# Converse and generalization of Wick's theorem

+ 3 like - 0 dislike
528 views

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.

+ 4 like - 0 dislike

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 (14,437 points)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverflo$\varnothing$Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.