perturbation theory around such backgrounds would involve dealing with Feynman hypergraphs?

I suppose a combinatorial proof of Wick's theorem can help clarify how far we can go in this direction, but unfortunately, I haven't been able to come up with one.

Answer 1

Wick's theorem for boson fields is fully equivalent to the Gaussian assumption.

However, there is also a Wick's theorem for generalized free fermion fields with a given 2-point function, with slightly different signs. These are not Gaussians, but also qualify as generalized free field, as the free fermion field is a special case.

There are no other instances of Wick's theorem. This can be seen by deriving a functional differential equation for the cumulant generating function.