# Converse and generalization of Wick's theorem

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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 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 mth 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.

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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.

answered Jul 12, 2014 by (15,518 points)

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