# Finite-dimensional Feynman Diagrams

## 7. Correlation functions

The way path integrals are used in quantum field theory is, very roughly speaking, that the probability amplitude of a process going from point v1 to point v2 is an integral over all possible ways of getting from v1 to v2. In our finite-dimensional model, each of these ways'' is represented by a point v in Rn and the probability measure assigned to that way is . The integral is what we called before a 2-point function

and what we will now call a correlation function.

We continue with the example of the cubic potential

.

By our previous calculations,

In terms of Wick's Theorem and our graph interpretation of pairings, this becomes:

where now the sum is over all graphs G with two single-valent vertices (the ends) labeled 1 and 2, and n 3-valent vertices.

This graph occurs in the calculation of the coefficient of in <v1,v2>.

The k-point correlation functions are similarly defined and calculated. Here is where we begin to see the usual Feynman diagrams.''

This graph occurs in the calculation of the coefficient of in <v1,v2,v3,v4>.