## Finite-dimensional Feynman Diagrams

**Feature Column Archive**

### 7. Correlation functions

The way path integrals are used in quantum field theory is, very roughlyspeaking, that the probability amplitude of a process going from point *v*_{1}to point *v*_{2}is an integral over all possible ways of getting from*v*_{1} to*v*_{2}. In our finite-dimensional model, each of these``ways'' is represented by a point **v** in **R**^{n} andthe 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, thisbecomes:

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 coefficientof in <*v*^{1},*v*^{2}>.The *k*-pointcorrelation functions are similarly defined and calculated. Hereis where we begin to see the usual ``Feynman diagrams.''

This graph occurs in the calculation of the coefficient of in<*v*^{1},*v*^{2},*v*^{3},*v*^{4}>.