Combinatorical aspects of the Schwinger-Dyson equation

In this work we analyse combinatorical aspects of the Schwinger-Dyson equation. This leads to generalizations of Wick's theorems on integrals with Gaussian weight to a larger class of weights which we call sub-Gaussian. Examples of sub-Gaussian contractions are that of Kac-Moody or Virasoro type, although the concept of a sub-Gaussian weight does not refer a priori to two-dimensional field theory. The generalization was chosen in such a way that the contraction rules become a combinatorical way of solving the Schwinger-Dyson equation. In a still more general setting we prove a relation between solutions of the Schwinger-Dyson equation and a map N, which in the Gaussian case reduces to normal ordering. Furthermore, we give a number of results concerning contractions of composite insertions.