Okay, so I gather from the link that $G^{(k)}$ in your notation refers to the correlation between field values at $2k$ points, with $\varepsilon^+$ inserted at half of them and $\varepsilon^-$ inserted at the other half.

This concept of an $n$-point correlation function is very similar to the $n$th moment of a random variable or statistical distribution. For simplicity, consider an example with 1 dimension: A field over 1 time dimension (at a single point in space), described by a random variable. Now, we can consider the probability distribution of this random variable and talk of it's moments. The *mean* value would be called the 1st moment and the *variance* would be related to the second moment (it is in fact called the second *central* moment). Similarly, you can generalize to higher order moments which help *characerize the distribution*. The moments help you characterize the distribution and also give an intuitive feel for the function.

Generalize this concept to random variables which are fields over many-dimensional spacetime. That is what your correlation functions are.

Btw, for a gaussian distribution (non-interacting fields i.e. quadratic action), all odd moments vanish. (That might be the motivation for $G^{(k)}$ to be defined as the correlation between field values at $2k$ points... even though the actual physical theory you're considering will probably be interacting, else all correlation functions are fairly trivial).
Also, all even moments beyond the 2nd-moment are completely specified by the 1st and the 2nd moment. Ref1 and Ref2

If you had an interaction term in the hamiltonian/lagrangian involving 4 fields, then the 4-point correlation function would have 2 kinds of contributions:

- 2 sets of 2-point correlation functions between pairs of points among those 4 points
- A nontrivial contribution from the interaction term with one of it's field insertions at each of the 4 points.

So you can see that higher order correlations functions give you very important (an unique) information in an interacting physical theory.

Update: The (many) answers to this SE question might also shed some light on the discussion.

This post imported from StackExchange Physics at 2014-04-24 02:35 (UCT), posted by SE-user Siva